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<|MaskedSetence|> Section 3 defines notation for cashflows based on stochastic representations of a lifetime. We leave aside multiple-state models and stochastic assets to focus on the structure of surplus; these extensions follow along clear lines laid down in the literature. Section 4 defines technical bases in term... | **A**: Section 5 defines assets and liabilities in the model balance sheet, leading to the definition of surplus, then in Sections 6 and 7 we introduce surplus based respectively on one or two technical bases, including a systematic notation.
**B**: Conclusions are in Section 14.
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**C**:
In Section 2 we start wit... | CAB | CAB | CBA | CAB | Selection 1 |
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. <|MaskedSetence|> <|MaskedSetence|> Finally, the privacy am... | **A**: And then, in the quantization phase, the CSI extracted by Alice and Bob is quantized into the raw bit sequences.
**B**: Next, during the information reconciliation, the error correction codes are used to correct discrepancies in the raw bit sequences between Alice and Bob.
**C**: For example, in an indoor quas... | ABC | ACB | ABC | ABC | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> The properties of the convex functions hh in detF\det F are important and influences the local invertibility and local monotonicity.
However, in order to check the regions on which the local monotonicity and the local invertibility are properly satisfied, we do... | **A**: The regions from Figures 12-12 are those on which the local monotonicity and the local invertibility are satisfies numerically.
There are also some other candidates for the part of the energy depending on detF\det F (the purely volumetric part).
**B**: For instance, for the energy considered before in the intro... | CBA | CBA | BCA | CBA | Selection 4 |
We note a few things about this process. <|MaskedSetence|> We saved these numbers for later processing and found that none of them were prime, therefore they are accounted for by some earlier prime pp which was filtered out. <|MaskedSetence|> <|MaskedSetence|> After each of these kk batches, we sorted the filters a... | **A**: Second, some filters are more efficient than others, in that they appear to filter more numbers.
**B**: 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.
**C**: First, not a... | CAB | CAB | CAB | CBA | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> In [4], the authors characterize when one Cantor minimal system is the speedup of another and provide topological analogues to the measure-theoretical results of [2]. <|MaskedSetence|> It is shown that the minimal bounded speedup of an odometer is a conjugate odometer, whereas the... | **A**: In [1], minimal bounded speedups of odometers and substitutions are investigated.
**B**:
There is a rich history of studying speedups of dynamical systems, particularly in the measure-theoretical setting; see, for example, [2, 11, 12].
**C**: More recently, there has been a focus on topological speedups.
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We also warm-start the solver using the Union of Stars construction, in line with best practices outlined in [13]. <|MaskedSetence|> In our formulation, the value of RR corresponds to the number of rows in the warm-start solution. <|MaskedSetence|> Rajakumar et al. [21, Theorem 8] prove that |𝐖r,r|≤(3n−2)(3n−1)/2... | **A**: This significantly improves the performance of Gurobi’s built-in heuristics.
**B**: Although the Union of Double-Stars construction yields tighter upper bounds in theory, it performs worse in practice with Gurobi.
Lastly, we discuss the choice of the big-MM constant.
**C**: However, in all our experiments, s... | ABC | ABC | ABC | ABC | Selection 2 |
<|MaskedSetence|> Organization
The paper is organized as follows. Section 2 reviews the necessary background and notation for the Ricci flow, which will be used throughout the paper. In Section 3, we introduce key preliminaries from interpolation theory that form the foundation for presenting Simonett’s stability the... | **A**:
1.3.
**B**: Appendices A, B provide supplementary proofs for Sections 5 and 6..
**C**: Section 6 then verifies the applicability of the linear theory.
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Another recent direction of relevance, grouped together under the term spectral deconfounding, includes work by Ćevid et al. <|MaskedSetence|> (2022); Fan et al. <|MaskedSetence|> (2024) among others, and considers estimation and inference in high-dimensional regression models with unmeasured confounders. <|Masked... | **A**: For example, Ouyang et al.
**B**: (2024); Wang and Shah (2025); Sun et al.
**C**: (2020); Guo et al.
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<|MaskedSetence|> <|MaskedSetence|> Similarly to the compact case, it is natural to ask whether the Hamilton-Perelman results can be extended to noncompact manifolds. <|MaskedSetence|> Second, we are interested in the stability of the Ricci flow at its fixed point, specifically the hyperbolic metric.
Bessières-Besso... | **A**: First, we need to determine if the existence theorems for Ricci flow apply in this context.
**B**: We will use it to extend [36] and [37] to this setting.
**C**:
Another focus of the paper is the application of Ricci flow to finite-volume hyperbolic 33-manifolds.
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<|MaskedSetence|> By reformulating ancient results, Descartes realized that straight lines were the only curves of degree 11, and the conic sections (circle, ellipse, hyperbola, parabola) were the curves of degree 22.404040The latter family also contains some exceptional cases, such as two perpendicular lines – see Vl... | **A**: and inaugurated the area now known as linear algebra.444444An contemporary introduction to this field can be found in Sheldon Axler, Linear Algebra Done Right, Third Edition (New York: Springer, 2015)..
**B**:
One example of the fruits of this method is a classification of curves by the degree of the associate... | BCA | BCA | BCA | BCA | Selection 1 |
<|MaskedSetence|> The experiments were executed in 2-dimensional and 3-dimensional regions for Laplace and Poisson equations. <|MaskedSetence|> <|MaskedSetence|> The experiment results indicate that WoS-NN performs better in various tests and examples with fewer sample paths. All experiments were done using Python 3... | **A**: The Finite Difference method (FDM) and the original WoS method were tested for comparison on the same problems.
