Question: How is the family-wise error rate (FWER) related to the maximum statistic in the context of multiple testing scenarios?

Answer: The family-wise error rate (FWER) is the probability that one or more of the component tests φi(Y ), i ∈I0, evaluate to 1 over the set of true null hypotheses I0. This probability is equal to the complementary probability of the event that all component tests evaluate to 0 over the set of true null hypotheses I0. This event corresponds to the probability that all relevant component test statistics assume values smaller than the critical value uFWE α′, which is identical to the event that the maximum statistic γ0 max(Y ) over the set of true null hypotheses is smaller than uFWE α′.


Question: How does the query optimisation process determine the correct sequence of operators for a given query on a factorised database?

Answer: The query optimisation process determines the correct sequence of operators by ensuring that the sequence satisfies three conditions: (1) For each selection condition Ai = Bi in the query, the sequence contains a selection operator that merges the equivalence classes of Ai and Bi and their nodes in the input f-tree. (2) The sequence contains the aggregation operator γF(U) for some subtree U whose set of attributes is T \ G, and it may be preceded by partial aggregations. (3) The output f-tree of the last operator must satisfy the condition of Theorem 2 for the order-by list L. These conditions ensure that the sequence of operators correctly implements the query.


Question: What are the specific conditions on the gain 𝜀, the dwell-time parameter 𝜅𝑢,1, and the time-ratio parameter 𝜅𝑢,2 that guarantee the solution of Problem 1 for the dynamic controller under attack?

Answer: The conditions are given by the inequality 2𝜇 > ln(𝜔)𝜅𝑢,1 + 2𝜅𝑢,2(𝜇 + ̄𝑀𝓁), where 𝜔 = max {1/𝜇, 𝓁²/𝜇}.


Question: What are some potential applications of the proposed architectures in the context of connectomic data?

Answer: The architectures can be extended to include various types of connectomic data, such as structural, multimodal, or gene co-expression data. This could lead to meaningful insights and personalized therapies, such as understanding the impact of drug responses and how pain variability is encoded in brain communication.


Question: How does the performance and training time of 2DMOINet compare to other state-of-the-art models on the Cora dataset?

Answer: On the Cora dataset, 2DMOINet is not directly compared in the candidate text. However, the state-of-the-art models are compared in Table 3. GraphSAGE achieves an F1 score of 85.0% in 18 seconds, FastGCN achieves 85.5% in 6.02 seconds, VRGCN achieves 85.4% in 5.47 seconds, L-GCN achieves 84.7% in 0.45 seconds, and L2-GCN achieves 84.1% in 0.38 seconds.


Question: What are the key challenges and solutions proposed for ensuring fairness in peer-to-peer energy trading?

Answer: Key challenges in peer-to-peer energy trading include the potential for network-wide benefits not being individually beneficial for each hub, and the risk of undesirable congestions in the power grid or disproportionate costs for prosumers due to naive choices of bilateral trading prices. To address these, the paper proposes a scalable and privacy-preserving price-mediation algorithm that converges to trading prices that are both locally beneficial and network-wide fair, thereby incentivizing active participation of energy hubs in autonomous peer-to-peer trading schemes.


Question: What does Lemma 6 imply about the relationship between (S, u) and (S, uj) when (S, uj) = i for every j?

Answer: If (S, uj) = i for every j, then (S, u) = i by Lemma 3.


Question: How does the action difference in the thin wall approximation relate to the bubble radius and what is the final action in this approximation?

Answer: The action difference in the thin wall approximation is given by B = −1/2π²¯r⁴∆V + 2π²¯r³σ. The first term corresponds to the energy difference in the interior of the bubble which drives the bubble to expand, and the second term corresponds to the tension of the bubble wall driving the bubble to contract. The bubble radius ¯r is fixed by extremizing B, leading to ¯r = 3σ/∆V, and the final action is BColeman = 27π²σ⁴ / 2(∆V)³.


Question: What properties must a morphism f: Y → Z satisfy to be considered a relative logarithmic orbifold?

Answer: A morphism f: Y → Z is considered a relative logarithmic orbifold if it is logarithmically regular and the sheaf of logarithmic derivations DY/Z is logarithmically separating.


Question: What computational method and software were used for the total energy calculations in the study of Fe2GeCh4?

Answer: Total energy calculations based on first principle density functional theory (DFT) were performed using the full-potential linear augmented plane wave (FP-LAPW) method as implemented in WIEN2k. The self-consistent calculations included spin-orbit coupling, and the Tran-Blaha modified Becke-Johnson (TB-mBJ) potential was used on top of GGA-PBE to reproduce the experimental band gaps.


Question: What future techniques and applications are proposed for investigating the orientation relationships and properties of nano-sized precipitates in materials?

Answer: Future techniques include resolving individual orientation relationship variants directly with 4D-STEM by finetuning the template matching procedure and data acquisition strategies. This technique shows promise for investigating complex nano-sized precipitates in other material systems, such as magnesium, aluminium, or titanium alloys. Investigating the early stages of precipitation or quantifying orientation relationship variants once they can be resolved may be interesting application domains.


Question: What are the characteristics of the brightest UDs and where do they typically emerge?

Answer: The brightest UDs, with a mean peak intensity greater than 0.65 Iph, typically emerge close to the penumbra, the light bridge, or the prominent UD chain (label A in Fig. 5.1).


Question: How is the integral of the shadow kernel against a test function f(x) evaluated in the limit ε → 0, and what is the resulting expression?

Answer: Integrating (C.2) against a test function f(x) yields the expression ∫Rd ddx lim ε→0 d/ε^2 (x^2 + ε^2)^(d/2 + 1) f(x) = d ∫Sd−1 dΩ ∫∞ 0 dr r^(d−1) (r^2 + 1)^(d/2 + 1) lim ε→0 f(εx). Taking the limit ε → 0, the integral simplifies to ∫Sd−1 dΩ f(0) = 2π^(d/2) Γ(d/2) f(0).


Question: How does the relaxation time τ∆ change with varying monomer friction constants γm, and what is the impact on the block mobility functions ΛAA and ΛAB?

Answer: The relaxation time τ∆ increases with decreasing γm, leading to a decrease in ΛAA and an increase in ΛAB. This is because the individual blocks relax more slowly, and the two blocks move more cooperatively at small qRG if the relative monomer friction is increased.


Question: What specific multiplets were used to analyze the C I velocity structure in DLA 0812+32, and why were other multiplets not used?

Answer: The 1656˚A and 1560˚A multiplets were used in the fit because of the serious blending with the Lyman α forest in lower wavelength multiplets.


Question: How does augmenting the support improve the quality of reconstructed marginals in the case of very few samples per time-point?

Answer: By augmenting the support with the scheme described in Section G.3, we add points to X that were not present before, improving the quality of the reconstructed marginals.


Question: What is the shape of the region conquered by species 2 when species 1 is strictly faster and starts at a single point on the boundary of the disc A2?

Answer: Species 2 conquers a 'heart-shaped' region bounded by two symmetric, partial logarithmic spirals.


Question: What was the final validation divergence after the greedy optimization process, and how does it compare to the original validation divergence?

Answer: The final validation divergence was 1.206, which is a modest improvement over the original validation divergence of 1.198.


Question: What are the misalignment angles between adjacent loops in the minimum energy model, and how do they compare to the predictions of the Parker model?

Answer: The minimum energy model yields small misalignment angles of ∆µ = 1.6°−1.8° between adjacent loops, which are consistent with observations of ubiquitous near-parallel loops seen in EUV. In contrast, the Parker model predicts substantially larger misalignment angles, with a threshold value of µcrit ≥20° required for nanoﬂaring.


Question: What is the local part of the quantum effective action for the gauged WZW theory with an integrable potential, and how does it contribute to the S-matrix in the case of H = U(1)?

Answer: The local part of the quantum effective action for the (axially) gauged WZW theory with an integrable potential, in the case of H = U(1), is given by the expression: Seff = −(k + cG) / (4π) * [1/2 * ∫d²x g⁻¹∂⁺g g⁻¹∂⁻g - 1/3 * ∫d³x εmnl g⁻¹∂m g g⁻¹∂n g g⁻¹∂l g + ∫d²x (A⁺∂⁻gg⁻¹ - A⁻g⁻¹∂⁺g - g⁻¹A⁺gA⁻ - (1 - 2cG / (k + cG)) A⁺A⁻) + m² ∫d²x (g⁻¹TgT - T²)]. This action, when supplemented with the potential, contributes to the one-loop S-matrix by integrating out A± and computing the resulting S-matrix.


Question: What conditions must be met for a market to have a market equilibrium according to the economy graph definition?

Answer: A market has a market equilibrium if it is strongly connected, meaning the economy graph G, which is a directed graph with vertices as traders and edges from Ti to Tj if Ti possesses a good which Tj wants, is strongly connected. Additionally, if all the parameters of the market are rational numbers, then it has a rational market equilibrium, and the number of bits needed to describe this equilibrium is polynomial in the input size of the market.


Question: How does nonlinear reconstruction improve the measurement of oscillation parameters in galaxy surveys, and what are the implications for future surveys like DESI?

Answer: Nonlinear reconstruction improves the measurement of oscillation parameters by substantially shrinking the measurement uncertainty, especially where the primordial features have been less well preserved pre-reconstruction, such as at 𝑧< 1. This improvement is equivalent to enlarging the survey area or increasing the 𝑘range, but at a much smaller cost. For future surveys like DESI, this means more robust constraints on the oscillation parameters, which can be achieved by using emulators of redshift-space galaxy or halo clustering, especially if the primordial oscillations are weak.


Question: How does the accuracy of the POD-DEIM model change with the number of DEIM interpolation points, and how does it compare to the QB-POD model in the tubular reactor model?

Answer: The POD-DEIM model's accuracy increases with more DEIM interpolation points, but it levels off after a certain number of points. In contrast, the QB-POD model does not suffer from the limitation of hyper-reduction interpolation error and continues to increase in accuracy as more basis functions are added. For the tubular reactor model, the POD-DEIM model with rDEIM = r approximates the original POD model well, while the QB-POD model maintains a steady decrease in error as the number of POD basis functions is increased.


Question: How does the adjustment of SAR contrast magnitude by wind speed exponent affect the correlation measures, and what does this suggest about the nature of the dependence between SAR contrast and wind speed?

Answer: The adjustment of SAR contrast magnitude by a wind speed exponent of 0.8 results in a reduction in the mean and variance of SAR contrast magnitude. Pearson and distance correlation are reduced to values of less than 0.2, suggesting that the adjustment primarily reduces linear dependence, but also to some extent nonlinear dependence. This indicates that the relationship between SAR contrast and wind speed involves both linear and nonlinear components, with the linear component being more dominant.


Question: What inequality is derived by combining (4.6) and (4.7), and what does it imply for the gradient of u?

Answer: Combining (4.6) and (4.7), and choosing sufficiently small ǫ0 > 0 and r1 > 0, we obtain that ∫_{B+^r(0)} |∇u|^2 ≤ Cr ∫_{S+^r(0)} |∇u|^2 + Cr, for all 0 < r ≤ r1. This implies that ∫_{B+^r(0)} |∇u|^2 ≤ r^α (r^{-α} ∫_{B+^{r1}(0)} |∇u|^2 + C/(1 - α)r1^{-α}), for some 0 < α < 1, for all 0 < r ≤ r1.


