Question: What is the structure of the top level in the hierarchical classifier, and how do the accuracies compare between the whole speech signal and the glottal waveform?

Answer: The top level of the hierarchical classifier contains 3 emotional states in the left branch (moderate joy, intense joy, and intense anger) and 4 states in the right branch (neutral, moderate anger, moderate sadness, and intense sadness). For the whole speech signal, the accuracy is 83.59%, and for the glottal waveform, it is 80.38%, with a difference of 3.21%.


Question: What specific performance improvements does the SOHMT device show over the conventional device?

Answer: The SOHMT device shows a 9.6% reduction in the FOM, 19.2% reduction in ON-resistance (averaged over 0 to 5 V of gate overdrive voltage), 20.2% improvement in peak transconductance, 32.2% improvement in breakdown voltage, and 25% reduction in switching delays.


Question: What are the specific forms of the polynomials Bk(m) and bk(m) for different values of (k1, ..., kn)?

Answer: For (k1, ..., kn) = (1, ..., 1), Bk(m) = mn and bk(m) = mn. For (k1) = (k), Bk(m) = (k+m-1 choose k) and bk(m) = (m choose k). For (k1, ..., kn) = (1, ..., 1, 2), Bk(m) = Σi=1 to m in and bk(m) = Σi=1 to m-1 in. For (k1, ..., kn) = (2, ..., 2), Bk(m) = S(n + m, m) and bk(m) = s(m, m - n). For (k1, ..., k2n) = (1, 2, ..., 1, 2), Bk(m) = T(2n + 2m, 2m) and bk(m) = t(2m, 2m - 2n).


Question: How does the direction of self-propulsion affect the stability and yield of self-assembly in the case of active triangular colloidal particles?

Answer: By choosing the direction of self-propulsion to be perpendicular to the repulsive face of the colloid, one can improve the stability of the final hexagonal aggregate and bias the formation of the target structure early on in the self-assembly process. This choice enhances the overall rate and yield of self-assembly, particularly within a narrow range of binding energy where the maximum yield observed is ∼90%.


Question: Quais foram os dois principais problemas que surgiram durante a formalização das ondas gravitacionais e como eles afetaram a estabilização da teoria?

Answer: Os dois principais problemas foram: 1. Como essas ondas se propagam?, visando não apenas a ideia matemática (como equações de ondas), mas uma forma conceitual; 2. Qual a fonte dessa irradiação gravitacional? Como ela funciona ou deveria funcionar? O segundo problema foi um dos mais complexos a ser examinado, devido à dificuldade observacional e experimental das ondas gravitacionais na época, o que levou a descrenças e descartes dessa teoria.


Question: What is the condition for a rational function f/g to be non-degenerate over Fq with respect to its Newton polyhedron Γ(f/g)?

Answer: The rational function f/g is non-degenerate over Fq with respect to Γ(f/g) if and only if the polynomial mapping (f, g) : Kn → K2 is non-degenerate over Fq with respect to Γ(f, g).


Question: What empirical method was developed to measure the value of γ in the relation dL = dA(1 + z)γ, and what was the result of this measurement?

Answer: The method Cosmic Redshift Inference was developed to measure the value of γ independently of any possible galaxy evolution. This method was applied to 1.2 million galaxies from SDSS DR15, obtaining the value γ ≃ 1.


Question: What is the main contribution of the proposed EFPN in terms of small object detection on the Tsinghua-Tencent 100K dataset?

Answer: The proposed EFPN introduces an extra high-resolution pyramid level specialized for small object detection, which improves small object accuracy by 3.4% and small object recall by 0.2% compared to Faster R-CNN with ResNeXt-101-FPN.


Question: How does the regularity of solutions for the Boltzmann transport equation change when the boundary is rectified, and what are the implications for the transport problem in rectified coordinates?

Answer: The regularity of solutions for the Boltzmann transport equation can be studied using the rectification of the boundary. When the boundary is rectified, the transport problem can be transformed using a diffeomorphism h, which maps the boundary to a simpler domain. This transformation helps in analyzing the regularity of solutions, particularly in the context of tangential (co-normal) regularity. However, full Sobolev regularity is not expected for characteristic problems, and even tangential regularity may not hold for problems with variable multiplicity. The regularity of solutions is strongly influenced by the properties of the escape time mapping t(x, ω) and the inﬂow boundary condition.


Question: What conditions ensure the regularity of the solution with respect to the parameter x in the context of the BSDE (36)?

Answer: Under the assumption in the previous theorem and the additional assumption (H) that g ∈ C3_p(Rd; Rm) and (x, y, z, v) → f(s, x, y, z, v) ∈ C3 with all derivatives bounded, the solution Σt,x = (Y t,x, Zt,x, V t,x) of the BSDE (36) has the regularity properties: (i) ∥Σt,x − Σt,x′∥p_Bp ⩽ Cp(1 + |x| + |x′|)q|x − x′|p; (ii) ∥∆i_hΣt,x − ∆i_h′Σt,x′∥p_Bp ⩽ Cp(1 + |x| + |x′| + |h| + |h′|)q(|x − x′|p + |h − h′|p).


Question: How does the linear matter-power spectrum produced by your Boltzmann solver compare to that of CLASS for cold dark matter (CDM)?

Answer: Our Boltzmann solver agrees with CLASS within the range of interest in k to a precision better than 1%, as shown in Fig. 6, where the relative difference between the two spectra is plotted.


Question: What are the main characteristics of the NEAT / µ-NEAT mission that relate to the formation flying configuration described in the query text?

Answer: The NEAT / µ-NEAT mission involves formation flying with two spacecraft (Telescope and Detector) at a distance of 40 m (NEAT) / 12 m (µ-NEAT). The mission is designed for high-precision astrometry to detect and characterize planets in the habitable zone, with up to 20 reorientation maneuvers per day and a telescope pointing accuracy of 3 arcsec. The detector positioning accuracy is 2 mm on all axes for NEAT and 1 cm for µ-NEAT.


Question: What are the main differences between the present study and the study mentioned in the query text regarding the gravitational red-shift?

Answer: The main differences are: (1) The Sun’s gravitational field is given by the Lense-Thirring solution of GR in free space. (2) The observer, on the Earth’s surface, is assumed to move in an elliptic trajectory about the Sun.


Question: How does the offline setting for representation learning differ from the online setting, and what are the practical advantages of using offline datasets?

Answer: The offline setting for representation learning focuses on learning from static offline datasets, which are separated from downstream task learning. This approach makes comparisons between different methods easier by avoiding confounding factors associated with exploration or nonstationary datasets. Additionally, the offline setting is more practical because static offline datasets are more common than cheap online access to an environment.


Question: What are the mean squared errors for each method in Case 1b (φ2(1) = −0.8, NT = 240)?

Answer: The mean squared errors for each method in Case 1b (φ2(1) = −0.8, NT = 240) are as follows: M1: 0.0105, M2: 0.0119, M3: 0.0157, M4: 0.0154, M5: 0.0406.


Question: How do the LL crossing points change with increasing displacement field D, and what additional crossing point emerges near ν=12?

Answer: At D=84 meV/nm, the LL crossing points first 'contract' vertically, occurring over a narrow range in n. For a larger displacement field, the crossing points 'expand' to cluster around two different n values. Additionally, an extra crossing point emerges near ν=12, as indicated by the black arrows.


Question: What does the candidate text reveal about the anticorrelation between proton bulk speed (Vp) and electron temperature (Te) observed by the Parker Solar Probe (PSP)?

Answer: Observations from PSP indicate that there is an anticorrelation between the proton bulk speed Vp and the electron temperature Te close to the Sun, which disappears as the wind expands, evolves, and mixes with different electron temperature gradients for different wind speeds. This anticorrelation is observed below 0.3 AU and is thought to be produced by non-thermal electron distributions in the corona, particularly in fast wind from polar coronal hole regions.


Question: How does the adversarial layer in adversarial-based deep transfer learning contribute to the transferability of features between the source and target domains?

Answer: The adversarial layer tries to discriminate the origin of the features extracted from the source and target domains. If the adversarial network achieves worse performance, it indicates a smaller difference between the two types of features and better transferability. The performance of the adversarial layer is considered to force the transfer network to discover more general features with higher transferability.


Question: What properties of the stable ∞-category KK are used to construct the spectrum-valued KK-theory functor as a cohomology theory?

Answer: The properties used include: 1. A functor C∗Algsep →KK which is the identity on objects. 2. An isomorphism of Z-graded groups π∗(mapKK(A, B)) ∼= KK∗(A, B), where KK∗ denotes Kasparov's bivariant K-theory groups. 3. The zero algebra 0 is a zero object in KK. 4. An equivalence of spectra mapKK(C, C) ≃KU, and more generally, mapKK(C, B) ≃KC∗(B). 5. An exact sequence of C∗-algebras in C∗Algsep gives rise to a push-out diagram in KK. 6. For a separable C∗-algebra B and a countable family of separable C∗-algebras (Ai)i∈I, mapKK(⊕i∈I Ai, B) ≃ ∏i∈I mapKK(Ai, B). 7. The upper left corner inclusion A →A ⊗K induces an equivalence in KK. 8. The inclusion A →C([0, 1], A) as constant functions induces an equivalence in KK.


Question: How does the proof demonstrate that σ2(θ′ζ) = θ′ζ using the properties of γ′ζ and γ′′ζ?

Answer: Using (9.16) and TrF2e/F2(t) = 1, we see that the sum of (tγ′ζ)2i from 0 to e-1 equals γ′ζ plus the sum of t2jβ2iζ from 1 to e-1 and the sum of t2j(δζ − s)(ηζ − b1)2i from 0 to e-1. By defining γ′′ζ = γ′ζ + the sum of t2jβ2iζ from 1 to e-1, and using c20 + c0 = b2e+10 and t = b0 + b2e0, we have σ2(γζ) − γζ ≡ c0 + the sum of (b2e0 (βζ + b0))2i from 0 to e-1 ≡ c2e0 + the sum of ((b0 + t)βζ)2i from 0 to e-1 mod pN′uζ. This leads to σ2(γ′ζ) − γ′ζ ≡ c2e0 + s(ηζ − b1) + the sum of (tβζ)2i from 0 to e-1 + the sum of b2i+2j0 from 0 to e-1 mod pN′uζ, which implies σ2(θ′ζ) = θ′ζ.


Question: What are the specific estimates for the integral terms that contribute to the proof of the inequality (42) for n = 5?

Answer:
For n = 5, the estimates for the integral terms contributing to the proof of the inequality (42) are given by:
∫_0^t (1 + t - τ)^(-5/8) (1 + τ)^(-3/4(p-1) + ε2(p-2)) dτ = O(t^(-5/8)) if 3/4(p - 1) - ε2(p - 2) > 1, O(t^(-5/8) log(e + t)) if 3/4(p - 1) - ε2(p - 2) = 1, and O(t^(-3/4(p-1) + ε2(p-2) + 3/8)) if 3/4(p - 1) - ε2(p - 2) < 1. Additionally, ∫_0^t (1 + t - τ)^(-3/2) (1 + τ)^(-3/4(p - 1/2) + ε2(p-1)) dτ = O(t^(-3/8)).


Question: What are the specific steps and results of the H2 ⊗ L2(P) reduction in the beam model example, and how does it compare to the intermediate model in terms of approximation quality?

Answer: In the beam model example, the H2 ⊗ L2(P) reduction was performed with nred = 20, reducing the intermediate model of order ns = 500 to a reduced model of order 20. The reduced model bHred(s, p) of Algorithm 3 matches the approximation quality of the reduced model bHF(s, p) of Algorithm 1, despite having a much smaller order. This is illustrated in Figure 9, which shows the frequency response comparison for two representative parameter values of p = 0.001 and p = 0.22. Additionally, for three levels of reduced orders (nred = 10, 20, and 30), the relative H2 error for the entire parameter domain was computed, and the results show that with nred = 20 and nred = 30, Algorithm 3 matches the approximation quality of Algorithm 1 over the entire domain.


