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Note that ‘loadings’, are mentioned, where today we would probably mention ‘expenses’. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> This became acute in the case of participating business, because premiums were often loaded well beyond mere considerations of risk, with a view to investment. Apart from that... | **A**: Safe-side premiums are larger than they would be based on a ‘realistic’ technical basis.
**B**: Most important was lack of realism.
**C**: If a valuation then takes a ‘realistic’ view, loadings in future premiums are identified and immediately capitalized as surplus, and if this ‘surplus’ is then disposed of i... | ACB | ACB | BAC | ACB | Selection 2 |
The basic process of PLKG consists of four main steps: channel probing, quantization, information reconciliation, and privacy amplification [5]. During the channel probing, Alice and Bob transmit the pilot signal to share the CSI in a channel coherence time. And then, in the quantization phase, the CSI extracted by Ali... | **A**: In other words, the randomness of the wireless channel determines the strength of the key’s entropy source.
**B**: The randomness of wireless channel is a critical internal condition for PLKG, which influences the key generation rate (KGR).
**C**: In recent years, the RIS has been used to improve the performan... | BAC | BAC | BAC | BAC | Selection 1 |
<|MaskedSetence|> The main goal was to obtain much finer control over the points where the obtained values fall into one of the cases presented above. Since a negative value can invalidate properties like local monotonicity or local invertibility, avoiding false negative evaluations is critical. <|MaskedSetence|> <|... | **A**: With this in mind, a second approach was implemented, using the C#C\# language for implementation.
The first step was to implement the BigNumber class, which corresponds to elements of the form a∗10ba*10^{b}, where aa and bb are of type Decimal.
**B**: The maximum value for the Decimal type in C#C\# is 7922... | CAB | ACB | CAB | CAB | Selection 1 |
We note a few things about this process. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We ran our C++ checking program over the first k=7100k=7100 batches and found that, for instance, S31S_{31} filtered out a majority of numbers, while most filters were successful 0 times. After each of these kk batches, w... | **A**: First, not all integers are filtered by filters in 𝒮\mathcal{S}.
**B**: Second, some filters are more efficient than others, in that they appear to filter more numbers.
**C**: We saved these numbers for later processing and found that none of them were prime, therefore they are accounted for by some earlier p... | BCA | ACB | ACB | ACB | Selection 3 |
<|MaskedSetence|> A bounded speedup of (X,T)(X,T) is a homeomorphism of the form (X,S)(X,S) where S(x)=Tp(x)(x)S(x)=T^{p(x)}(x) for some bounded function p:X→ℕp:X\to\mathbb{N}. We refer to the bounded function p(x)p(x) as the jump function.
In general, one can explore speedups where pp is not assumed to be boun... | **A**: Throughout this paper, we will always assume that pp is bounded and that (X,S)(X,S) is a minimal speedup.
**B**: Let (X,T)(X,T) be a minimal Cantor system.
**C**: Given a minimal bounded speedup (X,S)(X,S) of (X,T)(X,T), there is a constant c∈ℕc\in\mathbb{N}, called the orbit number, such that every TT-orbit i... | BAC | BAC | BAC | ACB | Selection 3 |
We also warm-start the solver using the Union of Stars construction, in line with best practices outlined in [13]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Rajakumar et al. [21, Theorem 8] prove that |𝐖r,r|≤(3n−2)(3n−1)/2|\mathbf{W}_{r,r}|\leq(3n-2)^{(3n-1)/2} suffices. However, in all our experimen... | **A**: This significantly improves the performance of Gurobi’s built-in heuristics.
**B**: In our formulation, the value of RR corresponds to the number of rows in the warm-start solution.
**C**: Although the Union of Double-Stars construction yields tighter upper bounds in theory, it performs worse in practice with ... | CBA | ABC | ABC | ABC | Selection 4 |
In this section, we outline the proof of the main theorem using interpolation theory. <|MaskedSetence|> In Section 4.2, we discuss its applications to compact manifolds as well as the difficulties encountered in the case of cusped manifolds. Finally, Section 4.3 introduces a new proof strategy based on Angenent’s maxi... | **A**: We also provide an overview of the structure of the remainder of the paper.
4.1.
**B**: We begin in Section 4.1 by presenting Simonett’s stability theorem for quasilinear parabolic equations.
**C**: Simonett’s theorem.
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Another recent direction of relevance, grouped together under the term spectral deconfounding, includes work by Ćevid et al. <|MaskedSetence|> <|MaskedSetence|> (2024); Wang and Shah (2025); Sun et al. (2024) among others, and considers estimation and inference in high-dimensional regression models with unmeasured ... | **A**: In contrast, we show that with multiple response variables, we can estimate the parameters in a collaborative way, improving the convergence rate compared to the univariate case.
.
**B**: (2020); Guo et al.
**C**: (2022); Fan et al.
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Another focus of the paper is the application of Ricci flow to finite-volume hyperbolic 33-manifolds. <|MaskedSetence|> Similarly to the compact case, it is natural to ask whether the Hamilton-Perelman results can be extended to noncompact manifolds. First, we need to determine if the existence theorems for Ricci flo... | **A**: We will use it to extend [36] and [37] to this setting.
**B**: Second, we are interested in the stability of the Ricci flow at its fixed point, specifically the hyperbolic metric.
Bessières-Besson-Maillot established the construction of Ricci flow with a specific version of surgery on cusped manifolds in [7], c... | ABC | ABC | BCA | ABC | Selection 1 |
Recall that in Boethius’ language,666His original description in the Proemium of the De Institutione Arithmetica [c. 500 A.D.] is well worth meditating on. A translation can be found in Michael Masi, Boethian Number Theory (New York: Rodopi, 1983), 72. the quadrivium consists of arithmetic (concerning number in its a... | **A**: They also extend into contemporary practice, and observing this continuity aids in clarifying the character of these arts.
**B**: Geometry also remains a pillar of the contemporary curriculum, and in modern research as topology.111111There is not scope here to elucidate this connection, but mathematicians will ... | ABC | ABC | CAB | ABC | Selection 1 |
Here, the Laplace equation has no known closed-form solution due to the irregular shape of the region. The approximation result from the WoS-NN method is as Fig. <|MaskedSetence|> To validate our results, we have experimented with the Finite Difference Method on the same environment as Fig. <|MaskedSetence|> The er... | **A**: 8(b).
**B**: 8(a).
**C**: The mean squared error over the 0.02∗0.020.02*0.02 intense grid on the valid region is MSE=0.000235MSE=0.000235.
.
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<|MaskedSetence|> <|MaskedSetence|> For example, the rings of diagonal and upper-triangular matrices are particular examples of incidence rings. In general, for any finite poset and any base ring, one can construct the incidence ring. There are interesting applications in combinatorics, see the classical book by [11]... | **A**: Such rings consist of matrices with special patterns of zeros.
**B**: Two finite posets are isomorphic iff their incidence algebras over a field are isomorphic [11, Theorem 7.2.2].
**C**: This paper studies generators of matrix incidence rings over finite posets.
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<|MaskedSetence|> <|MaskedSetence|> We wish to design a compensator that processes the measurements and delivers a bending moment and torque that dampens the bending and torsion oscillations. <|MaskedSetence|> After accomplishing this we convert the model of the beam into a model of a wing by adding the aerodynamic ... | **A**: We also assume that there are two point sensors, one measures the vertical velocity of the tip of the beam and other
measures the angular velocity of the tip of the beam.