Question: How is the Static Source BLoPS problem formulated to minimize the estimation error while adhering to system cost and bandwidth constraints?

Answer: The Static Source BLoPS problem is formulated as an optimization problem to minimize the trace of the MMSE estimation error covariance matrix, tr{Σθ|y(S)}, subject to constraints on the total system cost and total system bandwidth. The problem is mathematically written as: arg min{sl,k,b} tr{Σθ|y(S)} subject to ∑Ll=1 cTSl1B ≤ λ (system cost constraint), ∑Ll=1 [0 1TK]Slw ≤ W (bandwidth constraint), sl,k,b ∈ {0, 1} (binary selection constraint), and ||Sl||0 = 1 (at most one sensor per location constraint).


Question: What is the specific expression for the evolution of the longitude of the ascending node due to the Coriolis force?

Answer: The evolution of the longitude of the ascending node due to the Coriolis force is given by χΩ,Coriolis = −noutna2 √(1 − e2) [−3e2 sin 2ω − (1 + e2)/2 cos 2ω + B(e)].


Question: How does adjusting for time to clinical diagnosis affect the interpretation of cognitive impairment trajectories in Huntington disease patients?

Answer: Adjusting for time to clinical diagnosis shows that cognitive impairment worsens rapidly in the time period immediately before and after diagnosis, as indicated by the adjusted trajectories of SDMT scores (Figure 1B). This adjustment aligns with previous clinical findings, whereas unadjusted trajectories do not show this rapid decline.


Question: What are the key differences between the noise models used in [Krull et al., 2019, Prakash et al., 2020b, Prakash et al., 2020a] and the approach described in this paper?

Answer: In [Krull et al., 2019, Prakash et al., 2020b, Prakash et al., 2020a], the noise models are approximated using 2D histograms of denoised and noisy observed values, either using additional calibration data or a previously trained denoising function. These methods require several training procedures and calibration, and the denoising network predicts a whole distribution of 800 possible noisy values instead of a single point estimate. In contrast, the approach described in this paper emphasizes an efficient masking procedure at train-time, allowing the use of the central pixel at test time and requiring only a single prediction per pixel, resulting in simpler, more efficient, and more stable training and inference procedures.


Question: How do patterning and deposition defects affect the nanoparticle placement in the proposed design?

Answer: Patterning defects, such as a 10% deformation, cause a 3.3% particle placement error. Deposition defects, even if they reach unrealistic levels of 40 nm (four times the thickness of the Au layer), affect the nanoparticle placement by only 3.6%. This indicates that deposition defects have about 4 times less impact on the placement than patterning defects.


Question: What are the two strategies used to investigate the effects of neural text simplification systems on downstream NLP tasks, and what were the findings?

Answer: The two strategies used were: (a) augmenting data to provide machines with additional information during training, and (b) augmenting data to provide machines with additional information during training. The experiments indicated that the former strategy consistently helps multiple NLP tasks, regardless of the underlying method used to address the task or the neural approach used for text simplification. On input simplification, it was found that with a good simplification quality control method, text simplification can be used to reduce text difficulty for neural methods and eliminate the need for expensive retraining of large pretrained neural language models.


Question: What are the two online algorithms used for updating the power-performance models, and what are their computational complexities?

Answer: The two online algorithms used are the covariance form of recursive least squares (RLS) and the Dichotomous Coordinate Descent form of RLS (DCD-RLS). The covariance form of RLS has a computational complexity that grows quadratically, while the DCD-RLS has a complexity that grows linearly with the number of features.


Question: What are the experimental results of the teleported states and how are they reconstructed?

Answer: The experimental reconstruction of the teleported states |φexp⟩ is shown in Fig. 13, where the density matrices of |H⟩, |V⟩, |+⟩, |−⟩, |φ−⟩, and |φ+⟩ are presented. Each panel shows the real (upper subgraph) and imaginary (lower subgraph) parts of the density matrix. Additionally, the quantum process matrix χ for the teleportation protocol is reconstructed, with the real and imaginary parts shown in Fig. 14. The fidelity of the process matrix χ is calculated to be Fχ = (79.4 ± 1.6)%.


Question: What conditions must a piecewise normed cell complex satisfy to be uniquely geodesic and Busemann-convex according to Theorem A?

Answer: According to Theorem A, a piecewise normed cell complex X must satisfy the following conditions to be uniquely geodesic and Busemann-convex: (1) X is simply connected, (2) X locally admits a consistent geodesic bicombing, and (3) for any two intersecting maximal cells A, B of X, the union A∪B, with the induced length metric, is Busemann-convex.


Question: What specific wavelengths of excitation light provide the highest PL intensity in the dispersions, and how does this intensity vary with different wavelengths?

Answer: The PL excitation at λexc 405 and 457 nm provides the highest PL intensity, while excitation at either longer or shorter wavelengths produces a much lesser intensity of the emission.


Question: What are the specific mass differences and mixing angles that influence the νµ to νe oscillation probability in the three neutrino mixing scheme?

Answer: The νµ to νe oscillation probability in the three neutrino mixing scheme is influenced by the mass differences ∆m2_23 = (2.4±0.21)×10^-3 eV^2 and ∆m2_12 = (7.9±0.06)×10^-5 eV^2, and the mixing angles ϑ23 and ϑ13. The probability can be approximated as P = sin^2(2ϑ13) sin^2(ϑ23) sin^2(1.27 ∆m2_23 L/E), where ϑ23 and ∆m2_23 refer to the mixing of νµ to ντ, and ϑ13 is observed in νe reactor experiments and νµ to νe oscillations.


Question: How does the paper address the issue of agency-free external risk and what is the impact on optimal contracting?

Answer: The paper addresses the issue of agency-free external risk by allowing the principal to invest upon a halt to continue production at a degraded level. This investment decision is modeled as a control variable, and the optimal contract is characterized through a Hamilton-Jacobi-Bellman equation. The paper finds that investing to mitigate the effects of a halt is not always optimal for the principal, and when it is, it is only optimal up until a cutoff time related to the balance between the cost of investment, the agent’s rents, and possible remaining gain.


Question: What methods are proposed to bridge the gap between technical XAI techniques and end-users' diverse requirements?

Answer: To bridge the gap, the authors propose end-user-friendly explanation forms, which are abstracted structures of explanatory reasoning information generated by an XAI algorithm. These forms have end-user-facing properties for various visual representations and AI designer-facing properties for different underlying XAI algorithms. The explanation forms serve as a bridging interface to categorize the XAI algorithm space according to end-user-centered properties, such as feature-, example-, and rule-based explanations. Additionally, the authors conducted the EUCA (End-User-Centered explainable AI) user study to gain a deep understanding of user requirements and to evaluate XAI techniques for end-users' purposes.


Question: Which paper introduces the Msr-vtt dataset and what is its purpose?

Answer: Jun Xu, Tao Mei, Ting Yao, and Yong Rui introduced the Msr-vtt dataset in CVPR 2016. The purpose of the Msr-vtt dataset is to bridge video and language.


Question: What is the MobiCloud approach to enhancing mobile ad hoc networks (MANETs) and how does it utilize cloud computing?

Answer: MobiCloud proposes to use cloud computing to empower MANETs in a secure way by transforming MANETs into a service-oriented architecture. Each node in the network acts as a Service Node, which can function as a service provider or a service broker based on its computation and communication capabilities. These nodes are mirrored onto the cloud as Extended Semi-Shadow Images (ESSIs), which can be exact clones, partial clones, or images with extended functions of the physical device. This virtualized routing and communication layer assists the physical mobile nodes they represent.


Question: What are the partial-wave components and spectroscopic factors for the 0+ and 2+ states in the three-α configurations of the second Jπ = 2+ state in 12C?

Answer: The partial-wave components and spectroscopic factors for the 0+ and 2+ states are as follows: For the 0+ state, the (00) component has Pl1l2 = 0.352 and Sl1l2 = 0.193, and the (22) component has Pl1l2 = 0.351 and Sl1l2 = 0.175. For the 2+ state, the (02) component has Pl1l2 = 0.096 and Sl1l2 = 0.058, and the (22) component has Pl1l2 = 0.483 and Sl1l2 = 0.268. The total Pl1l2 and Sl1l2 for the 0+ state are 0.988 and 0.468, respectively, and for the 2+ state, they are 0.986 and 0.500, respectively.


Question: What is the definition of the resultant function and how does it relate to the roots of two polynomials?

Answer: The resultant function, Res : R[x] × R[x] → R, maps two polynomials f and g over a ring R to the determinant of a (m + n) × (m + n) matrix, where f and g have degrees n and m respectively. The resultant Res(f, g) is equal to am_n bn_m ∏(i=1 to n) ∏(j=1 to m) (αi − βj), where αi and βj are the roots of f and g in an algebraic closure of a field K. Thus, Res(f, g) = 0 if and only if f and g have a common root, and if f and g are relatively prime, Res(f, g) ≠ 0.


Question: What are the contributions to the one-loop effective potential from the gauge bosons in the given model?

Answer: The contributions from the three gauge bosons in the given model are included in the third line of the one-loop effective potential, which reads: 3/4 * (m_W^4 / 2) * [ln(m_W^2 / Q^2) + 3/2] + m_Z^4 / 4 * [ln(m_Z^2 / Q^2) - 3/2] + m_Z′^4 / 4 * [ln(m_Z′^2 / Q^2) - 3/2].


Question: What properties do homotheties representing elements of the stabilizer StabOut(FN)([T]) have, and how do they relate to the concept of eigenrays?

Answer: Such homotheties H represent elements of the stabilizer StabOut(FN)([T]), and they have a number of nice properties. If H fixes a branch point of T and a direction d at that branch point, then H possesses a well-defined eigenray ρ starting at the branch point in direction d such that H(ρ) = ρ, so that H acts on the ray ρ as multiplication by λ(ϕ) does on R≥0.


Question: How does the interpolation inequality (26) contribute to the boundedness of ∥ρ∥B1/p p,∞?

Answer: The interpolation inequality (26) implies that ∥ρ∥Ws,2 ≤ C ∥ρ∥B1/p p,∞. Since ∥ρ∥Lp is bounded uniformly in ℏ, it follows that ∥ρ∥B1/p p,∞ is also bounded uniformly in ℏ, which finishes the proof of Equation (18).


Question: What are the specific expressions for the coordinates X1 and X2 of the open string after the T-duality transformation, and how do they relate to the acceleration of the quark?

Answer: The coordinates X1 and X2 of the open string after the T-duality transformation are given by X1 = 2κϵ [sin(τ/2κ) + (n − 1)² sin(τ/2κ(n − 1))] and X2 = −2κϵ [cos(τ/2κ) − (n − 1)² cos(τ/2κ(n − 1))]. The modulus of the acceleration of the quark is a² = ϵ²/κ² sin²(nτ/4(n − 1)κ), and it vanishes at the points τm = 4κπ(n − 1)/n m, where m is an integer. These points correspond to the cusps of the small spiky string.


Question: What are the key criticisms of the Eigenstate Thermalization Hypothesis (ETH) and how have these criticisms influenced the field?