Question: What is the isomorphism established between the cyclotomic quotients of the semi-rationalization algebra and the non-degenerate affine Hecke algebra?

Answer: There is an isomorphism of algebras ˜Lq(Λ) ∼= Hq(Λ)e(C).


Question: What does Theorem 2.16 state about the upper envelope of a regular system that is 2-crossing, 4-crossing, or k-crossing for k > 4?

Answer: Theorem 2.16 states that if a regular system S is 2-crossing, then S is upper convex or only 2 paths appear on the upper envelope of S. If S is 4-crossing, then S is upper convex or at most 3 paths appear on the upper envelope of S. If S is k-crossing for k > 4, then S is upper convex or at most 4 distinct paths appear on the upper envelope of S.


Question: How is the fermion propagator of the lowest Landau level (LLL) with a finite chemical potential µ factorized, and what are the expressions for its longitudinal and transverse components?

Answer: The fermion propagator of the LLL with a finite chemical potential µ can be factorized into the longitudinal and the transverse parts as SLLL(r, r′) = S∥(r∥, r′∥) S⊥(r⊥, r′⊥). The transverse part of the propagator is given by S⊥(r⊥, r′⊥) = ∫ dky / (2π) * (1 / (π^1/2 * lq)) * exp(-1/2 * lq^2 * ((x - lq^2 * ky)^2 + (x′ - lq^2 * ky)^2)) * exp(iky * (y - y′)), where lq = 1 / sqrt(eqB) is the Larmor radius. The longitudinal part of the propagator can be written as S∥(r∥, r′∥) = ∫ d^2k∥ / (2π)^2 * exp(-ik∥ · (r∥ - r′∥)) * ˜S∥(k∥; µ), with ˜S∥(k∥; µ) = i / (2ϵk) * [θ(k3 - kF) / (k0 - (ϵk - iε)) + θ(kF - k3)θ(k3) / (k0 - (ϵk + iε)) - θ(-k3) / (k0 - (ϵk + iε))] * (¯k0γ0 - k3γ3) * P0, where kµ∥ = (k0, 0, 0, k3) with k3 = kz, ϵk = |k3|, ϵ±k = ±ϵk - µ, kF = µ, and P0 = (1 + iγ1γ2)/2 is a spin projection operator. Each term in the curly brace corresponds to particle (k3 > kF), hole (0 < k3 < kF), and antiparticle (k3 < 0) contributions, respectively.


Question: How are the classical fields a(t) and e(t) quantized in the context of the Yang-Mills theory, and what are the resulting quantum fields?

Answer: The classical fields a(t) and e(t) in B are quantized as the symmetric operators in K (H ∗) using I. Segal quantization. The resulting quantum fields are given by ba(t) := (bz(t)+bz(t))/√2 and be(t) := −i(bz(t)−bz(t))/√2, understood as local quantum fields on Minkowski space.


Question: How are the contours for the variables chosen to ensure the smallness of SL⃗ε[k1,k2,k3 | (k1,k2]]?

Answer: The contours for the variables are chosen as follows: γτ1 is w0(σ1, d(u)) with K := (∆1)3T, ensuring the estimate |G(ζ1(σ1) | i −nk1, ∆k1,r∗(m, a))|−1 ⩽C3e−C4σ21+Ψ(u/∆1). γτ2 is w0(σ2, d(λ)) with K := (∆2)3T, ensuring |G(ζ2(σ2) | nk2 −nk3−1, ∆k3,k2(m, a))|−1 ⩽C3e−C4σ22+Ψ(λ/∆2). γτ3 is w0(σ3, d(−v −λ)) with K := (∆3)3T, ensuring |G(ζ3(σ3) | nk3−1 −j + 1, ∆s,k3(m, a)|−1 ⩽C3e−C4σ23+Ψ(−(v+λ)/∆3). The ordering τ2 < τ3 is maintained due to the term (ζ2 −ζ3)−1 in the integrand.


Question: What is the asymptotic behavior of the m largest eigenvalues of the sample covariance matrix Sn under the given assumptions?

Answer: Under Assumptions (i) to (v), the m-dimensional random vector (√n(λ1/ψ1 - 1), ..., √n(λm/ψm - 1))^T and Tr(Sn) are jointly asymptotically normal and independent. Marginally, Tr(Sn) − Tr(Σp) converges in distribution to N(0, 2yγ2 + y(ν4 − 3)γd,2), and (√n(λ1/ψ1 - 1), ..., √n(λm/ψm - 1))^T weakly converges to an m-dimensional Gaussian vector (M1, ..., Mm)^T, with each Mk = uT_k G(ψ(αk))uk.


Question: What visual representation is used to show the nuclei size distribution across different tissue types in the PanNuke dataset?

Answer: Fig. 11 shows the nuclei size distribution per class within every tissue type in PanNuke, measured by pixel count within a segmentation mask for a particular nuclei.


Question: What are the implications of low values of ∆EW in the NUHM2 model compared to mSUGRA/CMSSM?

Answer: In the NUHM2 model, much smaller values of ∆EW, as low as ∼10, can be obtained, whereas in mSUGRA/CMSSM, ∆EW is typically bounded from below by about 100. Requiring low ∆EW ≲30 places strong restrictions on the allowed sparticle mass spectra, leading to distinctive predictions for collider and dark matter searches.


Question: What is the compatibility of the Kalton-Peck map with the group of isometries of L2?

Answer: The Kalton-Peck map is not compatible with the action of the group of isometries of L2.


Question: What does the training environment for the motion prediction model look like, and how does it contribute to the goal evaluation process in HPP?

Answer: The training environment for the motion prediction model is filled with randomly placed obstacles. All agents are given the same random goal and move towards it using their own P2P policies. During goal evaluation, the high-level policy Π simulates trajectories and evaluates goals based on how close the agents end up to each other. A better goal is one that brings the agents closer together.


Question: How does the deep tensor neural network (DTNN) iteratively refine atom-wise features, and what is the role of the update function in this process?

Answer: The deep tensor neural network (DTNN) iteratively refines a set of atom-wise features {xi} based on interactions with neighboring atoms. The update function, which takes the form xl+1_i = Fl(xl_i + Σ_j Gl(xl_i, xl_j, gl(|ri − rj|)) fcut(|ri − rj|)), updates each feature with information from all neighboring atoms through the interaction function G. This function depends on the neighbor features xj and the interatomic distance |ri − rj|, which is usually expressed in the form of a radial basis vector g. After the update, an atom-wise transformation F can be applied to further modulate the features.


Question: What are the mass ranges and mixing conditions that allow the Higgs boson to be hidden in a model with two additional scalar singlets?

Answer: In a model with two additional scalar singlets, the Higgs boson can be hidden if all three physical scalars are light, with masses mi <∼160 GeV, and have roughly equal mixing angles, V0i ∼1/√3.


Question: What is the lower bound for the a-number of a branched Z/pZ-cover of curves, and how is it defined in terms of the ramiﬁcation invariants?

Answer:
The lower bound for the a-number of a branched Z/pZ-cover of curves is given by \( aY \geq L(D) := \max_{1 \leq j \leq p-1} \sum_{Q \in B} L_j(\{d_Q\}) \), where \( L_j(\{d_Q\}) = \sum_{i=j}^{p-1} \left( \left\lfloor rac{id}{p}
ight
floor - \left\lfloor rac{id - (1 - rac{1}{p})jd}{p}
ight
floor
ight) \).


Question: What is the relationship between the configuration LP and the huge N-fold IP, and how does it help in solving the huge N-fold IP?

Answer: The configuration LP (ConfLP) is a relaxation of the huge N-fold IP (HugeIP). It is used to solve the huge N-fold IP by first solving the configuration LP, which provides a solution that can be used to derive a solution to the huge N-fold IP. The main idea is to prove a powerful proximity theorem that shows how one can drastically reduce the size of the input instance given that one can solve the corresponding configuration LP. This approach allows for a fast algorithm to solve huge n-fold IP instances, especially when the objective functions are separable quadratic and convex.


Question: What are the specific applications of the FFBRST transformation in gauge theories mentioned in the candidate text?

Answer: The FFBRST transformation has been used to connect effective theories in covariant gauges to theories in noncovariant gauges, regularize divergent energy integrals in the Coulomb gauge by modifying the time-like propagator, and connect different solutions of the quantum master equation in the field/anti-field formulation. It has also been applied to higher form gauge theories, such as Abelian rank-2 antisymmetric tensor field theories, and to resolve the Gribov problem in non-Abelian gauge theories.


Question: What numerical results demonstrate the efficiency and accuracy of the CGMsFEM compared to GMsFEM?

Answer: The numerical results show that the accuracy of the CGMsFEM is always better than that of the GMsFEM for Tests A and B, even in more complex cases. The energy error of the whole system remains stable, hardly affected by the complexity of the system, demonstrating that the CGMsFEM is computationally efficient and accurate with wide applicability in many scenarios.


Question: How does the G-invariance of Φh(X) relate to the SU(2)-equivariance of Φh(Xπ∗H) as discussed in the context of the Runge–Kutta method?

Answer: The G-invariance of Φh(X) is established by the fact that Runge–Kutta methods are equivariant with respect to affine transformations, and since Ψg is linear, it follows that Ψ−1g ◦Φh(X) ◦Ψg = Φh(Ψ∗gX). Given that the vector field X is G-invariant, this implies Ψ−1g ◦Φh(X) ◦Ψg = Φh(X), confirming that Φh(X) is G-invariant. This property is crucial for the SU(2)-equivariance of Φh(Xπ∗H) as it ensures that the method preserves the structure under the action of the group.


Question: How does the mediator particle's lifetime affect the DM relic abundance in the secluded DM scenario with a heavy mediator?

Answer: In the secluded DM scenario with a heavy mediator, if the mediator lifetime is short enough, the seclusion mechanism works well even with md ∼ mm. The departure of the mediator number density from its thermoequilibrium value causes non-negligible effects in the evaluation of the DM relic density. In the md = mm case with sufficiently strong d-m interaction, the evolution of the DM density exhibits a two-step decreasing behavior, where the first step is a 'fake' freeze-out and the second step starts when the mediator decay becomes active, with the true freeze-out occurring only after the DM particle decouples from the mediator.


Question: What are the components of the error bound in Theorem 4.3 for the nonconforming method, and what do they represent?

Answer: The error bound in Theorem 4.3 for the nonconforming method is given by |u − uL_h|1,Th ≤ c(SK,L) {A + B + C}, where: c(SK,L) is a constant possibly depending on the stabilization SK,L; A is the best approximation of u in the explicit space S0,L_p(Ω, Th); B is a term addressing the nonconformity of the global space; and C is a term involving the approximation of the boundary terms.


Question: How are the spin-3 and spin-4 operators defined in terms of the currents Ja and Ka, and what is their primary field status under the stress energy tensor?

Answer: The spin-3 operators Q(z) and R(z) are defined as Q(z) ≡ dabc JaJbJc(z) and R(z) ≡ dabc KaKbKc(z), respectively. The spin-2 operators Qa(z) and Ra(z) are defined as Qa(z) ≡ dabc JbJc(z) and Ra(z) ≡ dabc KbKc(z), respectively. The operators Q(z) and Qa(z) are primary fields of spin-2 and spin-3 under the stress energy tensor defined in the second numerator SU(N)1 factor, while R(z) and Ra(z) are primary fields of spin-2 and spin-3 under the stress energy tensor defined in the first numerator SU(N)k factor.


Question: How does the use of an analytical approximation simplify the calibration process for the CKLS model compared to using numerical solutions?

Answer: Using an analytical approximation simplifies the computation of the objective function, reducing the calibration to a one-dimensional optimization problem which can be quickly solved using simple algorithms, as opposed to the computationally demanding numerical solutions of the PDE (23) required for each evaluation of the objective function without the approximation.