**B**: We design this compensator using Linear Quadratic Gaussian (LQG) theory.
**C**:
We assume that the beam can bend and twist and that ... | CAB | CAB | CAB | CAB | Selection 1 |
<|MaskedSetence|> On the left is an attempt at creating a satisfying H4H_{4} as described in the proof; the only condition violated is Condition 5, that each of the monochromatic induced subgraphs (i.e. <|MaskedSetence|> However, any attempt to add edges to satisfy Condition 5 causes the graph to violate Condition 6.... | **A**: Each apex string intersects (apart from the jewel) three of the four strings that it needs to in order to achieve a string representation of N^4\hat{N}_{4}, and these three strings correspond to the path on three vertices in the monochromatic subgraph of the left graph, while the string not intersected by the ap... | CBA | CBA | CBA | CBA | Selection 1 |
Superconvergence. <|MaskedSetence|> <|MaskedSetence|> Before applying the filter, we observe the expected (k+1)(k+1)-th order of accuracy. <|MaskedSetence|> This observation matches the theories given in Sec. 3.
We also observe that the filtered solution is more accurate when the mesh resolution is sufficiently h... | **A**: 1, we present L2L^{2} error of the numerical solution before and after post-processing with the SIAC filter.
**B**: In Fig.
**C**: After applying the SIAC filter, we observe approximately 3.53.5 order of accuracy for Q1Q_{1} elements, and 2(k+2)2(k+2)-th order of accuracy for k=2,3k=2,3.
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For a precise definition, see [5, Definition 1.21]. <|MaskedSetence|> From condition (1) in the definition of an HHG, it follows that GG is quasiisometric to XX. <|MaskedSetence|> <|MaskedSetence|> Also, it is easy to show that the definition of HHG does not depend on the choice of the Cayley graph.
. | **A**: From here, one can give a hierarchical structure on the Cayley graph of GG, and the left action of GG on the Cayley graph also satisfies all the conditions for being an HHG.
**B**: Thus, to define an HHG, one can use a Cayley graph of GG itself.
**C**: In this case, we say that 𝔖\mathfrak{S} is an HHG structu... | CAB | CAB | CAB | CBA | Selection 2 |
While many studies have been done for wave operators but the complete understanding of sharp fixed-time estimates is still unknown in many settings. To the best of our knowledge, aside from classical Euclidean settings and compact Lie groups, the sharp fixed time estimates for wave operators on Hardy spaces in the ran... | **A**: In this article, we will study and prove the sharp fixed time estimates of the wave operators on Hardy spaces associated with the twisted Laplacian ℒ\mathcal{L} for the entire range 0<p<∞0<p<\infty.
The summary of our paper is as follows.
**B**: Theorem B is proved in Section 4.
**C**: Finally, in Section 5,... | CAB | ABC | ABC | ABC | Selection 2 |
(iii) Let ‖M‖A=‖N‖A.\|M\|_{A}=\|N\|_{A}. <|MaskedSetence|> <|MaskedSetence|> Since {xn}\{x_{n}\} and {yn}\{y_{n}\} are two AA-norming sequences of MM and N,N, then (xn0)\begin{pmatrix}x_{n}\\
0\end{pmatrix} and (0yn)\begin{pmatrix}0\\
y_{n}\end{pmatrix} are two 𝔸\mathbb{A}-norming sequences of T.T. Suppose S=(P000)∈... | **A**: Then ⟨Mxn,Pxn⟩A→λ(≠0)\langle Mx_{n},Px_{n}\rangle_{A}\to\lambda(\neq 0) and ⟨Nyn,Pyn⟩A→0.\langle Ny_{n},Py_{n}\rangle_{A}\to 0.
**B**: Let P∈𝔹A(ℍ)P\in\mathbb{B}_{A}(\mathbb{H}) be such that
⟨Mxn,Pxn⟩A→λ(≠0).\langle Mx_{n},Px_{n}\rangle_{A}\to\lambda(\neq 0).
**C**: Suppose {xn}\{x_{n}\} and {yn}\{y_{n}... | CBA | CBA | CBA | ABC | Selection 3 |
In many practical situations, the signal always sparse, therefore lasso method, as a useful shrinkage estimation, have been used to find the estimator of parameter β\beta (Han et al. (2015); Wang et al. (2011)).
Although lasso method, have ability of shrink the parameter estimates toward to zero under the natural spa... | **A**: (2023) proposed adaptive signal lasso, where appropriate weights are imposed to the penalty to promoting the estimation accuracy.
**B**: (2021) proposed signal lasso (SigL) method to solve the network reconstruction problem and they found SigL performed better than lasso and CS methods.
**C**: Furthermore Shi ... | BCA | BCA | BCA | ACB | Selection 3 |
The remainder of this paper is organized as follows. <|MaskedSetence|> <|MaskedSetence|> Based on these, a cost minimization problem of LEO satellite constellation is formulated and a satellite constellation optimization algorithm is proposed. Next, extensive simulation results are provided in Section IV to verify ... | **A**: In Section II, we introduce the system model of the LEO satellite IoT constellation, including the orbit elements, constellation model, coverage model and channel model.
**B**: Finally, Section V concludes the paper.
.
**C**: In Section III, we derive out the backhaul capacity expressions and introduce the co... | ACB | ACB | BCA | ACB | Selection 1 |
Graphs whose cyclomatic number is constant have a relatively simple structure. They are sparse (in the sense that they have a linear number of edges). <|MaskedSetence|> <|MaskedSetence|> In particular, in many cases, distance-related problems are computationally hard on graphs of bounded treewidth [KLP19, KLP22, KeKo... | **A**: Thus, they are studied extensively from the perspective of algorithms (for example for the metric dimension [EpsteinWeighted], the geodetic number [KeKo20] or other graph problems [CV85, DEHJLUU19, FKK01, KN12, UW13]).
**B**: They also have bounded treewidth (indeed the treewidth is at most the cyclomatic numbe... | BAC | BAC | BAC | ACB | Selection 1 |
(72)
where the dependence of κ\kappa on nn arises from its relation to the energy (see Equation (12)).
This condition causes the energy spectrum to become discrete but finite, with the maximum number of excited states determined by the fulfilment of the previous condition. The spectrum exhibits a linear behavior in k... | **A**: On the other hand, as in the previous case, when βν\beta_{\nu} approaches 11 (or −1-1) the energy levels tend to become degenerate at a value of −vdk-v_{d}k (see Figure 2b).
**B**: Regarding the energy current, it is also observed that the yy-component is nonzero due to the tilt of the Dirac cones (see Figure ... | CAB | CAB | BAC | CAB | Selection 4 |
<|MaskedSetence|> This was probably not realized in [6] since it was not stated in the description/manual of [10]. <|MaskedSetence|> Under additional assumptions (the distribution has a continuous density function), [6, Thm. 2.2, Cor. <|MaskedSetence|> | **A**:
(a version of the fixed point iteration for ψα\psi_{\alpha}) is also used by the codes in [10] to compute the empirical expectiles.
**B**: 2.3] provides quadratic convergence as expected from a Newton-type method..
**C**: On the other hand, [6] gives an interpretation of the fixed point iteration for ψα\psi_{... | ACB | ACB | ACB | CAB | Selection 1 |
The cargo types appear in Table 2 and are chosen so that they give interesting numerical results. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We also assume there is a zeroth type not appearing in the table with unit density and zero freight rate, representing ballast. It is conceivable that some ballast m... | **A**: For cargo types i=1,2,3,4i=1,2,3,4 the table gives the freight rates in monetary units per metric ton and density in tons per cubic meters.
**B**: For convenience, indexing is in decreasing density dj>dj+1d_{j}>d_{j+1}.
**C**: Water density ρ\rho is assumed to be one, thus cargo densities are relative to water... | ACB | ACB | ACB | CBA | Selection 2 |
<|MaskedSetence|> In section 2, we provide formal definitions for caterpillar trees and parking distributions and provide examples of them. <|MaskedSetence|> <|MaskedSetence|> In section 5 and 6, we prove our first main result that luck\mathrm{luck} (lucky drivers) and ω1\omega_{1} (first-space-loving drivers) have ... | **A**: In section 4, we introduce the first-return decomposition, which is our main tool for the main theorem.