Answer: The ETH has been criticized for its lack of predictive power, as it leaves open at least three important questions: what precisely are 'physical observables', what makes a system 'sufficiently complex' to expect that ETH applies, and how long it will take for such observables to reach the thermal expectation values. For this reason, a lot of effort has been focused on numerical investigations that validate the ETH in specific Hamiltonian models and for various observables, often including local ones. The ETH is generally found to hold in non-integrable systems that are not many-body localized and equilibration towards thermal expectation values usually happens on reasonable timescales. However, a satisfactory analytical explanation for ETH is still missing.


Question: What are the specific outcomes for n = 2 boson stars with small and intermediate values of Λ, and how do these outcomes differ?

Answer: For small values of Λ, the n = 2 boson star collapses into a black hole, as indicated by the abrupt drop in the minimum value of the lapse function α after a certain time (t ≈ 3000). For intermediate values of Λ, the star is still unstable but decays into a lower state, losing mass/energy to infinity and redistributing its mass internally, as shown by the minval(α) function and the radial profile at r = 200.


Question: What is the condition for the trivial steady state E0 to be globally asymptotically stable, and what does this imply for the predator and prey populations?

Answer: The trivial steady state E0 = (0, 0) is globally asymptotically stable if λ ≤ 1. This implies that both the prey and predator populations go extinct.


Question: How does the OpenHybrid framework address the issue of flow-based models assigning higher likelihoods to outliers?

Answer: The OpenHybrid framework empirically observes that the issue of flow-based models assigning higher likelihoods to outliers does not occur when learning a joint representation for discriminative and generative components.


Question: How does the Fourier transform handle timelimited signals, and what are the implications for reconstruction?

Answer: The Fourier transform deals with a truncated signal and is associated with an integral kernel {e-iωt} around ω=∞ composing a continuous frame. The transformed function doesn’t vanish at infinity, which is a deduction from the continuous frame property. For timelimited functions, the Fourier transform can reconstruct the signal using only frequency information in the infinity, but the lower bound (1-λ0) of the reconstruction error tends to zero quickly as the frequency support J gets larger, making numerical reconstruction worse.


Question: How does the BSSN formulation modify the ADM equations to improve numerical stability in black hole simulations?

Answer: The BSSN formulation modifies the ADM equations by adding constraints (23a), (23b), and (26b) to the ADM evolution equations (23c) and (23d). This results in a modified structure of the evolution PDE system, which includes new dynamical variables such as the conformal factor χ, the conformal metric ˜γ ¯m¯n, and the trace-free conformal extrinsic curvature ˜A ¯m¯n. The BSSN equations are derived by inserting these new variables into the modified evolution equations, leading to a system that is more stable for numerical simulations of black hole spacetimes.


Question: What is the formula used to calculate the dσAA→AAJ/ψ(y)/dy cross section, and how is it evaluated at different rapidities y?

Answer: The formula used to calculate the dσAA→AAJ/ψ(y)/dy cross section is given by dσAA→AAJ/ψ(y)/dy = Nγ/A(y) S2(y) dσγp→J/ψp(y, t = 0)/dt ΦA(y) + Nγ/A(−y) S2(−y)dσγp→J/ψp(−y, t = 0)/dt ΦA(−y). The first term is evaluated at Wγp(y) = 2√ωHEA = p2MJ/ψEAey/2, and the second term is evaluated at Wγp(−y) = 2√ωLEA = p2MJ/ψEAe−y/2.


Question: What specific observations were made regarding the growth of Ag on Co nanoislands on Cu(111)?

Answer: Ag preferentially nucleates on the Co nanoislands initially, leaving the free Cu sites clean. Furthermore, those Co islands which are capped with Ag are almost completely capped, forming a perfect multilayered system of Ag/Co/Cu(111). A (9×9) reconstruction of the Ag overlayer on Co/Cu(111) is observed.


Question: What does the intensity profile of the transmitted light show when the polarizer is oriented along the x and y directions at z = 50 µm?

Answer: The x-polarized profile closely reproduces the patterns obtained with no polarizer inserted, and the y-polarized light shows a null at x = 0, with side-lobes near ±2 µm due to the weak y-directed field components in the grating region.


Question: How is the file accessed in two different patterns, and what are the parameters used for the second access pattern?

Answer: The file is accessed in two different patterns. The second access pattern is defined with repeat=3, count=5, and stride=2. The gap between the first and second access patterns is 7 elements, so the offset is set to 7.


Question: What is the definition of a clean bundle and a clean allocation in the context of binary submodular valuations, and how does it relate to the properties discussed in the query text?

Answer: For any agent i ∈ N + 0, a bundle S is clean with respect to the binary submodular valuation βi if βi(S) = |S|. An allocation X is said to be clean if for all agents i ∈ N + 0, βi(Xi) = |Xi|. This definition is similar to the notion of non-wastefulness used in Akrami et al. [2022b] and cleanness used in Benabbou et al. [2021]. In the context of the query text, the concept of clean bundles and clean allocations is crucial because it helps in understanding the properties of allocations under submodular valuations, such as the leximin and MNW allocations, which may not coincide and may not always be EF1.


Question: How does the coaxiom help in filtering out undesired values in the derivation of the expression e∆?

Answer: The coaxiom plays a crucial role by allowing us to filter out all undesired values, leaving only the value ∞, which represents divergence. If we take into account the coaxiom, we have to construct finite derivations where the coaxiom can be used as an axiom. For the expression e∆, we can build such finite derivations only for the judgement e∆⇒∞. More precisely, if e∆⇒v∞ has a finite proof tree, then v∞ = ∞. This is proven by induction: if the coaxiom is applied, the thesis is immediate, and if the rule (app) is applied, then there is a premise e∆⇒v∞, hence v∞ = ∞ holds by induction hypothesis.


Question: What are the specific hadronic decay channels of the dark matter particle ψ, and how are the antideuteron fluxes from these decays calculated?

Answer: The hadronic decay channels of the dark matter particle ψ are ψ → Z0ν, ψ → W ±ℓ∓, and ψ → h0ν. The antideuteron fluxes from these decays are calculated using the coalescence model to simulate the nuclear fusion of an antiproton and an antineutron, and the event generator PYTHIA 6.4 to simulate the production of individual antiprotons and antineutrons.


Question: What are some significant results regarding the ergodic properties of billiards in polygons, and how do they relate to the weak mixing property?

Answer: Significant results include C. Boldrighini, M. Keane, and F. Marchetti's finding that billiard flows have at most a countable set of directions containing periodic trajectories, and A. Katok's result that the billiard flow has zero entropy, implying a subexponential bound on the growth of complexity. D. Scheglov improved this bound to 'weakly exponential' for almost all triangles. H. Masur showed that the counting function of generalized diagonals grows quadratically for rational tables. A. Avila and V. Delecroix, and D. Aulicino, A. Avila, and V. Delecroix proved weak mixing in almost all directions in certain rational polygons. A. Málaga and S. Troubetzkoy proved the weak mixing property in almost all directions for rational billiards in generic polygons with vertical-horizontal sides. J. Bobok and S. Troubetzkoy proved topological weak mixing for the billiard map in the generic polygon.


Question: How does the multiple reflections approach account for the conversion between longitudinal and shear waves in the plate?

Answer: The multiple reflections approach considers a ray-based method to calculate the contribution of all possible paths the waves could take when reflecting inside the plate. For example, an incident wave can transmit into the solid as a longitudinal wave, reflect twice as itself, and at the third and last reflection convert into a shear wave before coming out of the same side of the plate. The contribution to the final reflection coefficient of this path can be written as AR = AIl12rl221ml21ls21e3iδLeiδS, where AI is the amplitude of the incident wave, l12 represents the conversion to a longitudinal wave in the solid, rl21 represents reflection of longitudinal waves inside the solid, ml21 represents mode conversion from a longitudinal to a shear wave inside the solid, ls21 represents the conversion from a shear wave in the solid to a wave in the fluid, δL is the phase shift of a longitudinal wave traveling between plate walls, and δS is the phase shift for shear waves.


Question: What are the key differences between the non-consistent and self-consistent formulations of the EPF, and how do they impact the perturbed Reynolds stress tensors?

Answer: The non-consistent formulation of the EPF introduces a moderation factor to weaken the effect of perturbed Reynolds stress, which can cause unintended behavior and violate the EPF’s self-consistency. In contrast, the self-consistent formulation presented in this paper correctly represents the basic concept of the EPF without the need for a separate moderation factor. This ensures that the perturbation framework remains self-consistent and physically meaningful. The self-consistent formulation has been implemented in DLR’s CFD solver suite TRACE and has been shown to improve the interpretability of uncertainty estimates, particularly in turbulent channel flow.


Question: How does the intensity of the AB profile change with different forms of dispersion profiles, particularly in the logarithmic dispersion profile?

Answer: The intensity of the AB profile increases in the logarithmic dispersion profile. The number of peaks in the periodic localized structure increases in the logarithmic dispersion profile, while it decreases in constant, linear, and Gaussian dispersion profiles.


Question: How does the forward trajectory of the system (12) intersect the line {py = 1} and what is the specific point of intersection?

Answer: The forward trajectory of system (12), starting at any point in the specified set, intersects the line {py = 1} at (x, py, ε) = (xε, 1, ε), where xε = ελ˚η + O(ελ˚ + 1/(1+2k(n-1))).


Question: What evidence supports the idea that binary interactions are a significant channel for the formation of Type IIb supernovae?

Answer: Evidence supporting binary interactions as a significant channel for Type IIb supernovae includes: 1) clumping in stellar winds suggesting currently used mass-loss rates are too high, 2) recent observations indicating that massive stars are predominantly part of close binary systems, 3) observed SE SN rates being too high to be explained solely by single-star evolution, 4) pre-SN masses estimated from light curves indicating lower ZAMS mass for SE SN progenitors than those producing WR stars via the single-star channel, and 5) circumstellar medium densities for SE SNe inferred from X-ray/radio observations indicating pre-SN mass-loss rates higher than WR winds, implying additional mass loss processes close to core-collapse.


Question: What are the specific results and speedup when using p = 0.02 for sparsification in the Wikipedia 2005/09 dataset?

Answer: For p = 0.02, the sparsification procedure takes approximately 9 seconds, and the counting procedure averages 2.46 seconds with a variance of 0.1 second. The speedup using p = 0.02 is on average 2880 times compared to running the node iterator on the initial graph. The estimates of the number of triangles are 4.15x10^7, 4.25x10^7, 4.6375x10^7, 4.0875x10^7, 4.3625x10^7, 4.3375x10^7, 4.35x10^7, 4.375x10^7, 4.45x10^7, and 4.25x10^7.


Question: What is the distribution of the demand for leadership positions in law firms larger than 751 lawyers, and how does it compare to the 30% proportional demand?

Answer: The total demand for leadership positions in law firms larger than 751 lawyers is 12,358 individuals, with 21% of the demand for counsel and 79% for partners (22% non-equity partners and 57% equity partners). The 30% proportional demand is 3,708 individuals, representing 13% of counsel and 26% of partners (20% non-equity partners and 15% equity partners).


Question: What is the explicit expression for the error in the transfer function interpolation using rational Krylov subspace methods, and how does it differ from the moments matching theory?

Answer: The explicit expression for the error in the transfer function interpolation using rational Krylov subspace methods involves shifts and Ritz values. This error formula is advantageous over the moments matching theory in a way similar to how the Lagrange type remainder formula is advantageous over the Peano type remainder formula in Taylor theorem. The moments matching theory states that the k-th derivatives of the approximate and original transfer functions are the same at the interpolating points, implying that the error is of the order o((z - z0)^k) near z0. However, this Peano type remainder is not satisfied when analyzing a wide region Z (∋z), and the explicit error formula provides a more detailed understanding of the error behavior when z is far away from z0.