Question: How does the 60Fe abundance in MC2 evolve over time according to the simulation results shown in Fig. 2?

Answer: The 60Fe abundance in MC2 increases over time, with the average of 102 realizations showing a steady rise from 10^-8 to approximately 10^-3 solar masses over 20 Myr.


Question: How does the development of neuromorphic computing relate to the need for new hardware paradigms in AI, as discussed in the query text?

Answer: Neuromorphic computing takes inspiration from the brain to create energy efficient hardware for information processing, capable of highly sophisticated tasks. Building this new hardware necessitates reinventing electronics, with research in physics and material science being key to creating artificial nano-neurons and synapses, connecting them in large numbers, and organizing them in complex systems. This aligns with the query text's discussion on the need for fundamentally distinct, domain-specific hardware to scale up AI.


Question: What is the optimized surfactant composition for assembling s-SWNTs into raft arrays, and how does it balance depletion force and stability?

Answer: The optimized surfactant composition for assembling s-SWNTs into raft arrays is ~0.9 wt% SDS and ~0.1 wt% SC. This composition achieves a balance between depletion force and stability of the SWNT suspension.


Question: How is the Lq_w(Q0) norm of Tf(x) - mTf(t, Q0) estimated in the proof of the local version of (6.17)?

Answer: The Lq_w(Q0) norm of Tf(x) - mTf(t, Q0) is estimated by considering the norm of each summand separately. Specifically, |Tf(x) - mTf(t, Q0)| is bounded by 8 M♯_0,s,Q0(Tf)(x) + c ∑_{v,j} m♯_Tf(1 - (1 - t)/2n, cQv_j)1Qv_j(x). The first term is estimated using Theorem 4.1 or Theorem 4.3, which gives M♯_0,s,Q0(Tf)(x) ≤ c Mrf(x) = c M(|f|^r)(x)^{1/r}. This term can be further estimated in norm by ∥M(|f|^r)∥^{1/r}_{Lq/r_w} ≤ c ∥f∥_{Lp_v}. The second term is estimated using a geometric argument and duality, leading to the bound ∑_m λm ∑_{v,j} (1/|2mQv_j| ∫_{2mQv_j} |f(y)|^r dy)^{1/r} ∫_{Qv_j} w(x)^{1/q} h(x) dx.


Question: How does the model in Figure 2 differ in its approach to content encoding compared to the hierarchical poem model described in the query text?

Answer: In the model shown in Figure 2, two different Bi-LSTMs read the structural token sequence and previous sentences, while text generation is only modeled with one forward LSTM. The context vector 𝑐𝑖 provides partial information from input sequences that is used together with the aligned hidden state ℎ𝑖 for generating Chinese characters. The context vector 𝑐𝑖 in this model is calculated based on the concatenation of the two encoders’ hidden states, unlike the hierarchical poem model where the content vector of the next sentence is predicted by considering all previous sentences.


Question: What is the significance of the conjecture regarding the integrals I1 and I2 in the context of the Axisymmetric Bianchi IX model's non-integrability?

Answer: The conjecture states that, generically for λ and E, γ1 and γ2 are not simultaneously zero. By differential Galois theory, this means that either I1 or I2 do not belong to L1 = C(z, ξ1, ξ′1), i.e., these integrals are not expressed as algebraic functions of z, ξ1, and ξ′1. This implies that the Axisymmetric Bianchi IX cosmological model is not meromorphically integrable for generic values of (E, λ).


Question: What are the key differences between the proposed DB-Net and the existing VIN in terms of performance and training time?

Answer: Compared with VIN, the proposed DB-Net achieves better performance on Mars visual navigation and the training time is reduced by 45.8%.


Question: How is the data processed and managed in the GPU-based pipeline for the Medicina BEST-2 Array?

Answer: Data reception, interpretation, and buffering are performed on the CPU, which then forwards the data to the GPU. The GPU processes the data through several stages including power calculation, bandpass, and other processing tasks. Each beam is processed on a single GPU, with an associated CPU thread. The processed data is copied back to host memory where it is thresholded, clustered, and filtered using a separate lightweight thread per beam. Data buffers can be persisted to disk by sending signals to the Data Writer thread.


Question: What is the temperature dependence of the superconducting gap in CeIr3, and how is it approximated in the model used?

Answer: The temperature dependence of the gap can be approximated by δ (T/TC) = tanh[1.82 [1.018 (TC/T −1)]^0.51], and g(φ) gives the angular dependence of the gap function where φ is the polar angle for the anisotropy. The spatial dependence g(φ) is substituted by (a) 1 for an s-wave gap, and (b) |cos(2φ)| for a d-wave gap with line nodes.


Question: What is the Robinson’s constraint qualification (RCQ) and how is it used in the context of the cone constrained minimax problem (P)?

Answer: Robinson’s constraint qualification (RCQ) holds at a feasible point x∗ of the problem (P) if 0 ∈ int {G(x∗) + DG(x∗)[A − x∗] − K}, where DG(x∗) is the Fréchet derivative of G at x∗ and int C stands for the topological interior of a set C. RCQ is used to easily compute the contingent (Bouligand tangent) cone to the feasible set of the problem (P).


Question: What are the specific optical depth profiles and brightness temperatures for the positions G075.78 3 and G075.78 4 in the analyzed spectra?

Answer: For G075.78 3, the brightness temperature (TB) ranges from 0 to 20 K, and the optical depth (τ line) ranges from 0.0 to 0.3. For G075.78 4, the brightness temperature (TB) ranges from 0 to 20 K, and the optical depth (τ line) ranges from 0 to 20.


Question: How do the empirical risk (RS(h)) and divergence (Div) values compare for the 'ours' method and the 'blanchard' method on the MNIST dataset with σ² = 10⁻⁶ and learning rate lr = 10⁻⁶?

Answer: For the 'ours' method, the empirical risk (RS(h)) is 0.010 ± 0.000 and the divergence (Div) is 0.040. For the 'blanchard' method, the empirical risk (RS(h)) is 0.010 ± 0.000 and the divergence (Div) is 75.043 ± 11.586.


Question: What are the specific throughput gains of the proposed perpetual code over RLNC for the highest tested generation size of 2048?

Answer: For the highest tested generation size of 2048, the encoding throughput gain is approximately 1010% and the decoding throughput gain is approximately 809% over RLNC.


Question: What are the observed crack speeds in the supershear regime, and how do they compare to the theoretical predictions?

Answer: Trottet et al. (2022) observed that the crack speed \( c \) in the steady state is approximately 1.6 times the shear wave speed \( c_s \), which is close to the P-wave speed \( c_p \). Without considering the fracture process at the crack tip, the formula simplifies to \( c = rac{c_p}{\sqrt{2}} pprox 1.2 \cdot c_s \). The observed speeds are either of the order of, or greater, than the lower limit found in the model.


Question: How is the nebular color excess E(B - V)gas calculated from the Balmer decrement, and what does it represent in the context of dust attenuation?

Answer: The nebular color excess E(B - V)gas is calculated using the formula E(B - V)gas = 1.086τB / (kHβ - kHα), where τB is the Balmer optical depth, and kHα and kHβ are the kλ values at the wavelengths of Hα and Hβ, respectively. This represents the relative reddening of the ionized gas, which can be used to trace the stellar attenuation.


Question: What is the process for preparing the STM tip for bond-resolving imaging in the study of the synthesized molecules?

Answer: The STM tip was prepared ex situ by clipping a Pt/Ir wire (0.25 mm) and sharpened in situ by repeatedly indenting the tip few nanometers (1 to 4 nm) into the Au surface while applying bias voltages from 2V to 4V between tip and sample. In order to perform bond-resolving STM imaging, the tip apex was terminated with a CO molecule, directly picked up from the Au(111) surface, by positioning the sharp metal tip on top of it and applying a 500 ms bias pulse at -2 V.


Question: What does Figure 1 show regarding the impact of increased luminosity on the profile likelihoods and Bayesian probabilities for the WWH, ttH, and bbH couplings?

Answer: Figure 1 shows that the existence of correlations between the WWH, ttH, and bbH couplings is independent of the integrated luminosity. The relative weights of alternative solutions hardly depend on the luminosity, even when increasing from 30 fb−1 to 300 fb−1. The error on the Higgs rates does not decrease like 1/√S, and backgrounds are always included, making S + B the relevant number of events.


Question: What is the effect of varying the effective noise strength in the influencers on the synchronization of the whole network?

Answer: By changing the effective noise strength q in the influencers, synchronization of the whole network increases, reaches a maximum, and then decreases. Optimal synchronization of the whole network is achieved for an effective noise strength q ≈ 1 in the influencers.


Question: How does the ES-tree algorithm handle the update of node levels when an edge weight is increased, and what is the role of the global heap Q in this process?

Answer: When an edge weight is increased, the ES-tree algorithm inserts the nodes u and v into the global heap Q with keys ℓ(u) and ℓ(v), respectively. The UpdateLevels procedure then processes nodes in Q, updating their levels based on the minimum value of ℓ(z) + w(y, z) for each neighbor z. If the new level ℓ′(y) is greater than the current level ℓ(y), the level is updated, and the keys in the heaps N(x) for each neighbor x are also updated.


Question: What is the sequence of transformations used to generate the dipole black string background from the D1-D5 black string solution?

Answer: The dipole black string background is generated by applying a solution generating transformation known as TsT4, which consists of a T-duality on x5, a shift ψ → ψ + 2λ˜x5, and then a T-duality back on ˜x5, where ˜x5 is the coordinate T-dual to x5.


Question: What are the specific conditions and inequalities that demonstrate the log pair is not log canonical when Q ∈E4\(E3 ∪E5)?

Answer: If the point Q ∈E4\(E3 ∪E5), then the log pair K ˜X + λ ˜D + 2λa4E4 is not log canonical at the point Q and so is the pair K ˜X + λ ˜D + E4. By adjunction (E4, λ ˜D|E4) is not log canonical at Q and 4a4 −2a4 −4/3a4 ≥4a4 −2a3 −2a5 = ˜D · E4 ≥multQ(˜D|E4) = multQ(˜D · E4) > 4, and this implies that a4 > 6 which is false.


Question: What visual examples are provided to illustrate the performance of the model on the source and target domains, and how are the correct and incorrect classifications represented?

Answer: Figure 11 provides example images from the source SDSS data (top) and target DECaLS data (bottom). Each image contains the true label in the top left corner and the predicted label in the top right corner. The top two rows show the most often correctly classified examples, while the bottom two rows show the most often incorrectly classified examples for each domain.


Question: What is the relationship between the fixed points of the coordinate maps and the fixed points of the overall mapping in the context of countable IFS on R^n?

Answer: x is a fixed point for Fi if and only if xj is a fixed point for fij for j = 1, ..., n.


Question: How are the adapter components integrated into the encoder and decoder of the proposed model, and what is their role during pre-training?

Answer: The adapter components are integrated into the encoder and decoder of the proposed model by being placed after the Multi-head Self-Attention (MSA) and feedforward (MLP) layers. During pre-training, only the adapters of the current task are trained, while the rest of the model is frozen. Each adapter is composed of two Linear layers, with the first Linear layer preceded by a LayerNorm and followed by a ReLu activation. The role of these adapters is to efficiently adapt the model for a new script/language.


Question: How does the proof of Theorem 5 use Lemma 5 to bound the redundancy of optimal binary AIFV codes for pmax < 1/2?

Answer: In the case of pmax < 1/2, we note that |X| ≥ 3 and thus, q1 must be an internal node. It follows from Lemma 2 that q1 ≤ 2q2. Since q1 + q2 = 1, we get 1/2 ≤ q1 ≤ 2/3. By Lemma 5, we obtain rAIFV ≤ max(1/2 ≤ q1 ≤ 2/3) (q1^2 - 2q1 + 2 - h(q1)) = 1/4.