**B**: In section 3, we give a simple bijection linking caterpillar tree parking distributions and 𝐮\mathbf{u}-parking distributions.
**C**:
The rest of the paper is organized as follows.
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By Remark 1 the event {Rnm=n+m}\{R_{nm}=n+m\} has positive probability m(n+m)(n+m−1)\frac{m}{(n+m)(n+m-1)} and therefore cannot be ignored.
The paper is structured as follows: In section 2 we motivate the upper TnmT_{nm}-test (called VV-test) and determine the exact null-distribution of TnmT_{nm} in case that mm... | **A**: For the corresponding transformation, we also determine the limit distribution..
**B**: Moreover, we report on the results of a small Monte Carlo simulation study.
**C**: Here, the VV-test proves to be superior to the Smirnov-test (SS-test).
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<|MaskedSetence|> <|MaskedSetence|> Their framework extends to Kähler geometry via complexification techniques, allowing for systematic treatment of the Kähler curvature operator in compact settings [PW21b]. <|MaskedSetence|> For more related interesting works, we refer to [Wil13], [Yang18], [PW22], [NPW23], [Li23],... | **A**:
The curvature term Θ\Theta in the Weitzenböck formula is fundamentally instrumental in deriving new geometric results, particularly through its role in connecting analytic and topological properties of manifolds.
**B**: In their recent work, Petersen and Wink [PW21a] established novel vanishing theorems and e... | ABC | CBA | ABC | ABC | Selection 1 |
<|MaskedSetence|> Pekar [38] in 1954, for V(x)=0V(x)=0, F(u)=|u|2F(u)=|u|^{2}, α=2,g(x,u)=0\alpha=2,~g(x,u)=0 and N=3N=3, to describe the quantum theory of the polaron at rest. Later, Choquard [26] used equation (1.1) to study an electron trapped in its hole, and Penrose used equation (1.1) as a model for self-grav... | **A**: Carvalho, Silva, and Goulart [10] studied (1.1) with convex-concave type of growth and established the multiplicity of solutions using the Nehari method with a fine analysis on the.
**B**: In the last decade, Moroz and Schaftingen [34, 35, 36] studied the Choquard equations and proved the existence of ground st... | CBA | CBA | CBA | CAB | Selection 1 |
Several algorithms leverage the core Newton-type principle of using approximate second-order information for fast convergence. Sepcially, Proximal Averaged Newton-type method for Optimal Control (PANOC) [15] innovatively integrates the forward-backward (FB) method and the FBE method to overcome their individual drawb... | **A**: While PDOM provides theoretical guarantees for convergence to critical points and analyzes its local convergence rate, a significant limitation is that its convergence analysis, particularly the rate, is confined to the local domain,lacking established global convergence guarantees.
.
**B**: A key advantage of... | CBA | CBA | CBA | CBA | Selection 4 |
Mass lumping has always played a central role in explicit dynamics and became even more prominent after Leidinger [36] showed that the largest generalized eigenvalues for the row-sum lumped mass remained bounded regardless of how badly cut the elements were, provided the spline space was sufficiently smooth. <|Masked... | **A**: This finding has since then been thoroughly investigated, both numerically [44, 45, 46, 47] and more theoretically [41, 42] and is by now relatively well understood.
**B**: Instead, the most convenient way of eliminating them is to prevent them from surfacing in the first place.
Fortunately, spurious high-fre... | ACB | ACB | ACB | ACB | Selection 1 |
4.5 Analysis of a real-world graph dataset
We finally apply the proposed algorithm on a real-world network of political blogs (Adamic and Glance,, 2005). The network corresponds to the hyperlinks of blogs regarding U.S. politics after the 2004 presidential election. <|MaskedSetence|> After following the rule of thumb... | **A**: We apply MCMC and stochastic gradient Algorithm in 1 to compute the BE and SANVI estimates, together with ASE and OSE as the competitors.
**B**: These blogs are manually classified as either liberal or conservative, which we use as the true value of community labels.
**C**: The results are listed in Table 9, a... | BAC | BAC | BCA | BAC | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> These spherical functions can be viewed as multivariable special functions that depend on discrete sets of parameters. In essence, Dunkl operators are families of commuting differential-difference operators acting on Euclidean space. They are equipped with an additional structure d... | **A**: Notably, they are relevant in the study of quantum many-body systems of Calogero–Moser–Sutherland type, which are examples of algebraically integrable systems in one dimension.
**B**: One of the central motivations for studying Dunkl operators arises from the theory of Riemannian symmetric spaces, where spheric... | CBA | ABC | CBA | CBA | Selection 4 |
Homology objects of MM with respect to the homotopy tt-structure are Rost cycle modules given by diagonals in motivic homology of MM ([4]). Since MM is compact, we may
substitute it by the motivic cohomology of M∨M^{\vee} instead. <|MaskedSetence|> Hence, we will have only finitely many such non-zero diagonals if and ... | **A**: Hence, by (1), this object has only finitely many diagonal in motivic homology.
**B**: Since φ\varphi “moves” in non-diagonal direction, some power of it will be zero.
**C**: Since M∨M^{\vee} is compact, it has no motivic cohomology with numbers >d>d, for some dd.
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Most importantly, unlike geodesics, conformal circles can develop various singularities.
Based on many examples, Paul Tod conjectured [11] that one type of such singularities can be ruled out: that conformal geodesics cannot spiral. <|MaskedSetence|> The conjecture was shown in the case of Einstein metrics [11]. <|Ma... | **A**: Namely, that they cannot converge to a point without reaching it in a finite proper time (we will provide explicite definition later).
**B**: In this way, the original conjecture is resolved..
**C**: Other examples considered in [11] were highly symmetric, but in [12] more general manifolds were considered sup... | CAB | ACB | ACB | ACB | Selection 2 |
We first prove (c):
By Lemma 2.5, we can assume that the first row of E1E_{1} has exactly h2h^{2} entries equal to 11 and h2h^{2} entries equal to −1-1. <|MaskedSetence|> On the other hand, either all entries in the first row of E2E_{2} are equal, in which case we must have h2+h=2h2h^{2}+h=2h^{2}, i.e., h2=hh^{2}=h, ... | **A**: Therefore, h=1h=1, and hence n=4n=4.
To prove part (d), put R1:=[k1,…,kn/2]R_{1}:=[k_{1},\ldots,k_{n/2}].
**B**: This implies that E2E_{2} has h2+hh^{2}+h entries in its first row equal to 11 and h2−hh^{2}-h entries equal to −1-1.
**C**: Lemma 2.4, applied to each of the terms kjkj−1k_{j}k_{j-1} that sum up... | BAC | CAB | BAC | BAC | Selection 3 |
Our extrapolation approach differs from the existing one [51] by decoupling the extension of the derviative information from the assembly of ghost point values. <|MaskedSetence|> Our approach relies only on the normal derivatives at the boundary that are deliberately computed and stored at irregular grid points, in ... | **A**: We solve the homogeneous PDE (18) only once, instead of solving one homogeneous and two inhomogeneous equations (with different source terms).
**B**: We leave it to future work to extend our approach to a higher-order scheme at the expense of greater computational complexity.
.