Question: What is the relationship between the bordism theories Ωspin, Z/k-fb and Ωspin in the context of manifolds with fibered Z/k-singularities?

Answer: There is an exact triangle of (unreduced) bordism theories: Ωspin → Ωspin, Z/k-fb → Ωspin(− × BZ/k). The transformation i: Ωspin → Ωspin, Z/k-fb is given by considering closed spin manifolds as manifolds with empty fibered Z/k-singularity, and the transformation β: Ωspin, Z/k-fb → Ωspin(− × BZ/k) is of degree −1, determined by the Bockstein operator. The transfer τ: Ωspin(− × BZ/k) → Ωspin is of degree 0, and MSpinZ/k-fb is the cofiber of the map τ: MSpin ∧ (BZ/k)+ → MSpin.


Question: What are the specific kinship verification tasks supported by the FIW dataset, and how do they differ in terms of relationship types?

Answer: The FIW dataset supports kinship verification tasks that determine whether a face pair are blood relatives (KIN or NON-KIN). These tasks include verification over a broad range of relationship types such as parent-child (father-daughter (FD), father-son (FS), mother-daughter (MD), mother-son (MS)), siblings (brother-brother (BB), sister-sister (SS), brother-sister (SIBS)), and multiple generations (grandfather-granddaughter (GFGD), grandfather-grandson (GFGS), grandmother-granddaughter (GMGD), grandmother-grandson (GMGS)).


Question: What are the limitations of the full secular approximation in the context of weak qubit-qubit coupling and small detuning?

Answer: In the case of weak qubit-qubit coupling constant and small detuning, the full secular approximation is not able to detect important phenomena such as quantum beats and quantum synchronization, and does not produce entanglement during the evolution as depicted in Fig. 6.


Question: What are the limitations of using the parabolic approximation for the energy per particle in β-stable matter, and how does it impact the properties of neutron stars?

Answer: The parabolic approximation, which interpolates the energy per particle between neutron matter and symmetric matter, is unable to reproduce the exact asymmetry of β-stable matter in the range of densities considered. This discrepancy can impact the calculation of other neutron star properties, highlighting the importance of using the exact equation of state (EoS).


Question: What is the key idea behind computing an optimal schedule for smc-Unit on bipartite graphs, and how does it involve star forests and I, II, III-colorings?

Answer: The key idea is to find a spanning subgraph H of G for which AB-schedules with respect to H are among the optimal schedules for G. This subgraph H is a star forest of G, where each component of H is a star and H contains all vertices of G. The star forest allows for I-, II-, and III-colorings, which are used to schedule A-patterns and B-patterns on the respective vertex subsets, ensuring the resulting schedules are valid for both H and G.


Question: What is the formal condition for an edge to be not observable in a transition system (TS)?

Answer: An edge (u, n, v) ∈ E is not observable in (S, →) = TG(K) if for all s ∈ S, su ∈ [n−, n+), there exists k ∈ [0, ρ(v)] such that for all j ∈ [n−, n+), δ(F K_v)(su←j) = Sgn(k − (su←j)v).


Question: What is the significance of the finite thickness of SLMoS2 in the context of its bending modulus, and how does it compare to graphene?

Answer: The finite thickness of SLMoS2 is a key factor leading to a substantial enhancement in its bending modulus compared to monolayer graphene. Specifically, while each monolayer of S atoms contributes 1.75 eV to the bending modulus, similar to the 1.4 eV bending modulus of monolayer graphene, the additional pairwise and angular interactions between out-of-plane Mo and S atoms contribute 5.84 eV to the bending modulus of SLMoS2. This results in an overall bending modulus of 9.61 eV for SLMoS2, which is significantly higher than the 1.4 eV bending modulus of graphene.


Question: What are the specific locations and types of Wolf-Rayet stars detected in the central regions of NGC 5253, and how do they compare with previous studies?

Answer: In the central regions of NGC 5253, 1 WC star was detected in a 1′′×3′′ region covering the supernebula (cluster H04-1), and 10 WN and 2 WC stars were detected at the UV peak (cluster H04-4). López-Sánchez et al. (2007) found ∼1 WN star each in their regions A and B, and ∼10 WN stars in region C (equivalent to cluster H04-4). Monreal-Ibero et al. (2010) identified seven distinct regions containing blue-bump emission from WN-type stars. The location of their detections is consistent with those found by the above authors. Only two locations have coincident detections of WC and WN stars (regions WR3 and H ii 2 of Monreal-Ibero et al. 2010).


Question: What is the mechanism behind the giant magnetoresistance (GMR) effect in ferromagnet/non-magnet/ferromagnet trilayers?

Answer: In GMR heterostructures, the electrical resistance in a ferromagnet/non-magnet/ferromagnet trilayer is lower when the magnetization of the two ferromagnets is parallel than when it is antiparallel. This is because the conductivity is different for majority and minority spin channels in ferromagnets. When the magnetizations are parallel, one spin channel sees lower average resistance, leading to a difference in resistance for parallel and antiparallel magnetizations, and thus negative GMR.


Question: What are the key improvements and models in which the transshipment and SSSP problems are solved more efficiently according to the candidate text?

Answer: The key improvements include: 1. In the Broadcast CONGEST model, (1 + 𝜀)-approximate SSSP using e𝑂((√𝑛 + 𝐷)𝜀−3) rounds. 2. In the Broadcast congested clique model, (1 + 𝜀)-approximate transshipment and SSSP using e𝑂(𝜀−2) rounds. 3. In the Multipass streaming model, (1 + 𝜀)-approximate transshipment and SSSP using e𝑂(𝑛) space and e𝑂(𝜀−2) passes. These improvements are achieved by bypassing the hop set construction and using a spanner instead.


Question: What are the necessary and sufficient conditions for DMD without centering to model data generated by an affine system?

Answer: The necessary and sufficient conditions for DMD without centering to model data generated by an affine system are (1) having 'extra rank' available to capture the bias with an eigenvector of eigenvalue 1 and (2) a technical condition on the fixed point c and the eigenvectors of A as captured in V.


Question: How does the shipment planning problem incorporate side information and what is the structure of the optimization problem?

Answer: In the shipment planning problem, the decision maker has access to side information x (market trends, advertisements, etc.). The goal is to ship items from production facilities to multiple locations to satisfy demand at minimum cost. The optimization problem is structured as a two-stage process: first, the decision maker chooses an initial inventory quantity u1f to be produced in each facility at a per unit cost p1. After observing demands wℓ, the decision maker can produce additional units at a higher per unit cost p2. Finally, demand is fulfilled by shipping u2fℓ units from facility f to location ℓ at per-unit cost cfℓ, with each unit of satisfied demand generating revenue a. The problem is formulated as a minimization of the expected cost, considering the initial production, additional production, and shipping costs, subject to the constraint that the total shipped units must meet the demand in each location for all possible demand realizations.


Question: What does the presence of up to 8 phonon-dressed electronic state replicas suggest about the properties of 2D CrI3?

Answer: The exceptionally high number of phonon-dressed electronic state replicas (up to N = 8) suggests 2D CrI3 as an outstanding platform to explore nontrivial phases out of phonon-Floquet engineering, while the significant coupling to the spin degree of freedom adds an extra flavor whose impact on the phonon-Floquet states has not been studied.


Question: What are the specific flux and distance changes observed in the core, S1, and S2 components during the 2017 γ-ray outburst?

Answer: During the 2017 γ-ray outburst, the flux of S1 increased, coinciding with the rise of a flare. The core fluxes during the first two epochs (MJD 57859 and 57886) were consistent with each other within the errors, but increased rapidly thereafter. S2 also showed a tendency to move toward the core, and its flux increased rapidly after the 2016 γ-ray outburst.


Question: How does the acquisition process for KOSMOS data in the lab account for the impact of different grisms on throughput?

Answer: Both red and blue grisms were used in observations to gauge impacts on throughput throughout the entire bandpass. Spectra were taken using the krypton, neon, and argon internal calibration lamps on simultaneously, alternating between red and blue grisms.


Question: What are the specific transformations and their costs required to correct the generated semantics in the 4th frame to match the ground truth?

Answer: The transformations required are {’rubber’, ’sphere’} →{’metallic’, ’cylinder’} with an Edit cost of 4. This includes the replacement operation ’rubber’ →’metallic’ with an Edit cost of 2 and the transformation ’sphere’ →’cylinder’ with an Edit cost of 2.


Question: What is the relationship between the factorization homology of a locally constant commutative factorization algebra and the derived global sections of the associated cosheaf?

Answer: The factorization homology of a locally constant commutative factorization algebra F on X is isomorphic to the derived global sections of the cosheaf which, to any open U included in an euclidean disk D, associates the derived Hochschild homology CHU(F(D)).


Question: What specific visual and statistical methods were used to assess the out-of-sample predictive performance for the selected countries and years?

Answer: The out-of-sample predictive performance was assessed using the following methods: (1) Log-log plot of observed versus predicted population estimates for individual strata, (2) Histogram of standardized residuals plotted against a unit Normal reference distribution, (3) Observed frequency of a stratum having population>0 plotted against the modeled probability of this outcome, (4) Out-of-sample model estimates for population distribution by year of entry plotted against raw survey estimates, (5) Out-of-sample model estimates for population distribution by age at entry plotted against raw survey estimates, and (6) Out-of-sample model estimates for population distribution by current age plotted against raw survey estimates.


Question: How many meteor showers in the MDC SD have more than one set of radiants and orbital parameters, and what is the significance of these multiple solutions?

Answer: There are 195 showers with more than one set of radiants and orbital parameters in the MDC SD. These multiple solutions, or 'solutions,' strongly confirm the existence of a particular meteor shower.


Question: How does the performance of VGOS-INT-S compare to other Intensive programs, particularly during the period of superior precision?

Answer: During the period of superior precision (S22025 – S22081), VGOS-INT-S performed best compared to all other Intensive programs, including VGOS-INT-A, VGOS-INT-B, VGOS-INT-C, and the well-established S/X series IVS-INT-1, IVS-INT-2, and IVS-INT-3.


Question: How does the tidal deformability of NSs vary with mass for different EoSs, and what are the constraints from GW170817?

Answer: The tidal deformability Λ decreases with increasing NS mass and becomes very small at the NS maximum mass. For the EoSs DD-LZ1, DD-MEX, DDV, DDVT, and DDVTD, the tidal deformability at the canonical mass is 727.17, 791.60, 391.23, 337.51, and 281.05, respectively. The upper limit on the tidal deformability set by the measurement from GW170817 is shown, and all these values lie well below this limit.


Question: What is the proposed method for optimal sensor placement and how does it differ from traditional approaches?

Answer: The proposed method for optimal sensor placement is based on solving a sequence of reweighted ℓ1-optimization problems. This approach uses convex penalty functions and the A-optimal criterion, which is a convex function of w. In each step of the reweighted ℓ1 algorithm, a convex optimization problem is solved. This framework, based on reweighted ℓ1-minimization, is the first of its kind in the context of OED.


Question: How is the variational lower bound (V-LB) or evidence lower bound (ELBO) derived and expressed in the context of variational learning?