Question: How do the appearance probabilities of νe and ¯νe in the beams of νµ and ¯νµ vary with the coupling constant ǫ, and what is the impact of this variation on the experimental results?

Answer: The appearance probabilities of νe and ¯νe in the beams of νµ and ¯νµ vary with the coupling constant ǫ. For ǫ = 0.015, the contribution of sterile neutrinos has the character of smooth oscillations. With ǫ = 0.005, the relative outputs of νe and ¯νe increase by approximately two orders of magnitude (up to ∼10−4) in comparison with their values for ǫ = 0 (∼10−6).


Question: What are some examples of area-universal graphs mentioned in the related work, and how do they relate to the double stacking graphs discussed in the query text?

Answer: Examples of area-universal graphs mentioned in the related work include planar partial 3-trees (subgraphs of stacked triangulations or Apollonian networks), the octahedron graph, and plane 3-regular graphs. Additionally, the 1-subdivision of a plane graph and certain classes of plane quadrangulations with up to 13 vertices are also area-universal. These examples, like the double stacking graphs discussed in the query text, contribute to the broader understanding of area-universality in graph theory.


Question: How do the AIA observations and the magnetic topological structures in Figure 8 support the understanding of the eruptive process?

Answer: The 'v'-shaped structure in the 131 Å image, where the first phase of reconnection occurs, generally coincides with the null point identified in the reconstructed coronal NLFFFs. The main phase reconnection seems to occur below the null point, and the positions of the post-flare loops and flare ribbons correspond well with the bifurcated branches of the pre-eruption flux rope. The time-distance diagram of a slice extracted from the 131 Å images confirms the sequence of the eruption characteristics, showing that reconnection1 occurs at the null first, followed by the rise of the filament, and then reconnection2 occurs below the filament, after which the filament continues to rise and finally stops.


Question: How do the time intervals in frames A and B relate to the time interval in the reference frame F, and what is the significance of this relationship in the context of clock synchronization?

Answer: The time intervals in frames A and B are related to the time interval in the reference frame F by a factor k = k(u, c), where u is the speed of the frames and c is the speed of light. Specifically, the time at event 1 in A and the time at event 2 in B are both equal to k multiplied by the time at event p in F. This relationship ensures that good clocks in A and B, if synchronized at event zero, will record the same elapsed time at events 1 and 2, respectively, allowing for mutual confirmation of clock synchronization through the exchange of radio signals.


Question: What are the number densities of the most abundant atomic and molecular species in a brown dwarf atmosphere with Teff=1500 K, log g = 5.0, and solar abundances?

Answer: The most abundant atomic species include H, He, Li, C, N, O, Na, Mg, Al, Si, S, Cl, K, Ca, Ti, Cr, Mn, Fe, and Ni. The most abundant molecular species include H2, H2O, SiO, SiO2, SiS, MgH, MgOH, TiO, TiO2, AlOH, Al2O, Fe(OH)2, N2, NH3, CH4, and CO.


Question: What are the main drawbacks of using ODE/PDE models for complex biological systems, and how do these drawbacks affect their applicability?

Answer: The main drawbacks of using ODE/PDE models for complex biological systems include non-identifiable parameter sets due to incomplete and noisy data, experimental constraints that prevent measuring all needed parameter values, the need for parameter estimation leading to model overfitting, the lack of analytical solutions for detailed models requiring numerical methods, oscillating model behavior, long calculation runtimes, and the necessity to use integer arithmetic to avoid rounding errors. Additionally, differential equation models are sloppy, meaning they can be tuned to reality by adjusting one key parameter per stiff direction, independently of how reliably other parameters are estimated. Calibration of such models requires time series data, which are often not measured in Proteomics studies, and quantitative data in Proteomics are mostly given in relative terms, not absolute quantities. These drawbacks make it infeasible to apply such exhaustive differential equation-based techniques for modeling more complex biological systems like proteome-wide data sets on the molecular level.


Question: How does the on-current of the SDGTFET change with increasing Ge mole fraction, and what is the impact on the ITRS requirements?

Answer: The on-current of the SDGTFET increases by around 2 orders of magnitude for a Ge mole fraction of 0.5, and it meets the ITRS requirements. While the on-current increases with strain, the off-current also increases but remains much below the ITRS near-term guideline for low-standby power technology.


Question: How does the block coordinate descent method for Θ reduce cache misses and computational cost?

Answer: The block coordinate descent method for Θ reduces cache misses and computational cost by grouping computations to minimize cache misses. It divides Θ into p × kΘ blocks, where each block comprises a portion of a row of Θ. By pre-computing and storing (Sxx)i and ΣCr in memory, the method ensures that there are no cache misses while performing coordinate descent updates within each block. Additionally, it strategically selects partitions C1, ..., CkΘ to minimize the active set in off-diagonal blocks, further reducing cache misses. Row-wise sparsity in Θ is also exploited to skip computing elements in (Sxx)i if the corresponding row in Θ is all zeros, reducing the computational cost of each cache miss.


Question: How do the measurements of CP asymmetries in B0 → J/ψKS and B0 → φKS differ in the VdQ model compared to the SM?

Answer: In the SM, the difference between the asymmetries in B0 → J/ψKS and B0 → φKS is expected to be |sin(2βψK) − sin(2βφK)| ≤ sin(2χ) ≈ O(λ^2). In the VdQ model, these measurements are independent, with sin(2βψK) = sin(2˜β) and sin(2βφK) = sin(2˜β + 2˜χ). If the difference shows significant deviation from the SM prediction, it may reveal new physics effects.


Question: What factors contribute to the steeper than self-similar slope of the luminosity-temperature relation in the combined sample of clusters?

Answer: The steeper than self-similar slope for the combined sample is due to the combined effects of AGN feedback on the luminosity slope and non-thermal pressure support and temperature bias on the temperature slope, both of which lead to a steepening of the relation compared to the self-similar prediction.


Question: How does the conduction band minimum (E1M) of the Ge:P δ-layer compare to that of the Si:P δ-layer, and what factors contribute to this difference?

Answer: The CB minimum (E1M) of the Ge:P δ-layer is less shifted into the bulk band gap region compared to the Si:P δ-layer. This is due to the larger number of conduction states near the CB minimum in the Ge:P δ-layer, which decreases the confinement of any one state. Additionally, the larger Bohr radius of Ge results in a broader donor potential in the z direction, which also affects the CB minimum.


Question: How does the formation of a spin-gap in underdoped Ba(Fe1−xCox)2As2 affect the thermal conductivity κ(T) below Tc?

Answer: For underdoped Ba(Fe1−xCox)2As2, the magnetic excitation spectrum in the magnetically ordered state remains similar to that in the parent composition, characterized by a well-defined spin-gap of ∼10 meV. However, as the cobalt concentration increases, the spin-gap is either lost or strongly broadened, and the magnetic spectrum evolves to be similar to that in the paramagnetic phase. This explains the lack of an increase in κab at TSDW for x=0.043 and x=0.049, as well as the smooth κ(T) observed in the polycrystalline sample with x = 0.05.


Question: What boundary conditions are necessary to ensure the vanishing of the boundary term in the variation of the GYH action for nonzero α?

Answer: For nonzero α, the Robin condition (∇nϕ + ξK/α ϕ)|∂M = 0 is permissible. The non-vanishing Dirichlet condition can be implemented with the choice α = 1/2, and the Neumann boundary condition can be implemented with the limit α → ∞.


Question: What are the main topics associated with the subcategory 'Sociology of Education' (S5) based on user interactions?

Answer: The main topics associated with the subcategory 'Sociology of Education' (S5) are school, education, teaching, university, and Germany.


Question: How does the 3D arc diagram drawing method described in Theorem 4 improve the angular resolution of a 2D straight-line drawing of a graph?

Answer: Theorem 4 states that for a straight-line drawing of a graph, G, with an angular resolution of Ω(1/d), there is a 3D arc diagram drawing of G with the same vertex placements and an angular resolution of at least Ω(1/d1/2). This improvement is achieved by using a localized edge coloring and ensuring that all arcs project perpendicularly as straight lines onto the base plane.


Question: What is the significance of the subalgebra of highest-weight bivectors in Uq(sl2) ⊗Cq[SL2] and how is it generated?

Answer: The subalgebra of highest-weight bivectors in Uq(sl2) ⊗Cq[SL2] is locally-finite and is generated as an algebra by the set {v1, v2, v3, v4}. These bivectors play a very important role in the representation theory of D(Uq(sl2)).


Question: What are the limitations of the shape modification PEC technique compared to the dose modification PEC technique?

Answer: The shape modification PEC technique has a lower correction accuracy than the dose modification PEC technique and the minimum magnitude of the shape changes is limited by the resolution of e-beam lithography systems.


Question: How is the non-local but energy-independent potential defined in the HAL QCD method, and what is the role of the derivative expansion in this context?

Answer: In the HAL QCD method, the non-local but energy-independent potential is defined from the NBS wave function through the equation (Ek − H0)φk(r) = ∫ d³r′ U(r, r′)φk(r′), where Ek = k²/2m, H0 = −∇²/2m, and m = mN/2. The potential U(r, r′) is zero for r > R, where R is a length scale beyond which the NBS wave function satisfies the Helmholtz equation. The derivative expansion is used to define the potential uniquely by specifying how to extract it, with the lowest few orders for the NN with a given isospin channel written as V(r, ∇) = V0(r) + Vσ(r)(σ1 · σ2) + VT(r)S12 + VLS(r)L · S + O(∇²).


Question: How do the NICER X-ray observations and GW detections contribute to the constraints on the NS EoS, and what are the projected improvements from future observations?

Answer: NICER X-ray observations and GW detections provide constraints on the NS EoS by ruling out certain pressure and radius values. NICER measurements help rule out smaller pressures and radii, while GW detections provide upper bounds on the tidal deformability, translating into upper bounds on the NS pressure and radius. Future observations, such as additional BNS detections during upcoming LIGO-Virgo runs and hypothetical NICER mass-radius measurements, are expected to reduce the uncertainty in the NS radius by ∼30% in the next ∼5 years, with the fifth observing run potentially leading to O(1) km uncertainty in R1.4.


Question: How do the viable scenarios in the O-II orbifold model with δGS = −6 compare in terms of stop mass and Higgs mass, and what are the implications for LHC searches?

Answer: In the O-II orbifold model with δGS = −6, the stops are considerably lighter than in the previous O-I example, even for scenarios with the preferred relic density, making them more attractive for LHC searches (though still very challenging). The lighter stops also result in a lighter Higgs boson, though this does not exclude the scenarios. Both the LSP and NLSP are higgsino dominated (neutralino and chargino respectively) and separated by about a GeV.


Question: What are the key contributions of the paper in using public data to improve private query release?

Answer: The key contributions are: 1) Initiating the study of using public data to improve private query release in a realistic setting where the public data is related but not identical to the private data. 2) Presenting PMWPub, an extension of MWEM that incorporates public data. 3) Demonstrating that PMWPub is computationally efficient and practical for larger problem sizes. 4) Analyzing the theoretical privacy and accuracy guarantees of PMWPub. 5) Empirically evaluating PMWPub on the American Community Survey data to show strong performance even when public samples come from a different distribution.


Question: How does the choice of B = EG in the lifting-extension problem relate to the classification of equivariant maps X → Y up to equivariant homotopy?

Answer: For every free G-space Z, there is an equivariant map cZ : Z → EG, unique up to equivariant homotopy. If we set g = cX and ψ = cY in the generalized lifting-extension problem, it can be proved that [X, Y ]∅EG is in a bijective correspondence with equivariant maps X → Y up to equivariant homotopy. This is how we obtain Theorem 1.1.


Question: What are the specific simulation parameters that demonstrate the impact of host halo mass on the time MBHs spend off-center?