**C**: The simplicity of our ap... | BCA | ACB | ACB | ACB | Selection 4 |
<|MaskedSetence|> The theory of uniqueness of equilibrium states of uniformly hyperbolic dynamical systems was developed by Sinai, Ruelle and Bowen in the mid-1970s. Sinai initiated the study [Sin72] in the case of Anosov diffeomorphism. Ruelle and Bowen generalized to uniformly hyperbolic (Axiom A) system [Bow71, Rue... | **A**: In [Bow74], Bowen provided a criterion to guarantee the existence and uniqueness of equilibrium states.
**B**: Following Bowen’s approach, they are able to weaken the three hypotheses to weak specification with all scale, expansivity observed by all measures with large free energy, and Bowen property on a suita... | CAB | CAB | CAB | ACB | Selection 3 |
2. <|MaskedSetence|> We begin with a derived version of isogenies and prove that, in certain cases, the kernel of a derived isogeny behaves similarly as in the classical setting. <|MaskedSetence|> In Section 2.2, we define relative effective Cartier divisors in the setting of spectral algebraic geometry. <|MaskedSe... | **A**: Effective Cartier divisors of spectral Deligne–Mumford stacks
A main construct of this paper concerns derived level structures.
**B**: This gives evidence that our derived version of level structures must induce classical level structures.
**C**: We then use Lurie’s representability theorem to prove that cert... | CBA | ABC | ABC | ABC | Selection 3 |
In [19], the authors adopt the airborne RIS system of unmanned aerial vehicle (UAV) transmitted by non-orthogonal multiple access (NOMA). In addition, this work analyzed the system performance through the deduced interrupt probability and user traversal rate.
Liu et al. <|MaskedSetence|> <|MaskedSetence|> In [23], th... | **A**: A comprehensive solution using the H-function of multiple Foxes was proposed to solve this problem.
In [21], the performance of RIS-assisted wireless communication system was studied by considering both direct and reflective paths under the assumptions of perfect channel state information acquisition and optimal... | CAB | CAB | BAC | CAB | Selection 1 |
Once the approximate exchange operator is constructed, subsequent computations become computationally trivial. <|MaskedSetence|> To achieve this, we implement a two–level nested SCF iteration strategy: the outer–loop focuses on optimizing the Fock exchange operator (which contributes relatively little to total energy)... | **A**: This hierarchical approach confines approximate exchange operator construction exclusively to the outer–loop, and the reduced number of outer–loop iterations significantly enhances computational efficiency.
The rest of this paper is organized as follows: In Section 2, we introduce the preliminary knowledge rel... | BAC | BAC | BAC | BAC | Selection 4 |
<|MaskedSetence|> We also thank Christina Tønnesen-Friedman for useful comments on an earlier version of this paper.
C.S. <|MaskedSetence|> E.L. is partially supported by the grant PRCI ANR–FAPESP: ANR-21-CE40-0017. <|MaskedSetence|> | **A**: We would like to thank Eiji Inoue for a detailed discussion of his paper [Inoue_PerelmanEntropy] and some helpful remarks.
**B**: Part of this work was conducted during the Winter School on K-stability at CIRM.
.
**C**: is supported by a MSCA Postdoctoral Fellowship, funded by the Research and Innovation frame... | ACB | ACB | ACB | CAB | Selection 3 |
4. Growth Rates
We conclude by giving a table that compares the growth characteristics of the various methods we covered. <|MaskedSetence|> <|MaskedSetence|> Additionally, the Card. <|MaskedSetence|> Rate columns measure the growth rate between consecutive MSTD sets in the sequence generated by the minimal A1A_{1}... | **A**: columns give the smallest possible cardinality and diameter for a set A1A_{1} which a method can be applied to.
**B**: Note that the |A1|\big{|}A_{1}\big{|} and A1A_{1} Diam.
**C**: Rate and Diam.
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<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> This fact allows using transport sweep to solve the equation in each SI. Meanwhile, during each sweep, a dense block matrix of size O(r)×O(r)O(r)\times O(r) must be inverted for every spatial element. As shown in our numerical tests, for challenging problems th... | **A**: The core idea in [10] is to partition the angular domain into a small number of octant families, noting that all angles within the same octant share the same upwind ordering of spatial elements on a Cartesian mesh and thus the same sweeping pattern.
**B**:
Another closely related work is [10], which proposes a... | BCA | BCA | BCA | CAB | Selection 3 |
These two established dynamics naturally raise the question of what happens when the interaction rules are combined in a hybrid fashion. For example, one may consider a system where stronger particles push weaker ones backward while weaker particles jump forward over stronger ones, or, conversely, a system where strong... | **A**: As we demonstrate, the first hybrid leads precisely to the long-range swap model studied in this paper, which turns out to be integrable and structurally rich, whereas the second fails to yield a consistent multi-particle dynamics and is therefore non-integrable.
Integrability of an interacting particle system... | ACB | ACB | ACB | ACB | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Under suitable assumptions on the model coefficients, we derive an explicit expression
for R0R_{0}, and use it to numerically illustrate the convergence order of the numerical approximation of R0R_{0} for both the Fourier and Chebyshev methods (for the latter, we... | **A**:
In this section, we experimentally validate the theoretical results of section˜6 and discuss applications to epidemiological models.
**B**: The first one is a model obtained from the linearization of a dd-dimensional Susceptible–Infectious–Removed (SIR) model with seasonality in contact rates around the Diseas... | ACB | BAC | ACB | ACB | Selection 1 |
<|MaskedSetence|> For finite/spherical Coxeter groups, the conjecture is already known to hold because the weak order and extended weak order are isomorphic. <|MaskedSetence|> <|MaskedSetence|> This is the first non-spherical, non-affine Coxeter group for which
Dyer’s conjecture has been established. In fact, we pro... | **A**: Wang [Wan19] proved Dyer’s conjecture for affine Coxeter groups of rank 33, and Barkley and Speyer [BS23] proved it for all affine Coxeter groups.
**B**: In this article, we prove Dyer’s
conjecture for the universal Coxeter group of rank 33, which we
denote by U3U_{3}.
**C**:
In 1994, Dyer conjectured that th... | CAB | CAB | ABC | CAB | Selection 2 |
When we compare our constructions to the ones in 3, we noticed that some of our constructions are larger. For the 1818-distance set, the paper removed points from a regular hexagon with side length 44 to get a 4343 points construction. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> The main reason that we ar... | **A**: There was also a 7070 points construction for 2929-distance set in that paper.
**B**: On the contrary, we removed three corners from a (3,4)(3,4)-equiangular hexagon to get a 4545 points construction.
**C**: However, we were able to construct a set with 7575 points and only 2828 distances by taking a (4,5)(4,5... | ACB | BAC | BAC | BAC | Selection 2 |
The training data was split into 10 groups in order to define a set of 10 constraints. We began our experimentation by employing an implementation of an L-BFGS method with a backtracking Armijo line search in order to minimize the objective of (40), without any constraints. <|MaskedSetence|> <|MaskedSetence|> With a ... | **A**: Corresponding to this point, the training accuracy was determined to be 92.4% while the testing accuracy was determined to be 78.5%.
We then employed the same L-BFGS method to minimize a quadratic penalty function corresponding to (40) with a penalty parameter of 1e+02; see [28] for further details on penalty ... | BAC | BAC | BAC | ABC | Selection 3 |
II-A2 Improve Channel Correlation Among Adversaries
As demonstrated in Figure 2, we consider the scenario where threats arise from multiple eavesdroppers. In conventional FPA schemes, although transmitters are capable of suppressing signal reception at the eavesdroppers via beamforming, this inevitably results in a de... | **A**: This trade-off becomes increasingly severe as the number of eavesdroppers grows, since the beamforming flexibility has to be sacrificed to nullify the eavesdropping links, thereby depleting the available spatial DoFs for signal transmission.