Answer: The variational lower bound (V-LB) or evidence lower bound (ELBO) is derived using the variational inference approach. It is expressed as: L(x; θ, φ) = Eqφ(h|x) [logpθ(x|h)] − KL [qφ(h|x)||pθ(h)] or equivalently as L(x; θ, φ) = Eqφ(h|x) [logpθ(x, h)] + H [qφ(h|x)]. This lower bound is obtained by introducing an auxiliary inference model qφ(h|x) to approximate the exact posterior pθ(h|x).


Question: How is the Hamiltonian system represented in holomorphic coordinates, and what is the conserved quantity associated with horizontal translations?

Answer: The Hamiltonian in holomorphic coordinates is given by H(W, Q) = ∫(ℑ(Q ¯Qα) + 2σ(J^1/2 - 1 - ℜWα) + g(|W|^2 - 1/2(¯W^2Wα + W^2¯Wα))) dα. The invariance with respect to horizontal translations leads to the conservation of momentum, M(W, Q) = i∫(¯QWα - Q¯Wα) dα.


Question: How does the presence of flow affect the transition from the superconducting to the normal state, and what is the critical temperature T* for this transition?

Answer: For T < T* ≈ 0.556 T_0c, where T_0c is the critical temperature in the absence of flow, the transition from the superconducting to the normal state remains discontinuous. For T ≥ T*, the transition is continuous, associated with a continuously vanishing order parameter as T approaches the flow-velocity-dependent transition temperature Tc(v) from below.


Question: How are the Regge string scattering amplitudes explicitly calculated in the Regge regime, and what approximations are made in the high-energy limit?

Answer: The Regge string scattering amplitudes are calculated using the integral form involving the Mandelstam variables s and t, and the mass terms M2 and M3. The expression is given by A(s, t) ≃ ∫_0^1 dy y^(-s/2 + N - 2) (1 - y)^(-t/2 + N - 2) · ∏_(n>0) [(n - 1)! √(-t) / (1 - y)^n]^pn · ∏_(m>1) [-(m - 1)! ˜t / (2M2) / (1 - y)^m]^qm · ∏_(l>1) [(l - 1)! ˜t′ / (2M2) / (1 - y)^l]^rl · [s / (2M2) y - ˜t / (2M2) (1 - y)]^q1 [s / (2M2) y + ˜t′ / (2M2) (1 - y)]^r1. In the high-energy limit, the first term in the bracket with power of pn, and the first terms in the brackets with powers of qm and rl for m, l > 1 are dropped, leading to subleading order terms in energy in the Regge limit. The beta function in the expression is approximated as B(-s/2 + N - 1 - i - j, -t/2 - 1 + i + j) ≈ B(-s/2 - 1, -t/2 - 1) (-s/2)^(-i - j) ((-t/2 - 1)^i + j).


Question: What is the structure of a minimal violation trace that can be extracted from a program not robust against transactional causal consistency, and how is it related to the happens-before relation?

Answer: A minimal violation trace of the form τ = α·isu(p, t)·β·(p′, t′)·del(p′, t)·γ can be extracted, where (isu(p, t), (p′, t′)) ∈HB and ((p′, t′), del(p′, t)) ∈HB1. This means that the issue event isu(p, t) happens-before the event (p′, t′), and the event (p′, t′) happens-before the store event del(p′, t).


Question: How do different sink radii affect the orbital evolution of binaries in circumbinary disks, particularly in terms of anisotropic accretion torques?

Answer: In simulations with larger sink radii (rs = 0.03ab), the tidal distortion of the circumbinary disk at pericenter approach eliminates isotropic accretion patterns, leading to significant discrepancies between gravitational-only (g) and gravitational plus anisotropic accretion (g+a) calculations. This can result in qualitatively different interpretations of orbital evolution, such as 'outspiral' vs. 'inspiral'. In contrast, smaller sink radii (rs = 0.005ab) resolve the inner region of the circumbinary disk, maintaining isotropic accretion and showing good agreement between g and g+a calculations. The convergence study across different eccentricities and sink radii indicates that anisotropic accretion torques become negligible at sufficiently small sink radii (rs ≲ 0.010ab), and the small sink radius simulations converge with the g+a results from the fiducial simulations.


Question: How do the posterior means of the coefficients in the survival submodel compare between the two-stage approach, the corrected two-stage approach, and the full joint model?

Answer: The posterior means from the 500 datasets for the three coefficients in the survival submodel (the coefficient for the baseline group variable and the association parameters for the two longitudinal outcomes) are shown in Figure 1. The corrected two-stage approach closely aligns with the full joint model, while the uncorrected two-stage approach shows some deviation from the true values, indicated by the dashed horizontal line.


Question: What specific phonon mode shows a continuous decrease in energy with decreasing temperature, and at what temperature does a structural phase transition occur?

Answer: The [111] transverse acoustic mode shows a continuous decrease in energy with decreasing temperature from above room temperature to 200 K, where a structural phase transition from cubic to tetragonal symmetry occurs.


Question: How do the CPC codes obtained from Construction 3 and Corollary 23 compare in size to the LPC codes obtained from Proposition 9 for specific parameters?

Answer: For the parameters n = 10, t = 3, w = 6, and q = 16, Construction 3 yields a (160, 48, 6)-CPC code of size 5 · 16^5 ≈ 222.322, while Proposition 9 yields a (160, 48, 6)-LPC code of size 217. For n = 9, t = 2, w = 7, and q = 16, Corollary 23 yields a (144, 32, 7)-LPC code of size ∑6_i=0 16^i ≈ 224.093, while Proposition 9 yields a (144, 32, 7)-LPC code of size 218.


Question: What are the specific interpolating functions compared in Figures 6 and 7, and how do they relate to the quantum vacuum model?

Answer: Figures 6 and 7 compare MOND’s interpolating functions 𝑓𝑠𝑡𝑑(𝑎0 𝑔𝑁⁄) in blue and 𝑓𝑟𝑎𝑟(𝑎0 𝑔0⁄) in black, with the interpolating function 𝑓𝑞𝑣(𝑎0 𝑔0⁄) 'coming' from the quantum vacuum. The quantum vacuum function is represented with 𝛼1 = 5 12⁄ and 𝛼2 = 2.32 in Figure 6, and 𝛼1 = 1 2⁄ and 𝛼2 = 1.93 in Figure 7. The green line in the figures shows the interpolating function from the quantum vacuum, which is as good as MOND’s functions numerically but is considered superior from a fundamental point of view.


Question: What is the impact of the strong He-He repulsion on the one-particle wave function and the interaction parameters V and V' in the context of 4He adsorbed on graphene?

Answer: The strong He-He repulsion on the scale of the one-particle wave function effectively produces an infinite on-site Hubbard U and narrows the wave functions considerably compared to the Wannier functions. This leads to the hard-core t − V − V' model, where the nearest-neighbor V and next-nearest neighbor V' determine the relevant quantum phases of the system. The resulting enormous energy scales associated with these parameters are unphysical, suggesting that the spatial extent of the one-particle wave function is too large and fails to capture the correct interaction physics.


Question: How is the Cramer’s rule for the Drazin inverse solution of the matrix equation AX = B extended to the case where AW is Hermitian?

Answer: If AW ∈Hm×m is Hermitian, then the Drazin inverse solution xij of the matrix equation can be represented as xij = P β∈Jr, m{i} cdeti((AW)k+2 . i (f.j)) β β / P β∈Jr, m |(AW)k+2 β β|, where f.j is the j-th column of F = ¯VD = (AW)kAD.


Question: What is the maximum input throughput of SBT on an octa-core ARMv8 platform, and what is the associated overhead?

Answer: SBT delivers state-of-the-art performance by processing input events up to 140 MB/sec (12M events/sec) with sub-second delay. The overhead incurred by SBT’s security mechanism is less than 25%.


Question: What are the average diamond distances for the CX and RESET gates when using 10^6 shots per local Pauli observable in the deterministic map simulations?

Answer: For the CX gates, the average diamond distance is approximately 3×10−4, while for the RESET gates, it is approximately 1.4 × 10−2.


Question: What is the condition for a weight φ to be flat in a D(Q)-category A, and how does it relate to the coweight ψ?

Answer: A weight φ : A−→◦∗a is flat if and only if φ ◦(b ∧t) = b ∧a for all b ∈Q, and φ ◦(ψ1 ∧ψ2) = (φ ◦ψ1) ∧(φ ◦ψ2) for any coweights ψ1, ψ2 with tψ1 = tψ2.


Question: How do the estimates of the skill premium per year of schooling in DPEP districts change when the new data set, including the four districts absent in the K23 data, is used?

Answer: The skill premium per year of schooling in DPEP districts changes from .134 in the original data to .159 when the new data set, including the four districts absent in the K23 data, is used.


Question: How do the authors of the candidate text address the computational burden in video semantic segmentation, and what specific techniques are mentioned?

Answer: The authors address the computational burden in video semantic segmentation through adaptive processing schedules and asymmetric frame processing. Adaptive processing schedules include a clock-based network update mechanism, where deeper layers are updated less frequently than shallower ones. Asymmetric frame processing involves dividing video sequences into key and non-key frames, where key frames are processed by the entire architecture, and non-key frames pass through lighter architectures, with high-level features propagated from key frames. Specific techniques include optical flow-based feature and label warping, adaptive feature propagation using spatially-variant convolutions, and dividing the backbone into low-level and high-level networks for key and non-key frames.


Question: What is the definition of the set of 'free' maps M(0)free_f,σ in the context of the large N regime, and how does it simplify when l = 0?

Answer: The set of 'free' maps M(0)free_f,σ is defined as M(0)free_f,σ := {M ∈ Mp | g(M) = 0, g(Mf,σ) = 0, ∆f,σ(M) = 0, and Σ(Mf,σ) = 0}. When l = 0, the condition g(M) = 0 is redundant, and the set simplifies to M(0)free_f,σ := {M ∈ Mp | g(Mf,σ) = 0, ∆f,σ(M) = 0, and Σ(Mf,σ) = 0}.


Question: How does the vortex partition function in the 2d vortex worldsheet theory relate to the Nekrasov partition function in the 4d theory at the root of the Higgs branch?

Answer: The vortex partition functions in the 2d vortex worldsheet theory agree with the Nekrasov partition function at the so-called 'root of the Higgs branch' in four dimensions. This agreement is shown using the differential equation discussed in the previous section.


Question: How does the system maintain the flow achieved by the CAVs after they are turned back into human-driven vehicles?

Answer: When the CAVs are turned back into human-driven vehicles, the flow that was attained by the CAVs quickly diminishes, and the traffic state goes back to a level substantially lower than it was 20 seconds ago.


Question: How does the polarization of cyclotron and synchrotron radiation vary with the angle between the line of sight and the magnetic field?

Answer: For cyclotron radiation, when the angle 𝜃 is 0 (along the magnetic field), the wave is circularly polarized (𝐸𝑧= 0 and 𝐸𝑥= 𝑖𝐸𝑦). When 𝜃 is 𝜋/2 (along the line of sight for a parallel shock), the wave is linearly polarized (𝐸𝑧≠0 and 𝐸𝑥= 𝐸𝑦= 0). For synchrotron radiation, the amplitudes of the electric field components are given by 𝐴∥,syn and 𝐴⊥,syn, which depend on the angle 𝜗′ between the line of sight and the electron's trajectory plane in the comoving frame. The degree of linear and circular polarization at angle 𝜗′ can be calculated using the electric field amplitudes of synchrotron emission.