Answer: In the simulations, MBHs in a 10^10 M⊙ host halo spend on average 3.8 Gyr beyond 15 pc after the heating phase, compared to 1.9-2.5 Gyr in a 10^9 M⊙ host halo. This indicates that MBHs are off-center for a longer time in higher mass galaxies.


Question: How does the use of multiple physical layer attributes, such as CFO, CIR, and RSSI, enhance the authentication performance in the presence of an eavesdropper like Eve?

Answer: The use of multiple physical layer attributes, such as CFO, CIR, and RSSI, enhances the authentication performance because it is more difficult for Eve to simultaneously infer multiple attributes of a large search-space from a received signal. This combination provides a high grade of uncertainty for the adversaries and simultaneously improves multi-dimensional protection for the legitimate users.


Question: What are the research questions addressed in the study related to the FinSim challenges?

Answer: The research questions addressed in the study are: RQ1: How have the datasets and solution architectures of the FinSim challenges evolved over the years? RQ2: How to develop a system for ranking a set of hypernyms for a given financial term? RQ3: Does using domain specific embeddings improve model performance? RQ4: What is the impact of augmenting/adding data from other sources?


Question: How do ARCO-SG and ARCO-SAG perform on the MMWHS and LA datasets with different label ratios?

Answer: ARCO-SG and ARCO-SAG consistently produce more accurate predictions on anatomical regions and boundaries compared to all other SSL methods on the MMWHS dataset with 1%, 5%, and 10% label ratios. Similarly, on the LA dataset, they consistently yield more accurate and sharper boundaries with the same label ratios.


Question: How is the estimate for |bX2(v1 + c)| derived in the most difficult case, and what is the significance of the decay factor t in this estimate?

Answer: The estimate for |bX2(v1 + c)| is derived using (9.7) to replace bX(X ˚T) in (9.15). This leads to |bX2(v1 + c)| ⩽ |bX( bT 2)|t=δ / t + (1/t) ∫_δ^t |bX2(v1 + c)| dτ + G(t), with |G(t)| ≲ t|bX(˚y)| + ˚Mtε². The decay factor t on the right-hand side is significant because it provides extra decay due to the decay of y, which helps in closing the bootstrap assumption (B∞).


Question: What are the specific results of using CMA-ES state information in performance prediction compared to using ELA-features alone?

Answer: The average difference between true and predicted solution quality decreases from 14.4 to 12.1 when adding the state variables as features, where the average error is taken over all 24 benchmark problems from the BBOB suite of the COCO platform and over all performed CMA-ES runs.


Question: What were the findings regarding the effect of haptic control strategies on mental engagement, particularly for the non-dominant hand?

Answer: Among the evaluated strategies, EA (error-augmentation) is shown to be more mentally engaging when compared with the ER (error-reduction) under the non-dominant hand.


Question: What are the two important cases regarding the domain of σ in the context of characterization logics?

Answer: The two important cases regarding the domain of σ are: 1) dom(σ) = 2L (called full logics) and 2) dom(σ) = {T ∈ 2L | T is finite} (finite-theory logics), which is the restriction of L to finite knowledge bases.


Question: Which planets from the sample have known orbits that were removed to improve the radial-velocity data in Table 1?

Answer: HD 16417 b, HD 4308 b, HD 115617 b, HD 115617 c, and HD 115617 d.


Question: How does the generalized heat flux law (2.4) relate to the classical Fourier and Maxwell–Cattaneo laws?

Answer: The generalized heat flux law (2.4) q + τ aK ∗qt = −κ∇θ covers both the classical Fourier law and the Maxwell–Cattaneo law. The Fourier law (2.1) q = −κ∇θ is recovered by setting τ = 0 and K = δ0, while the Maxwell–Cattaneo law (2.2) q + τqt = −κ∇θ is obtained by setting K = δ0 and a = 1 in (2.4). The time-fractional law (2.3) (1 + τ αDα_t)q = −κ∇θ is also covered by choosing K as the fractional kernel and a = α.


Question: What are the conditions for the existence of a K-invariant hyperplane H through 0 with unit normal vector a, given that K is a simplicial cone and its dual K* is of form (20) with vectors u1, ..., um satisfying (21)?

Answer: The existence of a K-invariant hyperplane H through 0 with unit normal vector a is equivalent to one of the following situations: (i) The vector a belongs to cone{ep, -eq} ∩ cone{up, -uq}, for some p, q ∈ {1, ..., m}, p ≠ q. (ii) The inequality ⟨ep, ei⟩ ≤ 0 holds for some p ∈ {1, ..., m} and any i ∈ {1, ..., m} with i ≠ p, and a = ±ep/∥ep∥. (iii) The inequality ⟨up, uj⟩ ≤ 0 holds for some p ∈ {1, ..., m} and any j ∈ {1, ..., m} with j ≠ p, and a = ±up/∥up∥.


Question: What is the significance of the Quantum Speed Limit (QSL) in the context of time-dependent Hamiltonians, and how does it relate to the optimal evolution time in a driven two-level system?

Answer: The Quantum Speed Limit (QSL) establishes a fundamental bound for the minimum evolution time between two states of a quantum system, which is crucial in quantum control to perform operations rapidly and avoid environmental effects. For time-dependent Hamiltonians, the QSL bound is more complex and has been less explored. In the context of a driven two-level system, the QSL bounds are compared with the optimal evolution times to assess the time performance of control protocols. However, in some cases, no meaningful bound can be obtained even with precise knowledge of the physical evolution.


Question: What are the conditions under which a set is minimal for [∆1^1 \ pot(Π0^1), ≤r_B] according to Theorem 9?

Answer: The set ACS is minimal for [∆1^1 \ pot(Π0^1), ≤r_B] if for all p ∈ N, there exists k ∈ N such that for all q ∈ N, there exists c ∈ N ∩ [q, q+k] such that c + (S ∩ [0, p]) = S ∩ (c + [0, p]).


Question: What are the specific conditions and corresponding schemes that determine the randomness size h(N, M′) for the secure distributed linearly separable computation problem?

Answer: The randomness size h(N, M′) is determined by the following conditions and schemes: 1) h(N, 1) = N using the fractional repetition assignment. 2) h(N, M′) = h(N/GCD(N, M′), M′/GCD(N, M′)) using Scheme 1. 3) For N > 2M′, h(N, M′) = h(N −⌊N/M′ −1⌋M′, M′) + ⌊N/M′ −1⌋ using Scheme 2. 4) For 1.5M′ ≤N < 2M′ and M′ even, h(N, M′) = h(N −M′, M′/2) + 1 using Scheme 3. 5) For 1.5M′ ≤N < 2M′ and M′ odd, h(N, M′) = (N −3M′ −5)/2 using Scheme 4. 6) For M′ < N < 1.5M′, h(N, M′) = h(M′, 2M′ −N) using Scheme 5.


Question: How does the diagonalization of scalar fields in the (p+1)-dimensional SYM theory relate to the positions of Dp-branes and the open-string excitations between them?

Answer: When all scalar fields can be (almost) simultaneously diagonalized, the diagonal elements of the matrices are interpreted as the positions of Dp-branes in the transverse directions, and the off-diagonal elements are interpreted as the amount of the open-string excitations connecting the i-th and j-th Dp-branes. If the matrices are not (almost) simultaneously diagonalizable but can be taken block diagonal, each block is a bound state of D-branes and strings.


Question: What specific parameters were chosen to reproduce the angiogenic switch and the growth of the tumor to 1 cm³ in the simulation without therapy?

Answer: The parameters Chypo→V, CA,V, CV→A, and A0 were chosen to reproduce the angiogenic switch and the growth of the tumor to 1 cm³. The angiogenic cascade starts around 1-2 mm³, which translates to the range 10^6 - 2 * 10^6 cells. These parameters were selected to ensure that the simulation matches these specific events.


Question: What are the specific empirical relations between quark and lepton mixing angles that suggest a common origin in the context of the Quark-Lepton Complementarity (QLC)?

Answer: The empirical relations are θl12 + θus ∼ π/4 and θl23 + θcb ∼ π/4, where θl12 and θl23 are the solar and atmospheric mixing angles for leptons, and θus and θcb are the Cabibbo angle and the angle between the second and third generations for quarks, respectively.


Question: How does the bound on the integral involving RQ(t, x, y) and the Fourier transform of ψ relate to the final estimate in the proof of Lemma 11?

Answer: The bound on the integral involving RQ(t, x, y) and the Fourier transform of ψ is used to show that the expression is bounded by cλ^(d-2Q-2) times the integral of (Fdψ(λXu))^2 u^(2Q+1) du, which ultimately leads to the final estimate in Lemma 11, where the expression is bounded by C times the sum of (Kr ∑ ar,j)^2 for all r.


Question: How does the mean field rate equation (MFRE) approach compare to the kinetic Monte Carlo (KMC) simulations in predicting the island size distribution (ISD) for different coverages and Γ values?

Answer: The MFRE-ISD (solid lines) is in excellent agreement with the corresponding KMC-ISD (symbols) obtained from the KMC simulations for all coverages in the precoalescence regime. A variation of Γ does not affect the quality of agreement, as shown in Fig. 3b, where the scaled ISDs ns¯s²/Θ versus s/¯s for a fixed coverage Θ = 0.1 and four different Γ values are plotted. The scaled ISDs tend to approach a limiting master curve when Γ →∞. The agreement is seen over about four orders of magnitude of ns in Fig. 3a, demonstrating that the approximations involved in the MFRE are appropriate to predict the ISD with high accuracy for the hit and stick aggregation considered here.


Question: What are the detection efficiencies for the 334.0 keV and 406.5 keV γ-quanta, and how were they calculated?

Answer: The detection efficiencies for the 334.0 keV and 406.5 keV γ-quanta are 2.24% and 2.42%, respectively, and they were calculated using the EGSnrc simulation package.


Question: How does the breaking of U(1)Le−Lµ−Lτ symmetry contribute to the generation of neutrino mass and the dark matter candidate in the proposed models?

Answer: The breaking of U(1)Le−Lµ−Lτ symmetry is responsible for neutrino mass via the type-I mechanism. The real part of the U(1)Le−Lµ−Lτ breaking scalar field (called ϕ) plays the role of the freeze-in dark matter candidate. For ϕ to qualify as dark matter, its lifetime must be larger than the age of the Universe, implying that the relic of ϕ is generated through the freeze-in mechanism and its mass must be less than an MeV.


Question: How does the random initialization of node and relation embeddings in InGram affect the double equivariance of the generated representations?

Answer: Random initial features break the double equivariance of the generated representations, making them sensitive to the permutation of node and relation type indices. However, since the initial node V (0) and relation embeddings R(0) are randomly initialized, and by design of InGram architecture, the triplet score function ZInGram is distributionally double equivariant, meaning p(ϕ ◦τ ◦ ZInGram|ϕ ◦τ ◦A(∗)) = p(ZInGram|A(∗)).


Question: What is the relationship between the type and strength of a maximum point and the asymptotic behavior of the function f around that point?

Answer: For each global maximum point µp of the function f, there exists a positive integer kp (the type) and a negative real number λp (the strength) such that around µp, the function f can be written as f(x) = f(µp) + λp (x − µp)^(2kp) / (2kp)! + o((x − µp)^(2kp)).


Question: How does the choice of the meter QJP distribution affect the target QJP distribution in the context of conditioned measurements?

Answer: A different choice of the meter QJP distribution results in a different choice of the target QJP distribution. Specifically, the function ˜ωψ,α1(x, y) and its Fourier transform ˜W ψ,α1(x, p) are introduced for different values of the parameter α, where α = α1 + iα2. For α1 = 0, ˜ωψ,0 reduces to the original function ˜ωψ, and ˜W ψ,0 is the (yet-to-be-normalised) WV distribution.


Question: What are the key differences between the work presented in this paper and previous approaches in creating knowledge graphs from scientific publications?