**B**: Similarly, the anti-jamming capability can be enhanced by appro... | ACB | ACB | ACB | ACB | Selection 4 |
<|MaskedSetence|> Consequently, our approach serves as an equivalent reformulation, as shown in Table 1. Most importantly, the new model provides strong theoretical guarantees and enables the development of an efficient numerical algorithm. <|MaskedSetence|> <|MaskedSetence|> We introduce SPFs to equivalently transf... | **A**:
In this paper, we introduce a sharp-peak function (SPF) to derive the model (SPCO).
**B**: In other words, it offers a practical and effective continuous optimization approach for solving the discrete programming problem (UBIP).
**C**: The main contributions are fourfold.
1) The invention of SPFs.
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Fig. 8 compares the secrecy rate versus the normalized size of the antenna moving region, i.e., A/λA/\lambda, under different transmit powers PBP_{B}. The proposed scheme consistently outperforms both the RPA and FPA-based methods across all values of A/λA/\lambda, especially for higher transmit powers. <|MaskedSeten... | **A**: As A/λA/\lambda increases, the secrecy rate of the proposed, RPA, and FPAF scheme increases.
**B**: This is because enlarging the moving region provides two key benefits.
**C**: For the RPA scheme, the secrecy rate remains relatively stable as A/λA/\lambda increases and is consistently lower than that of the p... | ABC | BAC | ABC | ABC | Selection 3 |
The power-law halo structures have been conventionally explained by assuming self-similarity. The simplest self-consistent model for galactic globular clusters is the orbit-averaged Fokker-Planck (OAFP) model (Hénon, 1961; Cohn, 1979, 1980). <|MaskedSetence|> Cohn (1980) confirmed the self-similarity in the core and... | **A**: Hénon (1961) pioneered and assumed the infinite OAFP model to evolve self-similarly, showing that the self-similar state is stable against a linear perturbation, while the Boltzmann-Maxwell distribution function is not.
**B**: Later, (Heggie and Stevenson, 1988) revisited the self-similar OAFP model, assuming t... | ACB | BCA | ACB | ACB | Selection 3 |
Recently, progress has been made toward concretely trainable neural networks that achieve optimal approximation rates, albeit on idealized training data. Two parallel directions have emerged, both of which construct large dictionaries of basic functions (e.g., neural networks) and then perform linear regression over ... | **A**: These models enjoy near-optimal approximation rates, admit training via ridge regression, and can accommodate randomly sampled inputs.
**B**: The second direction [29, 99, 98] follows a similar strategy, but builds large dictionaries consisting of Riesz bases for standard function spaces that can be realized by... | ABC | BCA | ABC | ABC | Selection 4 |
This conjecture is referred to as “Odd Hadwiger’s Conjecture”. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> It remains open for t≥6t\geq 6. For graphs with no odd KtK_{t}-minor, the current best known upper bound of chromatic number is O(tloglogt)O(t\log\log t) due to Steiner [31].
. | **A**: Guenin [13] announced a solution of the case t=5t=5 at a meeting in Oberwolfach in 2005.
**B**: It is far stronger than Hadwiger’s conjecture.
**C**: The case t≤4t\leq 4 was proved by Catlin [4] in 1978.
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<|MaskedSetence|> Compared to some of the existing LIT methods, the DTB method exhibits noticeable improvements in accuracy with enhanced stability and robustness. Furthermore, it offers convenient adaptivity to combine multiple DNNs and theoretically provides error control. Meanwhile, several problems remain unanswer... | **A**: 4 Conclusion
We introduce a meshless and training-free method that leverages the traditional temporal schemes and the tangent bundle of DNN to compute the solution of evolutionary PDEs in high dimension.
**B**: A principled method is more desirable in practice.
**C**: For example, are there more systematic st... | ACB | ACB | BCA | ACB | Selection 2 |
In the absence of a full theory of quantum gravity, the semi-classical approach can serve as a first step towards comprehending black-hole evaporation and similar phenomena. <|MaskedSetence|> The background and the quantum field are coupled by considering the expectation value of the (renormalized) stress-energy tenso... | **A**: In this approach, quantum fields are considered on a classical background.
**B**: Unfortunately, a self-consistent solution of the semi-classical Einstein equations is still beyond our grasp outside of highly symmetric setups [6, 7, 8].
**C**: However, progress can still be made outside of these situations by ... | ACB | ABC | ABC | ABC | Selection 4 |
One can go further than this and prove a positive mass theorem in two spatial dimensions [7].
Indeed, in [7] we applied the Witten spinorial method [17] to n=2n=2, yielding either one or two inequalities for mass and angular momentum, depending on the topology of the manifold. <|MaskedSetence|> However, also in that c... | **A**: Although the Witten spinorial method had already been applied to n=2n=2 previously [18, 19], our work addressed previously ignored subtleties related to the definition of mass and inequivalent spin structures in two spatial dimensions.
Interestingly, there exist mass aspect functions that cannot be mapped to a... | BCA | ABC | ABC | ABC | Selection 3 |
Random termination is a common feature of many financial products. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> In some cases, the time cap is chosen to be unobservable; see [8].
Time-cap is closely related to cancellable options, which are terminated early when a specific event occurs. In fact, the pric... | **A**: 27] or [32]) or an asset-price-independent time cap following an exponential or Erlang distribution (see [4]).
**B**: For example, the random termination moment could correspond to the default time of a company (see, e.g., [2, p.
**C**: The latter case, which leads to the so-called Canadian approximation, has ... | BAC | BAC | ACB | BAC | Selection 4 |
Mirror symmetry for projective Calabi-Yau manifolds with h2,1(X)>0h^{2,1}(X)>0 has been the subject of intense research over the past 30 years; see for example [60, 97]. However, for non-Kähler, or rigid Calabi-Yau manifolds, the situation is still rather mysterious. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSet... | **A**: One of the early applications of conifold transitions, as suggested in the physics literature (see e.g.
**B**: [55]), was to extend mirror symmetry to some non-Kähler and rigid Calabi-Yau manifolds.
**C**: These two situations are related, since if XX is a rigid Calabi-Yau manifold, so that h2,1(X)=0h^{2,1}(X... | CAB | ACB | CAB | CAB | Selection 3 |
exists. So it is reasonable to conjecture that the limit exists for any P(x,y)P(x,y). <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> For many others we approximate the limit using various methods, some of them we have originally developed.
We are going to prove here the following
. | **A**: It is not our aim here to prove the conjecture.
**B**: In [3] Boyd and Mossinghoff, in their Chapter 6, have given some observations supporting the conjecture.
**C**: For some polynomials we determine the explicit value of the limit.
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The paper is organized as follows. In Section 2, we introduce the notation used throughout the paper and collect several results needed for the proof of the main theorem, with particular attention to the properties of the Codazzi tensor, which enable local computations involving principal curvatures with possible multi... | **A**: student at Lehigh University.
**B**: Xiaochun Rong for constant support and encouragement, as well as Guanhua Shao and Prof.
**C**: Ling Xiao for helpful discussions.
.
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One main obstacle for proving the preservation of the nn-entangledness of 𝔼\mathbb{E} is that at some point we will want to duplicate points in the second coordinates of good pairs, as well as ⩽n\leqslant n-size subsets of 𝔼\mathbb{E} via lemmas 2.3 and 2.4 respectively. Unfortunately, we can not apply these two lemm... | **A**: On the other hand, if we are dealing with a set of the second kind, we will prioritize duplications via Lemma 2.4.