Question: What are the specific results from the H1 Collaboration regarding jet production and the determination of the strong coupling constant at high Q2?

Answer: The H1 Collaboration measured inclusive jet, 2-jet, and 3-jet cross sections, normalized to the neutral current deep-inelastic scattering cross sections, as functions of Q2, jet transverse momentum, and proton momentum fraction. The strong coupling constant determined from these measurements is αs(MZ) = 0.1168 ± 0.0007 (exp.) +0.0046 −0.0030 (th.) ± 0.0016 (PDF).


Question: What is the proposed method for error correction in continuous-variable quantum key distribution (CVQKD) and how does it improve the key rates and transmission distances?

Answer: The proposed method is an efficient logical layer-based reconciliation technique that operates on the raw-data level to correct the noise of the quantum channel in the low-dimensional (scalar) space. This method allows the reconciliation to be extended to arbitrary dimensions and provides unconditional security with zero error probability in any practical CVQKD scenario, significantly improving the currently available key rates and transmission distances of CVQKD.


Question: What is the primary component responsible for the majority of power consumption in a radio transceiver, and how does reducing its power affect overall consumption?

Answer: The primary component responsible for the majority of power consumption in a radio transceiver is the Power Amplifier (PA). Reducing the transmit power causes a significant decrease in the consumed power in the PA.


Question: How does the optimal prior derived using the MSE criterion compare to the one derived using the KL criterion in terms of borrowing from historical data?

Answer: The optimal prior derived using the MSE criterion is more conservative and tends to discourage borrowing compared to the one derived using the KL criterion.


Question: How are the slow-roll parameters defined in the Einstein frame, and what are their values for ordinary Higgs inflation without criticality?

Answer: In the Einstein frame, the slow-roll parameters are defined as εV = M_P^2 (U'/U)^2 / 2, ηV = M_P^2 U'' / (2U), and ζ2_V = M_P^4 (U'''/U) (U'/U). For ordinary Higgs inflation without criticality, these parameters are given by εV ≃ 3 / (4N^2) and ηV ≃ -1 / N, where N is the number of e-foldings.


Question: How does the proposed method achieve a phase flip on the target qubits when the control qubit is in the state |1⟩, and what is the final state of the cavity mode?

Answer: When the control atom 1 is initially in the state |1⟩, a phase flip happens to the state |1⟩ of atoms (2, 3) while the cavity mode returns to its original vacuum state, after the three-step operation.


Question: What is the significance of the S/N for the CO2 feature at 15 µm, and how does it compare to the O3 feature at 9.6 µm in terms of characterizing habitable planets?

Answer: The S/N for the CO2 feature at 15 µm is significant for characterizing habitable planets, especially in nearby systems (<10 pc) and when hosted by M stars (<0.2 M⊙). The S/N for the CO2 feature is higher than that for the O3 feature at 9.6 µm, with an S/N of 3.5 for a single secondary transit with JWST-MIRI, making it more feasible to detect and characterize habitable planets.


Question: How does the paper approximate quasiconvex Lipschitz domains locally by convex domains, and what is the significance of this approximation?

Answer: The paper shows that a quasiconvex Lipschitz domain can be well approximated locally by convex domains at all small scales. Specifically, for any point x on the boundary of the quasiconvex domain Ω and a small radius r, the convex hull Vr(x) of the intersection of the ball Br(x) with Ω is such that the distance between the boundary of Br,+(x) and the boundary of Vr(x) is bounded by 2rω(2r). This approximation is significant because it allows the use of properties of convex domains to study the behavior of quasiconvex Lipschitz domains.


Question: What is the formulation of the personalized MCID c*(z) and how does it differ from the Bayes rule in classification?

Answer: The personalized MCID c*(z) is formulated as the solution to min_c P(Y ≠ sign(X - c(Z))) = min_c 1/2 E (1 - Y sign(X - c(Z))), where P is taken with respect to (X, Y, Z). It satisfies pz(c*(z)) = P(Y = 1|X = c*(z), Z = z) = 1/2, with pz(x) being a continuous and strictly increasing function in x for any value of z. The personalized MCID differs from the Bayes rule in classification in that the candidate function in the MCID formulation must take the form of x - c(z) to estimate c*(z), whereas a Bayes rule in classification searches for the optimal classification function g(x, z) that may not lead to an explicit estimation of c*(z).


Question: How does the proof of Theorem 4.1.1 utilize the inequalities established in Lemmas 4.1.4 and 4.1.5 to show that N T M,A<P> = N E M,A<P> for each P?

Answer: By combining the inequalities from Lemmas 4.1.4 and 4.1.5, we have NM,A ≤ ∑P P! N T M,A<P> ≤ ∑P P! N E M,A<P> ≤ NM,A. Since all inequalities are equalities, it follows that N T M,A<P> = N E M,A<P> for each P.


Question: How does the GIBS algorithm improve upon traditional variance decomposition methods in the context of the AMF model?

Answer: The GIBS algorithm improves upon traditional variance decomposition methods by abandoning the variance decomposition framework and not using PCA. Instead, it uses prototype clustering and high-dimensional methods, selecting basis assets as the 'prototypes' or 'center' of the groups they represent, which provides clearer interpretation and improved prediction accuracy. Additionally, it finds groupwise basis assets that may only affect stocks in specific groups, sectors, or industries, offering new insights into the low-volatility anomaly.


Question: How does the method of symmetry reduction proposed by Bödi, Grundhöfer, and Herr compare to the reduction lemma in terms of computational efficiency and dimensionality reduction?

Answer: The method of symmetry reduction proposed by Bödi, Grundhöfer, and Herr involves projecting the linear program to the intersection E of the 1-eigenspaces of all matrices Y such that (X, Y) ∈ Aut(L). This can reduce the number of variables if there are many automorphisms. However, the dimension of E can be expected to be much smaller than n. The reduction lemma, on the other hand, uses colour refinement, which is much more efficient than computing the automorphism group of a linear program. Experiments show that colour refinement reduces the dimension of the linear programs at least as much as the symmetry reduction and is significantly more efficient in practice.


Question: What are the conditions for ϕ to swap the nodal tangent lines and for f' to be N(G3)-conjugate to f?

Answer: ϕ swaps the nodal tangent lines if it has the form ϕ = [α β γ; 0 0 δ; 0 1 0]. For f' to be N(G3)-conjugate to f, the conditions are g' = g^(-1), b' = bg, δ^3 = 1, and α = (δg)^(-1).


Question: What is the physical explanation for the three-pronged fork-like structure observed in photoelectron momentum distributions using xenon in long, mid-IR driving pulses?

Answer: The three-pronged fork-like structure is nearly orthogonal to the laser polarization axis and is very sensitive to the pulse length and frequency. It is markedly enhanced with increasing wavelength and disappears for few-cycle pulses. Physically, it has been explained in terms of low-energy forward scattered trajectories. An SFA computation has shown that this structure is universal and does not necessarily require Coulomb-type potentials to form, but the divergent Coulomb scattering cross section makes it visible by allowing it to rise above the contributions of the direct electrons. The fork is of a classical nature, similar to the low-energy structures (LES), and is not caused by quantum-interference effects, but rather stems from the kinematical constraints imposed upon the electron’s returning energies, particularly for orbits whose excursion times are much longer than a field cycle.


Question: How does the transition semigroup P(t)ϕ(x) behave under the hypotheses of Theorem 5 for α = 1/2, and what is the bound on its derivative?

Answer: Under the hypotheses of Theorem 5 for α = 1/2, the transition semigroup P(t)ϕ(x) satisfies the bound ∥D1/2P(t)ϕ(x)∥1/2 ≤ eLF T √t ∥Q1/2∥L(X)∥ϕ∥∞.


Question: How does the behavior of A(x, u) change for large X+ and what method is used to evaluate it?

Answer: For large X+, the behavior of A(x, u) is determined by the singularities of the bulk-to-boundary propagator in the complex q+ plane, specifically the poles corresponding to quasi-normal modes. To evaluate this, steepest descent methods are used, which involve analyzing the integral expression for A(x, u) near the horizon.


Question: How does the Borel cardinality of the Fσ equivalence relation E ↾Y relate to different models in the context of a countable language?

Answer: For any two models A ⊆ M, N, the Borel cardinality of EM ↾ YM is Borel isomorphic to EN ↾ YN, allowing us to refer to the Borel cardinality of the Fσ equivalence relation E ↾ Y without specifying the model.


Question: How does intermediate timescale cycling affect the conversion of specialists to generalists in the context of the fitness landscape described in the supplementary information?

Answer: Intermediate timescale cycling enhances specialist-to-generalist conversions across fitness valleys without enhancing the time-reversed process. It allows evolution between specialists, introducing sequence variance even in initially monoclonal specialist populations, which enhances specialists-to-generalists transitions but not the reverse process.


Question: What is the statistical significance of the signal for BP1, and what are the corresponding masses of the doubly charged scalars?

Answer: The statistical significance for BP1 is 5.80, corresponding to mH±±1 = 416.37 GeV and mH±±2 = 216.13 GeV.


Question: What are the molecular hydrogen column densities and fractions for the components of DLA 2340−00?

Answer: For DLA 2340−00, the molecular hydrogen column densities and fractions for the components are as follows: comp 1 has log (total N(H2)) = 16.06 and fc = 1.03×10−4; comp 2 has log (total N(H2)) = 15.47 and fc = 2.63×10−5; comp 4 has log (total N(H2)) = 18.08 and fc = 1.06×10−2; comp 6 has log (total N(H2)) = 16.98 and fc = 8.51×10−4; comp 8 has log (total N(H2)) = 16.67 and fc = 4.21×10−4; comp 9 has log (total N(H2)) = 17.35 and fc = 2.00×10−3.


Question: What are the specific conditions and results for the cases where u1, u2, and u3 are all of type (I) in the context of the estimate (5.30)?

Answer: For the cases where u1, u2, and u3 are all of type (I), the estimate (5.30) is bounded by δθ(N 1)0+ times the square root of the sum over |n| ≲ N1 of the sum over Bn of the product of |gn1|/|n1|α, |gn2|/|n2|α, and |gn3|/|n3|α squared. This is further bounded by δθ−3/2β(N 1)−α+3/2ε+(N 2)−α(N 3)−α+1/2 ≲ N 3s−γ− ∏(N 0− j) for α > 1/3 + 1/6γ > 1/3 and sufficiently large N outside an exceptional set of measure < (N 1)0−e−1/δc.


Question: How does the EDP convergence theory apply to the family of GS with a small parameter ε > 0, and what are the conditions for this convergence?

Answer: A family (X, Eε, Rε)ε>0 of GS with a small parameter ε > 0 converges in the sense of the EDP to the limiting GS (X, Eeff, Reff) if the following Γ-convergences hold: Eε Γ→Eeff in X and Dε Γ→D0 in L2([0, T]; X). The dissipation functionals Dε are defined as Dε(u(·)) := ∫₀ᵀ (Rε(u, ˙u) + R∗ε(u, −DEε(u))) dt for ε > 0, and D0(u(·)) := ∫₀ᵀ (Reff(u, ˙u) + R∗eff(u, −DEeff(u))) dt for ε = 0.


Question: What is the proposed origin of the orange material in the diffuse ejecta deposits of Oppia and Octavia craters, and how does it differ from the orange material in smaller craters like Cornelia and Rubria?