Answer: The work presented here is the first to create a knowledge graph particularly tailored for analyses of software in science based on recognition performance that outperforms previous approaches and the first to include entity disambiguation for software mentions.


Question: How does the symmetry of the symplectic Ricci curvature tensor field σ relate to the operators Σ and Θ in the context of symplectic spinor valued exterior differential forms?

Answer: The symplectic Ricci curvature tensor field σ is symmetric, as stated in Lemma 6. This symmetry is crucial for the operators Σ and Θ, which are used to describe the action of the symplectic Ricci curvature tensor field on symplectic spinor valued exterior differential forms.


Question: How does the paper propose to handle the two scenarios of RIS-aided edge computation offloading, and what are the key differences in their deep learning architectures?

Answer: For scenario (a) without LoS direct links between UEs and AP, a CSI-based deep learning architecture is used, as shown in Fig. 3, to obtain the solutions of {φ, a, W}. The real and imaginary parts of the channel coefficients {hd,n}, {hr,n}, and HAP constitute the input feature of the DNN-CSI, and the normalized angles of the RIS reflecting coefficients φ and the energy partition parameters a constitute the output vector. For scenario (b) with strong LoS direct links between UEs and AP, the CSI-based deep learning architecture is still applicable, but a location-only deep learning architecture is also considered. The key difference is that in scenario (a), the architecture relies on channel state information (CSI), while in scenario (b), it can leverage the strong LoS direct links for more robust performance.


Question: How are the gradient and Hessian operators of the potential Φ computed in the context of the iSVGD approach?

Answer: The gradient DΦ(u) acting in any direction ˜u is given by ⟨DΦ(u), ˜u⟩ = ∫_Ω ˜ueu∇w · ∇pdx, where the adjoint state p satisfies the adjoint equation −∇· (eu∇p) = −1/σ^2 ∑_{j=1}^{Nd} (1/2πδ^2) e^(-1/2δ^2 ∥x−xj∥^2)(ℓxj(w) −dj) in Ω, and p = 0 on ∂Ω. The Hessian acting in direction ˜u and ˆu reads ⟨⟨D2Φ(u), ˆu⟩, ˜u⟩ = ∫_Ω ˆu˜ueu∇w · ∇pdx + ∫_Ω ˜ueu∇w · ∇ˆpdx + ∫_Ω ˜ueu∇p · ∇ˆwdx, where the state ˆw satisfies the incremental forward equation −∇· (eu∇ˆw) = ∇· (ˆueu∇w) in Ω, and ˆw = 0 on ∂Ω, and the state ˆp satisfies the incremental adjoint equation −∇· (eu∇ˆp) = ∇· (ˆueu∇p) − (1/2πδ^2σ^2) ∑_{j=1}^{Nd} ˆwe^(-1/2δ^2 ∥x−xj∥^2) in Ω, and ˆp = 0 on ∂Ω.


Question: What are the differences in the TEM contrast observed for the As-initialized and Ga-initialized GaAs/ZnSe interfaces at g = 200 and g = 400 reflections?

Answer: The As-initialized GaAs/ZnSe interface exhibits a continuous bright contrast for g = 200 reflection, while the bright contrast is absent for the Ga-initialized GaAs/ZnSe interface at the same g = 200 reflection. Additionally, the bright contrast is absent for g = 400 reflection in both the MEE GaAs/ZnSe interface layers.


Question: How does the cost of setting up a lightpath and allocating a wavelength contribute to the total network configuration cost in the context of MPLS over OTN?

Answer: The cost of setting up a lightpath (cLP) and allocating a wavelength (cλ) are components of the total network configuration cost. Specifically, cLP = 2(cP _IP + cP _OXC) and cλ = 2(cP _OXC + cT R), where cP _IP is the cost of IP/optical interface cards, cP _OXC is the cost of OXC ports, and cT R is the cost of transponders. The total cost is minimized by the optimization function: Minimize: cT T · ∑n ∆n + cLP · ∑(i,j) [w(i,j) + s(i,j)] + cλ · ∑(m,n) [w(m,n) + s(m,n)].


Question: How does the transformation of a 3-SAT problem into a system of polynomial equations help in determining the existence of a solution using the Buchberger algorithm?

Answer: The transformation of a 3-SAT problem into a system of polynomial equations allows the use of the Buchberger algorithm to find a Gröbner basis for the ideal generated by the polynomials. If the Gröbner basis is 1, it indicates that the zero locus is empty, meaning no solution exists. Otherwise, the existence of a solution can be determined, and specific solutions can be found using the polynomial system.


Question: What specific constraints are used to integrate the Direct and Positional models in the Essence specification?

Answer: The channelling constraints for integrating the Direct and Positional models in Essence include: 1) forAll i : number, forAll j : repetition, seq(pos(tuple (i,j))) = i; 2) forAll i : int (1..k*n), exists j : repetition, pos(tuple (seq(i),j)) = i; 3) forAll i : number, forAll j : int (2..k), pos(tuple (i,j-1)) < pos(tuple (i,j)).


Question: What is the focus of the study in the candidate text regarding the Richtmyer-Meshkov instability?

Answer: The study focuses on the late time flow structure of Richtmyer-Meshkov instability, particularly in the Atwood number unity limit where one material is far more dense than the other. It derives an analytic expression for the mass-velocity relation in the spikes and a global theory for the late time flow structure, comparing these results with fluid dynamical simulation.


Question: How does the sequence (bn)n∈N represent the transmission of maturity between successive generations, and what is the significance of Lemma 3.1 in this context?

Answer: The sequence (bn)n∈N represents the transmission of maturity between two successive generations, n and n + 1, where b0 = b and bn+1 is defined as ∆−1(bn) if bn ∈ [0, Θ(1)), and g(1) if bn ∈ [Θ(1), g(1)]. Lemma 3.1 states that if 0 < b < Θ(1) := ∆(g(1)), then there exists N ∈ N such that bN < Θ(1) ≤ bN+1 ≤ g(1). This lemma is significant because it ensures that the sequence (bn)n∈N will eventually reach a maturity level within the range [Θ(1), g(1)], which is crucial for understanding the long-term behavior of the cell population.


Question: What additional information is stored in the full event tree when the FULL parameter is set to 1 in GLISSANDO?

Answer: When the FULL parameter is set to 1, the full event tree stores the x and y coordinates (X, Y) and the RDS (W) of the source, as well as the event number (KK). This information is stored in a structure defined as typedef struct {Float_t X,Y,W; UInt_t KK;} SOURCE; and written to the branch 'full_source' of the TTree 'full_event'. Other useful information, such as the impact parameter in the given event, is stored in the events tree.


Question: What is the method used to compute the fundamental groups of the complements of rational cuspidal projective plane curves classified by Flenner, Zaidenberg, Fenske, and Saito?

Answer: The Zariski-Van Kampen algorithm is used to compute the presentations of fundamental groups of the complements of these curves, and the Cremona transformations used in the construction of these curves are exploited in the process.


Question: How does the action of the dg Lie algebra g on the SLie∞-algebra h manifest in terms of the SLie∞-derivations?

Answer: The action of g on h is given by operations νk(x; y1, ..., yk) for k ≥ 0, such that for a given x ∈ g, the (curved, pre-) SLie∞-derivation associated to x is x· = {νk(x; −)}k≥0. These derivations form a Lie algebra, and the SLie∞-structure on h is encoded by a SLie∞-derivation D. The action of g on h is explicitly described by the relations [D, x·] = (dx)· and [x·, y·] = [x, y]·.


Question: What are the specific open questions regarding the structure of optimal defects and the lengths of alternating layers in the original unperturbed Bragg reflector?

Answer: The open questions are: (i) about the structure of optimal defects and the lengths of alternating layers with permittivities n2_1 and n2_2 in the original unperturbed Bragg reflector, (ii) about symmetry and uniqueness of optimizers, and (iii) about the existence of a global minimizer of the decay rate Dr(·; ·) without constraints imposed on the frequency.


Question: What evidence is provided to support the formation of dark vector solitons with fluctuating darkness in different polarizations?

Answer: The zoom-in oscilloscope traces of dark vector solitons at different positions show that the relative shallowness ratio of each polarization is unrelated, indicating that the darkness for each polarization is neither invariant nor 'quantized' but fluctuant along the propagation. This is attributed to the uncorrelated formation of individual dark solitons from weak perturbations of the CW background pulse.


Question: How were the interaction forces between the user's finger phalanges and cylindrical objects measured in the grasping tasks?

Answer: FSR sensors were inserted on the objects, specifically on a water bottle and a mug, to measure the interaction forces between the user's finger phalanges and the cylindrical objects during grasping.


Question: How does the expected free energy functional G(π, τ) relate to the variational posterior and the generative model in the context of active inference?

Answer: The expected free energy functional G(π, τ) is defined as G(π, τ) = EQ(oτ, sτ|π)[log Q(sτ|π) − log P(oτ, sτ|π)]. Using the factorisation of the generative model, it can be approximated as G(π, τ) = DKL(Q(sτ|π)∥P(sτ)) + EQ(sτ)[H(P(oτ|sτ))]. This means that minimizing expected free energy involves both minimizing the divergence between the variational posterior and the prior, and minimizing the entropy of the observations given the states.


Question: What is the asymptotic expansion for the derivative of the Airy function Ai′(u2/3ζ) and what are the first two coefficients in this expansion?

Answer: The asymptotic expansion for Ai′(u2/3ζ) is given by Ai′(u2/3ζ) ∼ −u1/6ζ1/4 / (2π1/2) exp(−uξ + ∑(−1)s ˜as / (susξs)), for |arg(ζ)| ≤ π − δ, where ˜a1 = ˜a2 = −7/72, and subsequent terms also satisfy (A.2).


Question: What are the conditions for a cosmological bounce in cubic gravity in the absence of hydrodynamic matter, and how does this relate to the potential V1(φ) in the Einstein frame?

Answer: In cubic gravity, a cosmological bounce in the absence of hydrodynamic matter requires that V(φ) = 0 at the time of the bounce. This condition can only be satisfied by the potential V1(φ) in the Einstein frame, as it is the only potential that has a zero for non-zero φ. When V1(φ) = 0, the scalar field φ can climb to positive values of V1(φ) during the bouncing period, as shown in Fig. 5.


Question: In which regions of the parameter space (µ, ν) are there no ghost-free AdS solutions, and what are the implications for the AdS vacua in these regions?

Answer: There is no ghost-free AdS in region (c). Moreover, the region (g) does not admit any AdS solution. In region (c), there are no ghost-free AdS solutions, and in region (g), there are no AdS solutions at all.


Question: How does the concept of Fermi acceleration relate to the dynamics of billiards with moving walls, and what is the difference in energy conservation between nonautonomous and autonomous systems in this context?

Answer: Fermi acceleration takes place in nonautonomous Hamiltonian systems, characterized by unbounded growth of energy. In the context of billiards, a nonautonomous system would involve billiard dynamics inside a table with moving walls, where several results have established Fermi acceleration. In contrast, the system discussed here is autonomous, and the billiard map conserves energy. However, the energy conserved by the billiard map is different from that of the geodesic flow, and it is the energy of the geodesic flow that drifts when the billiard map is considered as a perturbation of a time shift of the geodesic flow.


Question: What is the bounded coreﬂection of a saturated preﬁlter F, and how is it related to the set of bounded elements in F?

Answer: The bounded coreﬂection of a saturated preﬁlter F, denoted ̺X(F), is the set of bounded elements in F. It is the largest bounded saturated preﬁlter contained in F, and it is defined as ̺X(F) = {λ ∨ ǫX | λ ∈ F, ǫ > 0}.


Question: What specific abundance variations and correlations are observed in the RGB sample of M62, and how do they compare to other Galactic GCs?