**B**: Naively speaking if a set is of the first kind, we will prioritize duplications via Lemma 2.3.
**C**: As it can bee seen in the next two lemmas, the main job of the chains of submodels form... | BAC | BAC | BAC | BAC | Selection 3 |
<|MaskedSetence|> However, prior work is confined to a narrow configuration, relies exclusively on simulations, and neglects the impact of phase noise. <|MaskedSetence|> This work generalizes the DSK framework and provides a theoretical analysis of its robustness to both rapid channel variations and phase errors, off... | **A**: Its full potential in general settings remains unexplored and unverified, lacking theoretical support and rigorous performance quantification.
**B**: Unlike SSK, which relies on CSI, DSK requires a pre-estimate of the DoA for coherent detection.
**C**: DSK represents the first class of SM demonstrated to exhib... | CAB | CAB | CBA | CAB | Selection 1 |
In the paper [8] has been studied the weighting problem. In the present work, we discuss Theorems 2.4 and 3.6 from the Ref. [8]. <|MaskedSetence|> <|MaskedSetence|> After that we derive correct proofs of the mentioned results, which are based on a different approach, because the scheme of the proofs of the authors ... | **A**: We point out an error in the proofs of these theorems.
**B**: A simple example shows that their proofs are not correct.
**C**: Following the scheme of the proofs of the authors of Theorems 2.4 and 3.6 in [8], we obtain two additional theorems, which concern the connection between the vector problem and its sca... | ABC | ABC | ABC | BCA | Selection 2 |
Next we consider (2). Set H:=G[V(P1∪P2)]\E(C)H:=G[V(P_{1}\cup P_{2})]\backslash E(C).
We claim that (2) holds when HH is connected. By (a), SS is the set of degree-1 vertices of HH. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: Hence, by Lemma 2.1, there is an induced tree TT of HH containing SS.
**B**: So (2) holds from Lemma 3.4.
We may therefore assume that HH is not connected..
**C**: By (a) and Lemma 2.3, we have that C∪TC\cup T is isomorphic to a graph pictured as Figure 2.
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We employ the method of sliding paraboloids to prove the weak Harnack inequality. <|MaskedSetence|> <|MaskedSetence|> This method has now become a standard approach to establish Harnack type estimates for non-uniformly elliptic equations, see [IS16]. It is worth mentioning that this method can also be employed to est... | **A**: Later, Savin [Sav03, Sav09] generalized it and used it to solve the celebrated De Giorgi conjecture.
**B**: Instead, we iterate by enlarging the opening of the touching paraboloids.
**C**: This method was first introduced by Cabré [Cab97] in his extension of Krylov-Safonov theory to manifolds with nonnegative ... | CAB | CAB | CAB | CBA | Selection 1 |
In [7] the class of generalized block graphs was defined as a generalization of the class of block graphs. <|MaskedSetence|> It is clear that every block graph is also a generalized block graph. An explicit formula for depth(S/JG)\operatorname{depth}(S/J_{G}) is known for any generalized block graph GG. <|MaskedSe... | **A**: Moreover, by a minimal cut set of GG, we mean a cut set which is minimal under inclusion.
**B**: Let GG be a chordal graph such that for every three maximal cliques of GG whose sets of vertices have a nonempty intersection, the intersection of the vertex sets of each pair of them is the same, (see for example [... | BCA | BAC | BCA | BCA | Selection 1 |
This paper addressed a class of nonsmooth fractional programming problems with fixed-point constraints, where the numerator is convex and the denominator is concave. To solve this problem, we proposed the FSSM,
which computes the subgradients for the convex numerator and concave denominator separately, updating each fu... | **A**: This simplification reduces the overall complexity of the optimization process by removing the necessity of selecting specific subproblem-solving techniques or fine-tuning parameters, thereby potentially reducing both computational time and complexity in certain scenarios.
**B**: We established the subsequentia... | ABC | BCA | BCA | BCA | Selection 3 |
See [CW94, CK95, JW02, BLdSN04, HW06, Ere07, LT20] for other proofs, and a recent survey [CL24].
In recent years, the study of partial differential equations on graphs has attracted increasing attention and has played an important role in various fields. <|MaskedSetence|> <|MaskedSetence|> Existence results for Ka... | **A**: In the case of nonlinear equations, Grigor’yan, Lin, and Yang [GLY16] investigated the Kazdan–Warner equation on finite graphs and established existence results, which were subsequently extended in [Ge17, FZ18, Ge20, LY20, CM22, PS22, SW22, LZ23, ZLY24].
**B**: For a given graph, there is a natural discrete Lap... | BAC | BAC | BAC | CAB | Selection 1 |
<|MaskedSetence|> The global existence of classic or strong solutions with arbitrary large initial data is not known generally. Only the one-dimensional theory is quite satisfactory.
The foundational results on global well-posedness for one-dimensional flows with large initial data away from vacuum trace back to the s... | **A**: Further developments in this direction were contributed by Li and Xin [30, 47].
**B**: Strong solution theory.
**C**: See also Huang and Li [22] for more general parameter case.
.
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<|MaskedSetence|> Using the discrete uniformization theorem [16], we first transform a given triangulated surface into a conformally equivalent hyperbolic surface equipped with a Delaunay triangulation. <|MaskedSetence|> The target hyperbolic surface is represented as a quotient of hyperbolic plane 𝔻/Γ\mathbb{D}/\Ga... | **A**: Taking the Euclidean harmonic map as the initializer, we implement a Riemannian gradient descent algorithm to find the unique hyperbolic harmonic map in the fixed homotopy class.
In summary, this paper introduces an algorithm for discrete harmonic maps that is both theoretically grounded and practically robust... | CBA | ABC | CBA | CBA | Selection 3 |
A common and powerful tool to study the convergence rate in Central Limit Theorems is Stein’s method. <|MaskedSetence|> However, is seems to not have employed so far for stochastic processes valued in Hilbert spaces, which is the case we treat here.
We finally mention that the convergence rate for (1.7) has been analy... | **A**: The basic idea is to view the limiting measure as an invariant measure of some process: it is widely used and has been applied in many problems, also for stochastic processes, and also in combination with Malliavin calculus; we refer to [APY21] for a recent survey.
**B**: We refer to
[Kol11, Prop.
**C**: 5.11.... | ABC | ACB | ABC | ABC | Selection 4 |
The reader may notice that the latter property (2.24) is not commonly required in the definition of Leray solutions. However, it comes naturally as a uniform bound of any reasonable approximate sequence and it will be needed for our later analysis. <|MaskedSetence|> We provide the proof since we could not find the ... | **A**: In fact, we are not aware of any result constructing Leray solutions satisfying the energy equality for a general initial datum with finite Hamiltonian.
**B**: By regularization and compactness arguments, Leray solutions exists.
**C**: Uniqueness can be obtained under additional assumptions [CW99, Mar08a, M08]... | BAC | BAC | BAC | BCA | Selection 3 |
Compared with classical NSPNP system, the new model (1) considers non-Newtonian viscosity and ionic steric interactions. <|MaskedSetence|> Developing efficient and structure-preserving algorithm is very challenging for this highly nonlinear coupled partial differential equation system (1). This work is inspired by sca... | **A**: Based on this idea, we introduce a suitable auxiliary variable and design an ordinary differential equation to deal with the nonlinear terms involved in the equations.
**B**: Besides the Coulomb force term coupling the electric potential and fluid velocity and the convection term coupling ionic concentration an... | BCA | BCA | BCA | BCA | Selection 1 |
It is clear that our assumption is considerably weaker than those commonly used in the literature for several reasons. <|MaskedSetence|> <|MaskedSetence|> Third, Assumption 2 itself is weaker than Abadie’s condition. Furthermore, while existing results in the literature provide only necessary or sufficient conditions... | **A**: Second, our definition of feasible directions is more general.