Answer: The orange material in the diffuse ejecta deposits of Oppia and Octavia craters is proposed to be ancient impact melt material excavated and deposited as breccia, rather than formed during the impact event. In contrast, the orange material in smaller craters like Cornelia and Rubria is more likely to be impact melt formed during the impact event, with volumes ranging from 0.00007 to 0.088 km³, depending on the impact velocity and the porosity of Vesta's surface.


Question: What are the constraints on the mass and distance of the lens system, and why are the posterior distributions very broad?

Answer: The posterior distributions of both mass and distance are very broad, and the lens system can almost equally well reside in the bulge or disk. This is a consequence of the fact that the only measured constraint is tE, while θE effectively has only a lower limit.


Question: What are the conditions for a submanifold to be an extrinsic sphere, and how does this relate to the mean curvature vector?

Answer: An extrinsic sphere is a totally umbilical submanifold with a nonzero parallel mean curvature vector. A submanifold is totally umbilical if its second fundamental form satisfies h(X, Y) = ⟨X, Y⟩H, and the mean curvature vector H is parallel if DH = 0 identically.


Question: How does the lattice constant and bulk modulus of the compounds in Table 1 compare with the trends observed in the candidate text?

Answer: The candidate text states that by growing the atomic number from Cl to Br, the lattice constant and bond lengths increase, which is consistent with Pauling's empirical rule. Additionally, the smaller the lattice constant, the larger the bulk modulus and the absolute value of the cohesive energy, which is consistent with the semi-empirical Cohen’s equation. This trend is reflected in Table 1, where compounds with larger atomic numbers (e.g., Br) generally have larger lattice constants and smaller bulk moduli compared to those with smaller atomic numbers (e.g., Cl).


Question: How does the choice of parameters θ and x in the independent process help approximate the conditioned combinatorial structures, and what are some specific examples of this approximation?

Answer: The choice of parameters θ and x in the independent process helps approximate the conditioned combinatorial structures by ensuring that E(Z1 + · · · + Zn) is close to k and ETn is close to n. For example, to study random permutations on n objects conditional on having at least 5 log n cycles, or exactly ⌊5 log n⌋ cycles, or at most 0.3 log n cycles, we propose using x = 1 and θ = 5 or 0.3. The independent process with these parameters should be a good approximation for both the conditioned random permutations and for the Ewens sampling formula.


Question: What is the significance of the Jantzen filtration in the context of restricted Verma modules at the critical level, and how does it relate to the Jantzen sum formula for baby Verma modules?

Answer: The Jantzen filtration for restricted Verma modules at the critical level is significant because it helps describe the structure of these modules. The Jantzen sum formula for restricted Verma modules at the critical level is analogous to the Jantzen sum formula for baby Verma modules over a modular Lie algebra, which describes the characters of the Jantzen filtration as an alternating sum formula of certain characters of baby Verma modules of weight 'lower' than λ. This formula also implies a new proof of the linkage principle, which was already derived by Arakawa and Fiebig.


Question: How does the synthetic caption pretraining method contribute to the performance of StreetCLIP in image geolocalization?

Answer: The synthetic caption pretraining method generates synthetic captions for contrastive learning, enabling the use of natural language to ground CLIP in the context of the image geolocalization task. This method is equivalent to generating a new domain-specific generalized zero-shot learner during every batch iteration, which learns to distinguish between classes seen and unseen during training. As a result, StreetCLIP, trained on 1.1 million Street View images, achieves state-of-the-art performance on the open-domain image geolocalization benchmarks IM2GPS and IM2GPS3K, improving geolocation prediction accuracy for some distance thresholds between 0.3 and 2.4 percentage points.


Question: What does the horizon value of the scalar field approach as temperature decreases, and why does it not increase further?

Answer: The horizon value of the scalar field approaches −1/√γ as temperature decreases. It does not increase further because if the horizon value of the gauge field is greater than −1/√γ, the gauge field becomes tachyonic near the horizon due to the wrong sign of the kinetic term of the gauge field, making the solution not allowed from the equation of motion.


Question: What are the key physical characteristics of the molecular clumps identified in the G305 star-forming complex?

Answer: The molecular clumps identified in the G305 star-forming complex have excitation temperatures ranging from 7–25 K, optical depths of 0.2–0.9, H2 column densities of 0.1–4.0 × 10^22 cm^-2, clump masses ranging from 10^2–10^4 M⊙, and a total molecular mass of > 3.5 × 10^5 M⊙.


Question: What are the advantages of using non-Cartesian lattices like BCC and FCC over the Cartesian lattice in terms of interpolation and data reconstruction?

Answer: Non-Cartesian lattices like BCC and FCC offer higher accuracy and smoother reconstructions compared to the Cartesian lattice. Specifically, the BCC lattice provides around 30% better approximations in 3D, and the FCC lattice offers more isotropic reconstructions with fewer visual artifacts, especially at lower resolutions. These improvements are due to the geometry and support of the chosen interpolants, which better attenuate high-frequency content in the Fourier domain.


Question: What are the characteristic densities ρc, ρmin, ρ∗, and the actual density ρ in terms of p and p0, and how do they relate to the density constraints for different values of p?

Answer: The characteristic densities are defined as follows: ρc = K / r^3, ρmin = 3.9 r / d p^(1/2) ρc, ρ∗ = 17.54 p^(3/2) ρc, and the actual density ρ = (r / ℓ)^3 ρc = p0 ρc. For p → 0, there are no density constraints. For 0 < p < 1, the treatment is valid for ρ ≥ ρmin. For p ≥ 1, the treatment is valid for ρ ≥ ρc and ρ ≥ ρ∗, but not in the region ρc < ρ < ρ∗.


Question: How does the intrinsic width of the Ca II line and the NBS bandpass FWHM affect the measurement of the ZL brightness?

Answer: The Ca II line has an intrinsic width of 70 km/s and the FWHM of the NBS bandpass is 245 km/s. Since the line is not fully resolved by the NBS and the effects of broadening are small compared to the intrinsic width, the dynamical effects have negligible impact on total ZL brightness measurements.


Question: How is the decoding delay minimized in the context of inter-session network coding, and what optimization problem is formulated to achieve this?

Answer: The decoding delay is minimized by balancing the delays for collecting the necessary number of innovative packets for every component session. This is achieved by formulating a min max optimization problem to determine the coefficients γt,t′i,s that minimize the maximum average delay ∆t i,s among the component sessions s ∈ St. The optimization problem is written as min γt i max s∈St ∆t i,s(γt i) = min γt i max s∈St di Ns qt i,s(γt i) subject to the constraints ∑ s′∈St′ γt,t′ i,s′ ≤ 1, γt,t′ i,s′ ∈ [0, 1], ∀t′ ∈ Tt.


Question: What does Theorem 4.5 state about the bijection induced by the (H, c)-lifting invariant for tuples g in G?

Answer: Theorem 4.5 states that there is a constant MH,c ∈ N such that for all ψ ∈ ZD satisfying min(ψ) ≥ MH,c, the morphism ΠH,c induces a bijection from the set of tuples g in G|ψ| where ⟨g⟩ = H, cH(g) = c, and µH,c(g) = ψ, to the set of elements x in U(H, c) that have image ψ in ZD.


Question: What are the key factors that influence the mode spectrum and dynamical properties of nanomagnetic elements, as discussed in the introduction?

Answer: The key factors that influence the mode spectrum and dynamical properties of nanomagnetic elements include the non-uniform demagnetizing field, the aspect ratio of the elements, the effective internal magnetic field, and the static magnetization. These factors introduce additional complexity into the magnetization dynamics, leading to a richer mode spectrum and less uniform magnetic response to a pulsed magnetic field.


Question: How does the angle distribution and convergence history of the proposed method compare to the previously proposed SPH-based mesh generator [10] and its variant [10.v2]?

Answer: The proposed method shows better convergence and mesh quality compared to [10] and [10.v2]. Specifically, [10.v2] fails to converge within the plotted time range and has the worst mesh quality. The proposed method maintains stability at the boundary, while [10.v2] experiences particle losses at the domain boundary, requiring remapping of particles into the positive phase.


Question: How does the degree of polarization in HBLs and IBLs compare to the theoretical maximum from synchrotron polarization, and what does this imply for γ-ray polarimetry?

Answer: In IBLs and HBLs, where X-rays (in IBLs only the soft X-rays) are dominated by synchrotron emission, the degree of order/disorder of the magnetic field can be determined by comparing the measured Πsy to the theoretical maximum from synchrotron polarization. This re-scaling factor can be used to assess whether the expected γ-ray polarization is within the capabilities of Fermi-LAT to determine polarization. However, HBLs typically exhibit very hard photon spectra in the Fermi-LAT range, with very low photon fluxes below 200 MeV, making the detection of γ-ray polarization infeasible in the near future.


Question: What are the key properties of the predicted spin defects in calcium oxide that make them suitable for quantum technologies?

Answer: The predicted spin defects in calcium oxide have electronic properties similar to the NV center in diamond, emit in the telecommunication range, and exhibit improved coherence times due to the presence of clock transitions. The high-spin nucleus of each dopant (Sb, Bi, or I) strongly couples to the electron spin, leading to many controllable quantum levels and the emergence of atomic clock-like transitions that are well protected from environmental noise. Specifically, the Hanh-echo coherence time increases beyond seconds at the clock-like transition in the defect with 209Bi.


Question: What evidence is provided to show that the pepducin PP-001 reaches and interacts with primary sensory neurons in the dorsal root ganglia (DRGs)?

Answer: Following lumbar i.t. injection of a TAMRA-tagged ICL1-derived pepducin (PP-001), diffuse TAMRA staining was detected in the different subtypes of sensory neurons (small, medium, and large) at the level of the L5-L6 vertebrae of pepducin-treated animals, as observed via confocal microscopy. This fluorescence was absent in vehicle-treated animals, indicating that PP-001 reaches the primary sensory neurons in the DRGs.


Question: What conditions must be met for a stochastic map Φ to preserve the f-divergence of two operators A and B, and what does this imply about the reversibility of Φ?

Answer: If a stochastic map Φ preserves the f-divergence of two operators A and B corresponding to an operator convex function which is not a polynomial, then it preserves a certain set of 'primitive' f-divergences, corresponding to the functions ϕt(x) := −x/(x + t) for a set T of t’s. Moreover, if this set has large enough cardinality (depending on A, B, and Φ) and Φ is 2-positive, then there exists another stochastic map Ψ reversing the action of Φ on {A, B}, i.e., such that Ψ(Φ(A)) = A and Ψ(Φ(B)) = B.


Question: How is the backward propagation of errors performed in the neural ODE, and what is the role of the adjoint system in this process?

Answer: Backward propagation of errors is performed through the backward solve of an adjoint system given by dbg/dt = −bTg ∂f(ar, θ)/∂ar, where bg is the augmented state vector given by bg = [ ∂E/∂ar, ∂E/∂θ, ∂E/∂t ]T. This value of the gradient can then be used to update the parameters θ using an optimization algorithm.


Question: How does the nonstationary GEV model account for changes in extreme rainfall, and what are the different forms of the GEV distribution based on the shape parameter?