Answer: In the RGB sample of M62, the Mg abundance is constant within the uncertainties, while O, Na, and Al show significant star-to-star variations. Oxygen and sodium are anti-correlated, and aluminum and sodium are positively correlated. The Na-O anti-correlation is more extended than in most Galactic GCs, with two discrete groups of stars: one with [O/Fe]∼+0.2/+0.3 dex and [Na/Fe]∼+0.1 dex, and a second group with [O/Fe]<0.0 dex and [Na/Fe] at ∼+0.5 dex. The sub-solar O-poor component (the so-called 'extreme population') is prominent in M62, while these stars are usually rare in other GCs.


Question: How does the polydifferential functor O act on the prop closure of the operad of degree d shifted Lie algebras, and what are the key features of the resulting operad O(Lied)?

Answer: The polydifferential functor O applied to the prop closure of the operad of degree d shifted Lie algebras results in the operad O(Lied). Elements of O(Lied) are obtained by joining the outputs of an element of the prop closure of Lied to a new white vertex and partitioning all input legs into groups, each joined to a new labelled white vertex. The black vertices are called internal vertices, and edges between internal vertices are called internal edges. For d even, an ordering of the internal edges and in-edges of the white out-vertex is assumed (up to a sign). For d odd, an ordering of the internal vertices and out-edges attached to each white in-vertex is assumed (up to a sign). The compositions in O(Lied) involve erasing white vertices, summing over all possible attachments of hanging edges, and summing over all attachments of remaining in-edges to white in-vertices.


Question: What are the specific parameters and techniques used to improve the signal-to-noise ratio (SNR) of the acquired THz signals in the data processing algorithms?

Answer: A digital FIR low-pass filter with a passband frequency of 2.5 THz and a stopband frequency of 3.5 THz (attenuation: -80 dB) was applied to the raw data. Consecutively acquired THz signals were filtered by a moving average filter with a window size of five signals, which further reduces the SNR.


Question: How does the economic aspect of crypto art relate to the historical attempts of conceptual artists to disrupt the traditional art market?

Answer: Crypto art has 'materialized' Duchamp’s and the conceptual artists’ attempt to get rid of material objects in order to expose and disrupt the traditional art market. In doing so, it has created a much more rapid market for digital artifacts, whose velocity makes it akin to financial trading in some respects, according to data scientists.


Question: What are the specific outcomes of the instabilities for the first solution at k = 0.2 when β = 0.05 and |T| = 0.4?

Answer: The instability is non-oscillatory, resulting in a slight decrease of the amplitude on the transmitting side (|A2| ≈0.39 at time 400).


Question: What is the reason for the discrepancy between the weak lensing mass estimate of the West cluster by C12 and the previously published estimates?

Answer: C12 misread the Abate et al. (2009) weak lensing estimate of the Subcluster as applying to the West cluster, which is incorrect. The only previously published weak lensing mass estimate of the West cluster (S08) is actually higher than the dynamical estimate. The difference is not statistically significant, but it suggests that a systematic analysis choice, such as the use of distance ratio weighting using source photometric redshifts, had a substantial influence. A 1–2σ shape-noise fluctuation at low redshift would affect the C12 result and the previously published convergence maps, but would be nearly invisible to the method of S08 and the weighted convergence map presented in this paper.


Question: How does the flux calibration of the IGRINS spectra compare to the IRTF/SpeX spectra for the G8 III star HD 27277?

Answer: The flux calibration of HD 27277 (G8 III) shows that the black lines represent the observed IGRINS spectra (R ∼45,000), the red lines show the IRTF/SpeX spectra (R ∼2,000), and the cyan line represents the resolution-convolved IGRINS spectrum (R ∼2,000). The flux ratio of the convolved IGRINS spectrum to the IRTF/SpeX spectrum is provided as the scale factor in the legend.


Question: What are the two main proposals of the paper and how do they improve contrastive learning?

Answer: The two main proposals of the paper are (1) semantically conditional negative sampling, a method that uses pretrained networks to define a negative sampling distribution, and (2) latent mixup, a simple strategy to form hard negative samples. Both proposals consistently outperform previous knowledge distillation methods and improve contrastive learned models in the standard supervised learning setup.


Question: What is the efficiency at maximum power extraction for the heat engine, and how does it relate to the Curzon-Ahlborn efficiency?

Answer: The efficiency at maximum power extraction, ηm, approaches the Curzon-Ahlborn efficiency, ηca = 1 − √(Ts/Tg), for small temperature differences, so ηm = ηc/2 + O(ηc^2). As the temperature difference increases, ηm becomes dependent on the configuration, particularly on the temperature of the conductor: the lower Ts, the closer ηm is to the Carnot efficiency. In the range of validity of the model, kTs, kTg ≫ ℏΓ, the efficiency at maximum power is maintained around ηc/2 far from the linear regime.


Question: What is the structure of the centralizer Zg(s) of s in g, and how does it relate to the root spaces gα for roots α in Ω?

Answer: The centralizer Zg(s) of s in g is given by Zg(s) = t ⊕ (⊕α∈Ω gα). This means that Zg(s) is the direct sum of the Cartan subalgebra t and the root spaces gα for roots α in Ω.


Question: What recent observations have shown significant circular polarization in FRBs, and what mechanisms might explain this phenomenon?

Answer: Recent observations of FRB 20201124A have shown significant circular polarization that in many cases exceeds what can be produced from Faraday conversion between linear and circular polarized modes. Mechanisms such as inverse Compton scattering or cyclotron resonance absorption may produce this anomalous circular polarization.


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

Answer: For IH, the final fluxes are Fνe ≈ cos2 θ12F 0 νX + sin2 θ12F 0 νe + freg(F 0 νe − F 0 νX) and F¯νe ≈ F 0 ¯νX. For NH, the final fluxes are Fνe ≈ F 0 νX and F¯νe ≈ cos2 θ12F 0 ¯νe + sin2 θ12F 0 ¯νX + ¯freg(F 0 ¯νX − F 0 ¯νe).


Question: What is the interpretation of the absence of thermal emission in the region expected to have a water plume, and how does it relate to the surface properties of Europa?

Answer: The absence of thermal emission in the region expected to have a water plume is interpreted as due to its temporal variation. The thermal emission distribution can be explained by modeling the surface heat conduction, taking into account the spatial inhomogeneity of the surface albedo, solar radiation flux, and surface thermal inertia. The thermal inertia distribution is likely associated with oceanic water that erupted and accumulated on the surface, with differences in crystal structure, porosity, and particle size of the fresh surface material newly created by the sedimentation of erupted water.


Question: How does the DRL algorithm demonstrate robustness in controlling the chaotic dynamics of the KS system, and what specific experiments were conducted to verify this?

Answer: The DRL algorithm demonstrates robustness through numerical experiments where the initial conditions of the control were varied 15 times, and the controller successfully drove the system towards the target solution in about 10 time-units. Additionally, a second set of experiments with 10 different initial conditions randomly chosen in the phase space showed that the controller could control the system in about 15 time-units in all but one case, where it took about 40 time-units. The phase-space view and time evolutions of the corresponding values further confirmed the robustness and convergence of the trajectories towards the target solution.


Question: What are the thermal stability and ionic conductivity values of the ternary polymer electrolytes incorporating ionic liquids in the PEO-LiTFSI system as reported by Vries et al?

Answer: Vries et al reported that the ternary polymer electrolytes incorporating ionic liquids in the PEO-LiTFSI system are thermally stable up to 200 °C, and the ionic conductivity obtained was 10−4 S cm−1 at room temperature and 10−3 S cm−1 at 60 °C.


Question: What additional figures show the clustering of signals for different proteins translocating through a nanopore at the same sampling rate and bandwidth as Figure S6?

Answer: Figures S7, S8, S9, and S10 show the clustering of signals for Hb, HSA, BSA, and con A, respectively, translocating through a nanopore at 40 Msps (BW = 10 MHz) under applied biases of 300 mV and 400 mV.


Question: How does the von Neumann entropy of a subsystem evolve over time, and what does this indicate about the thermalization process in the system?

Answer: The von Neumann entropy of a subsystem initially starts at zero and increases as the modes couple with one another. For initial energy near the ground-state energy, the von Neumann entropy quickly approaches the Gibbs entropy in a single-step relaxation. For initial states closer to the middle of the spectrum, a two-step relaxation process is observed, where the von Neumann entropy first relaxes to a quasi-stationary value different from the thermal entropy, forming prethermalization plateaus. The lifetime of these prethermalized states increases with the initial energy of the system, indicating the presence of metastable states that temporarily break the parity.


Question: What is the surface density distribution of dust and gas components in the generalized disk model based on the MMSN model?

Answer: The surface density distributions of dust (rock/ice) and gas components are given by Σd = 10ǫicefd (r/1AU)^(-3/2) gcm^(-2) and Σg = 2400fg (r/1AU)^(-3/2) gcm^(-2), where fd and fg are multiplicative factors to scale disk surface densities of dust and gas components, and ǫice is a step function that is 1 inside the ice line aice and 4.2 outside aice.


Question: What are the dark matter candidates in the model, and how are they stabilized?

Answer: The dark matter candidates in the model are the lightest among the CP-even and CP-odd neutral components of the SU(2)-doublet scalar h (scalar DM candidate) and the lightest among the right-handed Majorana neutrinos (fermionic DM candidate). These candidates are stabilized due to the preserved Z(1)2 symmetry.


Question: What is the running time of the algorithm for (ν(F) + k)-SAT, and what does it imply for variable-matched formulas F?

Answer: The algorithm for (ν(F) + k)-SAT has a running time of O((2e)2k+O(log2 k)(n + m)O(1)), and it implies fixed-parameter tractability of MaxSat parameterized by δ(F) for variable-matched formulas F.


Question: What is the significance of the monolayer AlOx coating on GaN in terms of surface properties and its potential applications?

Answer: The monolayer AlOx coating on GaN increases surface hydrophilicity and basicity, facilitating the formation of a stable self-assembled monolayer (SAM) with a coverage approaching the theoretical limit. This allows for additional tuning of interfacial energetics and the controlled introduction of chemical moieties at the terminal surface, which is crucial for applications in chemical sensing, (photo)catalysis, high electron mobility transistors, and advanced optoelectronic devices.


Question: What is the relationship between a formal CM point and its supersingular reduction, and how does this relate to the p-adic discriminant?

Answer: A CM point E is a formal CM point if and only if it has supersingular reduction. In this case, the p-adic discriminant of E is the unique p-adic discriminant containing the discriminant of E.


Question: What are the key observational tests that have been used to validate general relativity, and how do they relate to the Shapiro time delay?

Answer: The key observational tests that have validated general relativity include the deflection of light, the perihelion shift of Mercury, and the Shapiro time delay. The Shapiro time delay, first noticed in 1964, has been a significant test of Einstein's theory. A notable improvement in measuring this effect was reported in 2003 from Doppler tracking of the Cassini spacecraft, achieving a precision of γ − 1 = (2.1 ± 2.3) × 10−5, where γ is a parameter in the parameterized post-Newtonian (PPN) formulation of gravity. The bending and delay of photons by the curvature of spacetime are proportional to γ + 1, with γ being unity in general relativity and zero in Newtonian theory. The sensitivity in the Cassini experiment approaches the level at which deviations of 10−6 − 10−7 are expected in some cosmological models, making it crucial to investigate the Shapiro time delay with high accuracy.


Question: What are the specific conditions for the emergence of zero modes in the dynamical phase boundaries for N odd and N even?

Answer: For N even, the condition for the emergence of zero modes is cos(ϕ±) = 1/2 * (s + s−1)sech(Nr), specifying one angle ϕ+ = ϕ− in the interval [0, π], smaller than π/2. For N odd, the conditions are cos(ϕ±) = ±2 sech(Nr), resulting in two distinct angles ϕ± symmetric about each side of π/2. Both phase boundaries host zero modes for N odd, and just the left boundary for N even.