**B**: In fact, if at the point x¯\overline{x} under consideration there are no active constraints, then this point is an interior point of the feasible set (in the case of continuous variables), and thus Assumption 2 is trivially satisfied.
.
*... | CAB | CAB | CAB | ABC | Selection 3 |
Viscoelastic fluid is a type of complex fluid which has both viscous and elastic properties, as encountered in cosmetics industry gallegos1999rheology , food processing industries bistany1983comparison and blood montecinos2014hyperbolic . Compared with the Newtonian fluid, the total stress for the viscoelastic flow ... | **A**: lee2006new used the direct discretization of the objective derivative.
**B**: hulsen1990sufficient rewrited the stress equation in terms of conformation stress tensor and showed that conformation stress tensor is symmetric positive-definite.
**C**: The logarithm conformation representation (LCR) method was... | ABC | BAC | BAC | BAC | Selection 2 |
Suppose that MM is a faithful simple AA-module. Then 𝔯M=0\mathfrak{r}M=0 (by the simplicity of MM and definition of the radical). This implies that 𝔯=0\mathfrak{r}=0, by the faithfulness of the AA-module MM. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Therefore, the algebra AA is a simple algebra.
∎
.... | **A**: For a semisimple but not simple finite dimensional algebra, every simple module is unfaithful.
**B**: it is a finite direct product of simple finite dimensional algebras.
**C**: So, the algebra AA is a semisimple finite dimensional algebra, i.e.
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The existing stochastic models for dormancy can be roughly categorized into two approaches: population genetics models and population dynamics models. <|MaskedSetence|> <|MaskedSetence|> In [BHS21], the authors discuss three switching strategies between dormant and active types: stochastic (or: spontaneous; simultan... | **A**: Following a brief example in the book [HJV07], a two-type branching process (without migration) in a fluctuating random environment has been introduced in [BHS21], which served as a motivation for this paper.
**B**: The first approach assumes a constant population size and focuses on the genealogical implicatio... | BAC | BAC | BAC | ACB | Selection 1 |
The remaining part of the paper is organized as follows. We provide an overview of relevant frame theoretic background in Section 2. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> The Appendix contains the probabilistic tools and results used in this paper.
. | **A**: Finally, Section 6 contains numerical results on the frame bounds of random Gabor frames with small cardinality and their robustness to erasures.
**B**: In Section 3, we develop general analysis tools for frame bounds estimation, which we then apply to obtain our results for random and deterministic Gabor frame... | BAC | CBA | BAC | BAC | Selection 4 |
<|MaskedSetence|> (2024); Shen et al. (2025) and other optimizers attempt to address these issues through heuristic momentum and learning rate schedules Zaheer et al. (2018); De et al. (2018); Chen et al. <|MaskedSetence|> <|MaskedSetence|> However, these approaches lack theoretical grounding in the geometry of the ... | **A**: (2018).
**B**: (2019; 2022); Zaheer et al.
**C**:
Current adaptive optimization methods like Adam Kingma & Ba (2017), AdamW, Muon Jordan et al.
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6 Conclusion
In this paper, we have confirmed that the cubic G1G^{1} parametric approximant given in [6], whose curvatures at the middle and at the boundary point coincide, is indeed, under some conditions, the best fit for a circular arc in terms of curvature. This only holds for a circular arc whose inner angle is... | **A**: Also, the problem of finding the optimal quartic G1G^{1} interpolant is a challenging issue, and we do not know the characterization of optimal approximants..
**B**: We have shown that all other circular arcs’ optimal approximants satisfy a different condition.
**C**: A similar characterization for the optimal... | BCA | BCA | BCA | BCA | Selection 4 |
The theoretical analysis demonstrates that the method maintains stability and optimal convergence rates through a novel interpolation strategy that avoids extrapolation even near boundaries. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> The method handles ellipses and diamonds with extreme aspect ratios, sha... | **A**: The numerical result reveals mesh-independent conditioning for double layer formulations with Dirichlet conditions and controlled growth for Robin conditions, comparable to continuous boundary integral methods.
**B**: In contrast, our unfitted method avoids evaluations of singular integrals
Numerical experime... | CAB | CAB | CAB | ACB | Selection 1 |
in kCG(u)eukC_{G}(u)e_{u}, it follows (e.g. from [9, Theorem 4.7.1]) that
⟨u⟩\langle u\rangle has a unique local point on iBiiBi. Let JJ be a primitive
decomposition of ii in (iBi)⟨u⟩(iBi)^{\langle u\rangle}. <|MaskedSetence|> If j′j^{\prime} belongs to
ϵ\epsilon, then χ(uj)=\chi(uj)= χ(uj′)\chi(uj^{\prime}... | **A**: By.
**B**: If jj does not belong to ϵ\epsilon, then
jj belongs to a point of ⟨u⟩\langle u\rangle which is not local.
**C**: Let j′∈Jj^{\prime}\in J.
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<|MaskedSetence|> A higher level of critical sparsity indicates improved empirical reconstruction performance. The simulation results reveal that the critical sparsity of JS-gOMP algorithms is much larger compared to OMP and gOMP algorithms. <|MaskedSetence|> <|MaskedSetence|> 3. As more unnecessary components start... | **A**: In Figure.1, we provide the recovery performance as a function of the sparsity level KK.
**B**: But JS-gOMP gives the best results in reduced noise in the reconstructed sparse signals as per the error plots in Figure.
**C**: As the external Gaussian noise is introduced (SNR↓\ \downarrow) the reconstruction per... | ACB | ACB | ACB | ACB | Selection 1 |
<|MaskedSetence|> In statistics and data science, it is used in statistical inference from data samples, as discussed by [12] and [28]. Additionally, ME techniques are utilized in entropic optimal transport and Schrödinger bridges, which are foundational for score-based generative modeling, as explored by [27], and [7... | **A**: In probabilistic machine learning, the ME principle underpins entropic regularization techniques, providing principled methods for deriving optimal posterior distributions during model training, as detailed by [5] and [1].
**B**: Notable works include [4], [29], and [26].
**C**: The ME principle and its associ... | CBA | ABC | CBA | CBA | Selection 4 |
Sketch of proof. A key ingredient in the proof of Theorems 1.1 and 1.2 is some (explicit) uniform bounds on the Cartier index of Weil divisors in a general type MMP [HQZ25, Theorem 3.2, Corollaries 3.18 and 3.20]. By Birkar’s inductive approach [Bir07], if termination holds in dimension n−1n-1, then it also holds for ... | **A**: To turn this into an explicit bound on the number of steps in the terminal MMP, we show in Section 4.2 that the above lexicographic sequence of invariants decreases in a bounded way, in the sense that each time an individual term drops by one, the terms thereafter increases by at most a uniformly fixed amount.
... | CBA | BAC | BAC | BAC | Selection 3 |
The first implication (a) ⇒\Rightarrow (b) will be one of the central results of this paper. <|MaskedSetence|> Therefore we devote Section 22 to introducing these tools and discussing how these tools are to be used. In Section 33, we will jump right into working out the details for the first implication. This can be f... | **A**: This will be a simple consequence of a limiting argument on the boundary of the unit disk.
**B**: For completeness, this will be formulated in Theorem 3.
.
**C**: It involves the use of analytic properties of conformal mappings from the unit disk onto a symmetric quasidisk, and the use of the modulus of the Te... | CAB | CAB | CAB | ACB | Selection 2 |
Usually, for RMSD calculation algorithms performance comparison, a lower RMSD will be considered as a more accurate RMSD calculation, because the wrong alignment (translate, rotate, swap, etc.) will probably lead to a larger RMSD. However, if the atom mapping algorithm ignores the bonds or chemical environment, such as... | **A**: The molecule with different (R) and (S) chiral properties can be totally regarded as a different molecule.