Answer: The nonstationary GEV model accounts for changes in extreme rainfall by allowing the location parameter (μ) to covary with different physical processes. The GEV distribution can take one of three forms based on the shape parameter (ξ): i) Frechet, or lower bounded with a heavy upper-tailed, if ξ is positive; ii) Gumbel, or unbounded with heavy upper tails, if ξ is zero; iii) 'Reverse' Weibull, or upper bounded with heavy lower tails, if ξ is negative.


Question: What methods are used to handle the ultra-violet (UV) divergences in the calculation of the mass quadrupole moment at the 4PN order?

Answer: The ultra-violet (UV) divergences due to the modeling of compact objects by delta-functions are treated with dimensional regularization (DR).


Question: What are the conditions and properties of the matrix Y in the Riemann-Hilbert problem as described in the candidate text?

Answer: The matrix Y in the Riemann-Hilbert problem is characterized by the following conditions: (Y1) Y is analytic in C \ [−1, 1]. (Y2) On (−1, 0) ∪ (0, 1), Y possesses continuous boundary values Y+ (from the upper half plane) and Y− (from the lower half plane), and Y+(x) = Y−(x) * [[1, wc(x)], [0, 1]]. (Y3) As z → ∞, Y(z) = (I + O(1/z)) * [[z^n, 0], [0, z^−n]], where I is the identity 2 × 2 matrix. (Y4) Y has specific asymptotic behaviors at the end points of the interval and at the origin.


Question: What is the maximum value of G(c, r, p) on the face p = 1 and at which point is it attained?

Answer: The maximum value of G(c, r, 1) is 15.5264, attained at the point (c, r) = (1/2(60 - √30), 1/105(25 - √30)).


Question: What is the range of the power-law exponent for which the boundary layer problem for Ostwald-de Waele power law fluids has a solution, and how does this compare to the range discussed in the candidate text?

Answer: The boundary layer problem for Ostwald-de Waele power law fluids has a solution in the range 0.5 < α ≤ 1, with an analytical solution obtained in the range 0.74 ≤ α < 1. In contrast, the candidate text focuses on dilatant fluids with the exponent n in the range (1, ∞).


Question: What does the NUV/NIR flux ratio of TOI 451 B indicate about its age and membership in the Psc-Eri stream?

Answer: TOI 451 B sits above the Hyades sequence and shows an NUV flux excess that is broadly consistent with the Pleiades sequence, which is similar in age to Psc-Eri. This supports the membership of the TOI 451 system to the Psc-Eri stream.


Question: What are the specific criteria for categorizing a device as defective in the 200 mm wafer analysis, and what is the yield of functional devices?

Answer: A device is categorized as defective when the gate leakage (Igs) exceeds 100 nA, a short circuit channel (Ids > 5 mA) or an open circuit channel (Ids < 100 nA) is detected, or when the drain current Ids modulation as a function of Vgs is < 10%. Out of the 648 devices randomly picked from the 200 mm wafer, 624 were categorized as functional and 24 were categorized as defective, corresponding to a yield of 96%.


Question: What is the impact of the PRS on the omnidirectional path loss in the measured scenarios?

Answer: The PRS increases the omnidirectional received power by a maximum of 22 dB and a minimum of 0.5 dB with a mean of 6.5 dB. This increase depends on the PRS and other existing multipath components (MPCs). A slight increase indicates that other MPCs dominate the total received power, typical for the LOS scenario, while a significant increase often occurs at NLOS and near-outage locations, where the PRS-enhanced path becomes the strongest path and dominates the channel.


Question: What are the specific experimental settings used to evaluate the methods in Section 6.1 and 6.2?

Answer: In Section 6.1, synthetic data sets are used by CE-based methods to estimate causal effects and by the local ITC method to perform conditional independence tests. In Section 6.2, the methods are evaluated based on structures learned from data using three global structure learning algorithms: PC, stable PC, and GES, and the MB-by-MB variant to learn parents and siblings of the vertices of interest. The significance level α of statistical independence tests is 0.001. The experiments were run on a computer with an Intel 2.5GHz CPU and 8 GB of memory. The experiments involved generating 5,000 weighted DAGs for each combination of n (50, 100) and d (1.5, 2.0, 2.5, 3.0, 3.5, 4.0) with edge weights drawn from a Uniform([0.8, 1.6]) or Uniform([−1.6, −0.8] ∪ [0.8, 1.6]) distribution. For each experiment parameter setting, the experiment was repeated 5,000 times.


Question: How does the concept of regular interval exchange transformations relate to the minimality of an interval exchange transformation, and what is a necessary condition for minimality?

Answer: A d-interval exchange transformation is said to be regular if the orbits of the points µi, 1 ≤ i < d, are infinite and disjoint. A regular interval exchange transformation is minimal, as stated by Keane's theorem. However, the converse is not true. A necessary condition for the minimality of an interval exchange transformation is that the pair of permutations (π0, π1) is indecomposable, meaning π0({1, ..., k}) ≠ π1({1, ..., k}) for every k < d.


Question: What is the new heuristic algorithm presented in the candidate text, and how does it compare to other existing algorithms in terms of finding reset words?

Answer: The new heuristic algorithm presented is called CutOﬀ-IBFS, which is based on inverse breadth-first-search in the power automaton. Experimental investigations show that this method generally finds shorter reset words in an average case and works well in practice compared to other algorithms existing in literature.


Question: How does the velocity dispersion function (VDF) of the BOSS RS at z ∼0.55 compare with the low-z VDF, and what are the implications of this comparison?

Answer: The BOSS RS VDF at z ∼0.55 is in overall agreement with the z ∼0 early-type galaxy (ETG) VDF at log10 σ ≳2.47. However, the number density of z = 0.55 RS galaxies is larger than that of z = 0 ETG galaxies at 2.35 ≳log10 σ ≳2.47. This disagreement may be due to the extrapolation of an intermediate-mass L-σ relation towards the high-mass end in previous low-z works. Additionally, differences in sample selection methods can affect the comparison, and these can be mitigated by analyzing z = 0 SDSS data using the same framework presented in this paper.


Question: What are the implications of the observed gender differences in self-efficacy for physics, and how can they be addressed in secondary schools and universities?

Answer: The observed gender differences in physics self-efficacy have important implications for research and practice. Counter programming in secondary schools should be introduced to broadly address stereotypes and improve women’s self-efficacy for physics, such as using growth mindset interventions. If more narrowly targeted to incoming engineering first-year students, prior biases might be lessened just in time to reduce stereotype-threat effects within the early coursework. Physics-specific messages could be a good pragmatic choice, and such interventions could shift the large gender imbalance of students selecting physics-heavy engineering majors.


Question: What is the formula for the optimal transport cost when both states are represented by diagonal density matrices?

Answer: The formula for the optimal transport cost when both states are represented by diagonal density matrices is T Q(ρcl_r, ρcl_s) = 1/2 max{ (√r - √s)^2, (√(1-r) - √(1-s))^2 }.


Question: What are the specific conditions that lead to the inadmissibility of the cases (E.116a) to (E.116d) in the proof of Lemma E.8?

Answer: For case (E.116a), if A(1)2 = B(1)2 = A(3)2 = B(3)2 = 0, then from C(13) = 0 and C(31) = 0, we obtain A(2)1 = B(2)1 = A(2)2 = B(2)2 = 0, leading to an inconsistent result with the square property. For case (E.116b), if A(1)2 = B(1)2 = A(3)3 = B(3)3 = 0, then from C(13) = 0, C(15) = 0, C(31) = 0, and C(22) = 0, we obtain A(2)3 = B(2)3 = 0, A(3)4 = B(3)4 = 0, A(2)4 = B(2)4 = 0, and A(1)1 = B(1)1 = 0, leading to an inadequate result. For case (E.116c), if A(1)4 = B(1)4 = A(3)2 = B(3)2 = 0, then from C(13) = 0, C(15) = 0, C(31) = 0, and C(22) = 0, similar contradictions arise. For case (E.116d), if A(1)4 = B(1)4 = A(3)3 = B(3)3 = 0, similar contradictions also arise, making all cases inadmissible.


Question: What are the key differences between the Nagy-Soper and Catani-Seymour subtraction schemes in terms of polarization handling?

Answer: The Nagy-Soper subtraction terms contain generic spinors and polarization vectors, which enable them to treat both fixed helicities and random polarization states simultaneously. In contrast, the Catani-Seymour dipoles, as implemented, can only work with helicity eigenstates. An extension of the Catani-Seymour formalism for randomly polarized partons has been recently presented, characterized by the introduction of additional subtraction terms.


Question: What is the form of the angular part of the eigenfunction Ψ(r, ϕ) and how does it depend on the eigenvalues of the reflection operators R1 and R2?

Answer: The angular part of the eigenfunction Ψ(r, ϕ) is given by Φ(s1,s2)_m(ϕ) = ηm cose1 ϕ sine2 ϕ P(µ2+e2−1/2,µ1+e1−1/2)_m−e1/2−e2/2(cos 2ϕ), where P(α,β)_n(x) are the classical Jacobi polynomials, ηm is a normalization factor, and e1, e2 are the indicator functions for the eigenvalues of the reflections R1 and R2. Specifically, ei = 0 if si = 1, and ei = 1 if si = −1, for i = 1, 2. The constant m is a positive half-integer when s1s2 = −1, and a non-negative integer when s1s2 = 1.


Question: Which algorithms perform best in the robot pushing experiment, and why might GS be slower in this context?

Answer: The PG and EG algorithms perform best in the robot pushing experiment. GS is slightly slower due to increased exploration, which may be of less benefit for good-action identification compared to regular optimization.


Question: How does the Weyl scalar ψ2 for the rotating dark matter solution differ from the non-rotating case, and what does this imply about the space-time structure?

Answer: The Weyl scalar ψ2 for the rotating dark matter solution is given by ψ2 ≡ −Cabcdlamb ¯mcnd = −m/¯R ¯R ¯Riar cos θ, which shows the coupling of the rotational parameter a with the mass m of the dark matter. When the rotational parameter a = 0, the Weyl scalar ψ2 vanishes, leading to a conformally flat space-time as in the non-rotating dark matter solution. This implies that the rotating dark matter solution is type D in the Petrov classification, with a repeated principal null congruence ℓa that is geodesic, shear-free, expanding, and has non-zero twist.


Question: How does the theoretical value of the center-to-limb variation in red-shift compare to the observed value, and what does this discrepancy suggest?

Answer: The theoretical value of the center-to-limb variation in red-shift (Λtheo = 7.58 × 10−7 km/sec) is greater than that obtained in the previous study (Λ = 4.8 × 10−7 km/sec), but both are still too small compared with the observed value (Λobs = 0.3 km/sec). The ratio between the theoretical and observational values of Λ is 2.5 × 10−6, indicating that there is some parameter missing in the theoretical treatment. The order of magnitude of this ratio is the same as that of the square value of the fine structure constant (α = 1/137), suggesting that the spin-torsion interaction might be the missing parameter.


Question: How do the adversarial lighting and geometry perturbations generalize to different models and viewing directions?

Answer: Adversarial lighting perturbations generalize well across different models, with success rates ranging from 43.5% to 81.2% on AlexNet, VGG, SqueezeNet, and DenseNet. However, adversarial geometry perturbations are less generalizable, with success rates ranging from 40.1% to 71.1%. When evaluated on unseen viewing directions, adversarial lighting perturbations are more generalizable, achieving success rates of 64.2% when computed using 5 views, while adversarial geometry perturbations have a much lower success rate of 3.6%.