Question: What additional experimental and fabrication details support the realization of multiple optical frequency-conversions in few-layer GaSe?

Answer: The fabrication of the integrated GaSe-PC cavity, growth recipes of the single-crystal GaSe, characterizations of GaSe’s crystal structure, experimental arrangement, Raman spectroscopy of the GaSe flake integrated on the PC cavity, estimations of the SHG enhancement factors by the PC cavity, frequency-upconversion signals from the bare silicon PC cavity, and spatial mappings of the OFC signals are provided in the supporting information.


Question: How are the primary constraints and the time derivatives of the Abelian p-form fields derived in the canonical analysis?

Answer: The primary constraints are given by G(1)_{i1...ip-1} ≡ π_{0i1...ip-1} ≈ 0, and the time derivatives of A_{i1...ip} are expressed as ˙A_{i1...ip} = −p!π_{i1...ip} −(−)p ∂[i1A_{i2...ip}]0.


Question: What are the specific SDEs that describe the non-colliding square Bessel processes and their square root, and how do they differ from the Bessel2,α interacting Brownian motion?

Answer: The SDEs for the non-colliding square Bessel processes are given by dY N,i t = 2√Y N,i t dB i t + 4{−Y N,i t /8N + (α + 1)/2 + ∑_{j=1, j≠i}^N Y N,i t / (Y N,i t − Y N,j t)} dt, and for the infinite system, dY i t = 2√Y i t dB i t + 4{(α + 1)/2 + ∑_{j≠i}^∞ Y i t / (Y i t − Y j t)} dt. For their square root, the SDEs are dZ N,i t = dB i t + {−Z N,i t /4N + (α + 1)/2Z N,i t + ∑_{j=1, j≠i}^N 2Z N,i t / ((Z N,i t)^2 − (Z N,j t)^2)} dt, and for the infinite system, dZ i t = dB i t + {(α + 1)/2Z i t + ∑_{j≠i}^∞ 2Z N,i t / ((Z i t)^2 − (Z j t)^2)} dt. These differ from the Bessel2,α interacting Brownian motion in the coefficients aN(x, y) and a(x, y), which are 4x for the non-colliding square Bessel processes and 1 for their square root, compared to the Bessel2,α interacting Brownian motion where aN(x, y) = a(x, y) = 1.


Question: How does the KL divergence between two members of the same ϕ-exponential family change when the natural parameters are scaled by the Ω norm?

Answer: The KL divergence between two members of the same ϕ-exponential family, when the natural parameters are scaled by the Ω norm, is given by KLϕ(θΩ∥θ′Ω) = 1/Ω(θ) · Dϕ†(θ′∥θ).


Question: How does the proposed observation-level Frobenius loss function differ from traditional Frobenius loss functions in terms of its application and formulation?

Answer: The proposed observation-level Frobenius loss function operates over random vectors and is formulated as L(X; ψ, η0) := ∥XX⊤−ψ∥2_F,η0 = ∑_{j=1}^J ∑_{l=1}^J η(jl)_0 (X(j)X(l) −ψ(jl))^2, where X(j) is the jth element of a random vector X, ψ(jl) is the entry in the jth row and lth column of an arbitrary covariance matrix ψ, and η0 is a J × J matrix acting as a scaling factor. This formulation is more amenable to Dudoit and van der Laan (2005)'s estimator selection framework compared to traditional Frobenius loss functions that operate on the sample covariance matrix.


Question: What is the relationship between the associative and commutative localizations of a commutative ring spectrum A with respect to an element a ∈ πkA, and how does this relate to the Bott element β in the ring spectrum bu?

Answer: The morphism c : Aass[a−1] → Acom[a−1] between the associative and commutative localizations of a commutative ring spectrum A with respect to an element a ∈ πkA is an equivalence. This is because there exists a commutative diagram where the morphism of A-modules is an equivalence, and the categories of modules Ho(Aass[a−1] −Modr) and Ho(Acom[a−1] −Mod) are identified with the subcategory of Ho(A −Mod) consisting of A-modules for which multiplication by a is an equivalence. For the ring spectrum bu and the Bott element β ∈ π2bu, the localizations bu[β−1] in the associative and commutative senses are equivalent, and bu[β−1] is equivalent to the standard colimit colim(bu ∪β / bu ∪β / · · · ). The structure of bu[β−1] as a ring is the usual one, and there is an isomorphism bu[β−1] ∧S HC ≃ HC[u±1] in the homotopy category of HC-algebras.


Question: What experimental evidence supports the presence of plateau-like features in the echo signal, and how can these features be used to determine the coupling strength and switching rate of the fluctuator?

Answer: The echo signal measured in Ref. [45] shows a plateau-like feature at τ12 = 3.5 ns at the height ⟨m(e)+⟩ = 0.3, which yields v ≈ 143 MHz, and γ ≈ 27 MHz. Measuring the position and height of the first plateau can determine both the fluctuator coupling strength v and its switching rate γ.


Question: What are the specific scenarios where the CUSUM procedure fails to achieve an EDD linear in log(ARL) under misspecification of post-change parameters?

Answer: For the network of size 8, the post-change parameters are estimated to perceive a change in the influence from 6 →1, 1 →7, with a magnitude of 0.4. For the network of size 100, the misspecified post-change parameters select 200 edges randomly (independent of the true topology of the change) with a magnitude of 0.2. In these scenarios, the CUSUM procedure fails to achieve an EDD linear in log(ARL).


Question: What are the F1 scores of different classifiers when using the All pair generation algorithm and the KPOne algorithm?

Answer: The F1 scores for different classifiers are as follows: K Nearest Neighbours (99.39 for All, 99.39 for KPOne), Support Vector Machine (99.57 for All, 99.51 for KPOne), Neural Network (99.62 for All, 99.51 for KPOne), and Logistic Regression (99.39 for All, 99.39 for KPOne).


Question: How does the calibration of the thick filter group compare to the thinner filters in terms of the observed and expected intensities?

Answer: The black plus signs for the thick filter group (med-Al, med-Be, thick-Al, and thick-Be filters) deviate from the line depicted by orange bars, indicating that the thick filter group is not well calibrated. In contrast, the black plus signs for the thinner filters (Al-mesh, Al-poly, and C-poly) follow nicely the line where Iexp = Iobs, suggesting that the thicknesses of the thinner filters are well calibrated.


Question: What are the key findings from the studies of FRB host galaxies that have influenced progenitor models?

Answer: The key findings include the association of FRB 20121102 with a dwarf, rapidly star-forming galaxy and a persistent radio source, suggesting a young magnetar progenitor. Additionally, the discovery of a repeating FRB 20200120E associated with a globular cluster of M81 indicated that the progenitor was formed in a compact binary coalescence event. These findings have been actively incorporated into proposed progenitor models.


Question: How do the torsion classes W1, W2, W3, W4, and W5 appear in the SU(3) decomposition of the exterior derivatives of J and Ω?

Answer: The torsion classes appear in the SU(3) decomposition of the exterior derivatives of J and Ω as follows: dJ = 3/2 Im(W1* Ω) + W4 ∧ J + W3, and dΩ = W1 J ∧ J + W2 ∧ J + Ω ∧ W5*.


Question: How does the sub-periodicity of I or J affect the containment of Axo and Cy in the context of the joining trick?

Answer: If I is sub-periodic in K, there exists a subset Ao ⊂ Axo and an extreme invariant mean λ on G such that λ(Ao) = λ(Axo) and Ao ⊂ P = τ−1_o (Iot−1_o), where P is a proper subset invariant under the finite-index normal subgroup Go = τ−1(U). If J is sub-periodic, then GoCy ⊂ τ−1(UJ′π(x,y)−1) ̸= G, with Go = τ−1(U), for ν-generic y ∈ Y, and thus GoC cannot be ν-conull.


Question: How do β-arrestins contribute to the temporal and spatial control of GPCR-mediated signals, particularly in the context of FSHR?

Answer: β-arrestins act as multifunctional scaffolds and activators for a large number of signalling proteins, allowing GPCRs like FSHR to signal independently of heterotrimeric G proteins through ligand-induced β-arrestin 1 and 2 recruitment. They control the temporal encoding of GPCR-induced signalling pathways, such as the dual mechanism of ERK activation. G protein-mediated ERK activation is rapid and transient, while β-arrestin-mediated ERK activation is slower in onset (~5-10 min to reach maximum) but protracted (t1/2 > 1 hour). Additionally, β-arrestin-dependent signalling is tightly regulated by GRKs, with GRK5 and 6 promoting β-arrestin-dependent ERK activation and GRK2 and 3 inhibiting it.


Question: How does the Geo-I algorithm perturb a user's location to ensure privacy?

Answer: The Geo-I algorithm perturbs a user's location by drawing a random angle θ uniformly in [0, 2π) and a random value p uniformly in [0, 1). It then calculates the perturbation distance d using the formula d = −1/ϵ(W−1(p−1/e) + 1), and updates the location coordinates to x′ = l.x + cos(θ) · d and y′ = l.y + sin(θ) · d, returning the perturbed location l′ = (x′, y′).


Question: What are the calculated values of the Heisenberg exchange constants for the fcc and hcp stackings of Mn on Re(0001), and how do they compare to the energy difference between the 3Q and RW-AFM states?

Answer: For the fcc stacking, the Heisenberg exchange constants are J′1 = -22.4 meV, J′2 = -3.4 meV, J′3 = 0.88 meV, B1 = -1.56 meV, Y1 = -2.29 meV, and K1 = -0.43 meV, with an energy difference ∆E = -0.7 meV between the 3Q and RW-AFM states. For the hcp stacking, the constants are J′1 = -18.7 meV, J′2 = -4.2 meV, J′3 = -1.38 meV, B1 = -1.25 meV, Y1 = -2.49 meV, and K1 = -0.66 meV, with an energy difference ∆E = -0.4 meV.


Question: How does the SDSP model perform in predicting spatial features compared to the Gaussian process in the scene shown in Figure 9?

Answer: The SDSP model adapts to different regimes of ice and cloud, providing an almost constant prediction over the ice with very low prediction variance, and spatially-varying anisotropy over the region containing clouds. In contrast, the Gaussian process does not adapt as well to these different regimes.


Question: What does the long-term light curve of PSR J0007+7303 show regarding flux variations, and how were the data analyzed around the glitches?

Answer: The long-term light curve of PSR J0007+7303, shown in the top panel of Figure 9, is well fitted by a constant line with a reduced χ2 of 1.36, indicating no significant flux variation. Around the glitches, the Fermi-LAT data were split into four bins, and a likelihood analysis was performed using a power law with an exponential cutoff model. No changes in the integral flux above 100 MeV were detected in the four epochs.


Question: What are the specific conditions and parameters used to fabricate single-mode waveguides in the ion-exchange process?

Answer: The conditions and parameters used to fabricate single-mode waveguides include a fixed temperature of 400°C for both the first ion-exchange and the second diffusion (burial process) in a soda-lime glass with an index of 1.5104 for λo = 633nm. The diffusion time for the first diffusion is 30 minutes, and the time for the second (burial) diffusion process is 10 minutes. The effective index of the first planar ion-exchange is Neff = 1.5125, and the effective index of the planar waveguide after the second ion exchange is Neff = 1.5117.


Question: What are the specific forms of the infinitesimal generators G(z) that deliver boundary points of Vτ(F, Λ)?

Answer: The infinitesimal generators G(z) that deliver boundary points of Vτ(F, Λ) are of the form G(z) = Gζ,σ(z) = (τ −z)(1 −τz) / (σ + z) (σ −z) Reℓζ + i Im ℓζ + p0(z; F, Λ), for ζ ∈ int Z, σ ∈ ∂D, or G(z) = Gζ(z) = (τ −z)(1 −τz) / (i Im ℓζ + p0(z; F, Λ)), for ζ ∈ ∂Z \ {0}, or G(z) = G0(z) ≡ 0.