**B**: Also, for water clusters, sometimes the oxygen and two hydrogen atoms of a certain water molecules are mapped to other water molecules respectively (Figure 8).
Only under the condition that the mo... | ABC | CAB | ABC | ABC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> was partially supported by NSF grant DMS-2200804. S.M. <|MaskedSetence|> We also thank Bhargav Bhatt, Kęstutis Česnavičius, Brian Conrad, Hai Long Dao, Eike Lau and Shizhang Li for helpful comments and conversations.
. | **A**: M.
**B**: was supported by the Institute for Advanced Study, University of British Columbia and a start up grant from Purdue University.
We thank Luc Illusie for sharing with us Grothendieck’s letter [24].
**C**: K.
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<|MaskedSetence|> Then we take the last value τ¯≤1\bar{\tau}\leq 1 such that the angle
between γ(τj)γ(τ¯)\gamma(\tau_{j})\gamma(\bar{\tau}) and LL is α\alpha. <|MaskedSetence|> Note that if the angle between
γ(τj)γ(1)\gamma(\tau_{j})\gamma(1) and LL is at most α\alpha, such a point exists by continuity. Assume ... | **A**: Consider the case that γ(τj)γ(τj+1)\gamma(\tau_{j})\gamma(\tau_{j+1}) is steep.
**B**: We add τ¯\bar{\tau} to the set of division points of [0,1][0,1], and if
τk≤τ¯<τk+1\tau_{k}\leq\bar{\tau}<\tau_{k+1}, we proceed to the segment γ(τ¯)γ(τk+1)\gamma(\bar{\tau})\gamma(\tau_{k+1}).
**C**: Then we stop the p... | ABC | ABC | ACB | ABC | Selection 1 |
3. Collective and geometry-dependent (e.g. <|MaskedSetence|> Importance of the topology of the configuration space. Topological data analysis leverages topological structures defined on discrete data sets by associated simplicial complexes. <|MaskedSetence|> Simplicial complexes can also be associated with point clo... | **A**: It is defined in a similar way to the Lusternik-Schliemann category of XX and represents the Schwarz sectional genus of the fibration of the continuous paths in XX over the space of endpoints X×XX\times X Schwarz66 .
**B**: Standard algorithms based on metric affinities or other relations allow the construction... | CBA | CBA | CBA | CAB | Selection 2 |
The dynamical systems perspective brings new information into the description of these events, providing a dynamical template that helps to understand the long-time behavior of fuel blobs, and thus having the capability to identify potentially dangerous regions for oil spill disasters. In particular we show that the i... | **A**: In the movie it is observed that the reattachment point is a moving saddle which evolves towards the north of the coast of Gran Canaria.
**B**: A full visualization of the time evolution of these transport barriers is available from movie S1.
**C**: 6 displays the velocity field of that day superimposed, offer... | CBA | CBA | CBA | CAB | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> the function to be recovered is constant in the distance to the gliding surface. However, we do not make such assumption for proving kernel characterizations.
The geometry of the integration paths for the head wave transform are the ray paths giving by propagation of singularitie... | **A**: Our reconstruction formulas employ the corresponding assumption i.e.
**B**:
Usually one assumes that the seismic refraction imaging is done to a shallow enough region below the surface to that the material parameters can be assumed constant in depth.
**C**: A formula for the critical angle is given in [7] and... | BAC | BAC | BAC | BAC | Selection 3 |
2.1 Geometric Triangulations and Local Moves
For background on geometric triangulations, we refer the reader to Chapter 4 of [15]. Let TT be an ideal triangulation of an orientable cusped hyperbolic 3-manifold MM. A 2-3 move333This is sometimes called a Pachner move or a bistellar flip. <|MaskedSetence|> <|MaskedS... | **A**: is a local modification of the triangulation which replaces a bipyramid consisting of two distinct tetrahedra which share a common face with a union of three distinct tetrahedra which share a common edge, illustrated in Figure 1.
**B**: The inverse of a 2-3 move is called a 3-2 move, and can be applied at a deg... | ACB | ACB | ACB | BAC | Selection 2 |
<|MaskedSetence|> In Section 2, we present preliminary concepts and technical results concerning the P3P_{3}-convexity in graphs, with a particular focus on Hamming graphs. <|MaskedSetence|> In Section 3, we derive upper bounds for the Carathéodory number in Hamming graphs. <|MaskedSetence|> Section 4 contains the m... | **A**: This includes key lemmas on the structure of the P3P_{3}-hull number and Carathéodory sets.
**B**: We also establish a recursive inequality that relates the Carathéodory number to the dimension of the graph.
**C**:
The article is organized as follows.
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In response to this architectural limitation, various extensions have been proposed to improve representational capacity, including SVD-DeepONet [31], the R-adaptive DeepONet designed for discontinuities [41], multi-input EDeepONet [29], and the Separable Operator Networks [40]. These models reflect an active line of r... | **A**: While this design demonstrates improved predictive accuracy across a range of PDE benchmarks, it typically requires two to three times longer training time.
**B**: The original “vanilla” DeepONet, on the other hand, remains attractive for its computational simplicity.
**C**: These contrasting approaches reflec... | ABC | ABC | CAB | ABC | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> Moreover, Figure 6 demonstrates that PBRWP particles are less noisy than BRWP particles at low iterations.
Figure 5: Evolution of the KL divergence between baselines and BRWP-based methods for the bimodal distribution, initialized with large variance 𝒩(0,6I){\mathcal{N}}(0,6I)... | **A**:
We observe a different compression phenomenon with the high-variance initialization, when the samples are initially distributed around the modes instead of between the modes.
**B**: In this case, the preconditioning accelerates the convergence.
**C**: Figure 5 demonstrates that in this case, the preconditioni... | ACB | ACB | BCA | ACB | Selection 1 |
We show that the same dynamic plays out on the whole plane when the initial data is made up of such a pair of patches plus their reflections across the horizontal axis (again with opposite strengths, yielding a “doubly odd” scenario). This means that we have two pairs of patches that touch initially, which seems to pos... | **A**: Of course, the latter just coincides with the no-touch half-plane setting and is not a proper well-posedness result on ℝ2{\mathbb{R}}^{2}, but it also a priori excludes certain high frequency “unstable modes” as we discuss at the end of this introduction.
Nevertheless, our other main result — and second motiva... | ABC | BCA | ABC | ABC | Selection 1 |
It would be interesting to compare our result with the pioneering work in semilinear PDE by Zaag [Zaa02a, Zaa02b, Zaa06]. Zaag proved the C2C^{2}-regularity of the singular set of semilinear PDEs under the assumption that the singular set is a submanifold. <|MaskedSetence|> The curvature of this singular set is due t... | **A**: The mechanism is similar to getting higher regularity of 𝒮k(𝐌)0\mathcal{S}_{k}(\mathbf{M})_{0} for mean curvature flows, which we do not address in the current paper.
**B**: In contrast, Remark 1.7 suggests that the curvature of the degenerate cylindrical singular set originates from the spherical part, whic... | CAB | ABC | ABC | ABC | Selection 2 |
Proof.
The argument here is standard and goes back to Arens [A]. <|MaskedSetence|> Note that the spectrum of an element x∈Ax\in A may possibly change when calculated in some larger algebra only if some elements non-invertible in AA become invertible in BB. This, of course, cannot happen if every non-invertible elem... | **A**: We present a sketch of the proof for the convenience of the readers.
**B**: In the other direction, we need to prove that if a non-invertible element x∈Ax\in A is not a topological divisor of zero, then there exists a unital commutative Banach algebra BB containing AA as a closed subalgebra such that xx is inve... | ABC | ABC | ABC | BAC | Selection 1 |
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