**B**: We also trained a simple feed-forward neural network using WoS results directly as a reference.
**C**:
4 Experiments
In this section, WoS-NN is tested with different parameters and factors u... | CAB | CAB | CAB | ACB | Selection 1 |
<|MaskedSetence|> Such rings consist of matrices with special patterns of zeros. 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 c... | **A**: A natural probabilistic interpretation of this result shows that the algebra is generated by two random matrices.
.
**B**: As a consequence, the probability of generating a finite incidence ring by a tuple of matrices is determined.
**C**: This paper studies generators of matrix incidence rings over finite pos... | CBA | ABC | CBA | CBA | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> To find such a feedback we set up and solve a Linear Quadratic Regulator (LQR). We are certainly not the first to use
LQR to find a state feedback that stabilizes a bending and twisting beam. <|MaskedSetence|> We stabilize the whole beam with two actuators at the root of the beam.... | **A**: One actuator affects the bending of the beam at its support and other affects the torsion of the beam at its support.
We first seek a state feedback control law which drives the oscillations to zero.
**B**: To the model of [2] we add two actuators located where the horizontal beam is joined to its support.
**C... | BAC | ACB | BAC | BAC | Selection 4 |
Figure 4: An example demonstrating the idea of the reduction in the proof of \creflem:nonstring. <|MaskedSetence|> the subgraph on all red vertices and edges, and the one on all blue vertices and edges) is connected, as each is the disjoint union of a path on 33 vertices and an isolated vertex. <|MaskedSetence|> On ... | **A**: However, any attempt to add edges to satisfy Condition 5 causes the graph to violate Condition 6.
**B**: 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.
**C**: Each apex stri... | ABC | BAC | BAC | BAC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> However, unlike the one-direction linear advection equation, the RTE equation involves both left- and right-going particles coupled through an integral term. As a result, the ODE argument in [4] is not applicable for kinetic equations. To bypass this difficulty, ... | **A**:
We use the same framework for the superconvergence analysis of the linear advection equation as used by Adjerid et al.
**B**: [4] and match the coefficients of the polynomial expansion of the error equation.
**C**: A key step in [4] is to utilize the theory for the Runge-Kutta method by viewing the one-dimen... | ABC | ABC | ABC | CBA | Selection 3 |
<|MaskedSetence|> In this case, we say that 𝔖\mathfrak{S} is an HHG structure for the group GG and use the pair (G,𝔖)(G,\mathfrak{S}) to denote the hierarchically hyperbolic group GG equipped with the specific hierarchically hyperbolic group structure 𝔖\mathfrak{S}. <|MaskedSetence|> From here, one can give a hier... | **A**:
For a precise definition, see [5, Definition 1.21].
**B**: Also, it is easy to show that the definition of HHG does not depend on the choice of the Cayley graph.
.
**C**: From condition (1) in the definition of an HHG, it follows that GG is quasiisometric to XX.
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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**: Theorem B is proved in Section 4.
**B**: Finally, in Section 5, we present some technical results that are used in the proof of the main theorems..
**C**: In Section 2, we provide background on the Heisenberg group and establish Taylor’s formula for the twisted translation.
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Let T∈𝔹A1/2(ℍ)T\in\mathbb{B}_{A^{1/2}}(\mathbb{H}) be such that ‖T‖A≠0.\|T\|_{A}\neq 0. Then TT is said to be AA-smooth if for any S1,S2∈BA1/2(ℍ)S_{1},S_{2}\in B_{A^{1/2}}(\mathbb{H}) such that T⟂ABS1T\perp_{A}^{B}S_{1} and T⟂ABS2T\perp_{A}^{B}S_{2} imply T⟂ABS1+S2.T\perp_{A}^{B}S_{1}+S_{2}.
In this work, we begi... | **A**: Finally, we examine AA-smoothness in the context of block diagonal matrices..
**B**: We also explore the relation between the AA-smoothness of operators in 𝔹A1/2(ℍ)\mathbb{B}_{A^{1/2}}(\mathbb{H}) and the smoothness of operators in 𝔹(R(A1/2))\mathbb{B}(\textbf{R}(A^{1/2})).
**C**: We establish that an AA-... | CBA | CBA | CBA | CBA | Selection 1 |
To considered the effects of un-classification in Table 1, the success
rates for the detection of existing links (SREL) and non-existing links (SRNL) can be defined to study the performance of network reconstruction (Wang et al. (2011); Han et al. (2015); Shi et al. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSete... | **A**: (2021)).
**B**: (2023), which has form of
.
**C**: Another more efficient measure in this case to study the estimated accuracy when unclassified set remain is the adjusted MCC(MCCa)MCC(MCCa) defined by Shi et al.
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Firstly, we introduce the orbit elements in the LEO satellite IoT constellation. <|MaskedSetence|> For ease of analysis, this paper adopts the commonly used spherical Earth model. <|MaskedSetence|> As shown in Fig. <|MaskedSetence|> | **A**: 2, six orbital elements are defined in the Earth-Centered Inertial (ECI) reference frame to describe the size, shape, and orientation of a LEO satellite’s orbit as follows:
.
**B**: Following previous relevant works [30]-[32], we assume that Earth is an ideal sphere and the satellite orbit is circular222Accoun... | BCA | BAC | BCA | BCA | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> The smallest size of such a set, denoted by c(G)c\left(G\right), is the cyclomatic number of GG [berge1973graphs]. It is sometimes called the feedback edge (set) number or the cycle rank of GG. For a graph GG on nn vertices, mm edges and kk connected components, it is not difficul... | **A**: To find an optimal feedback edge set of a connected graph, it suffices to consider a spanning tree; the edges not belonging to the spanning tree form a minimum-size feedback edge set.
.
**B**: Our goal.
Our objective is to study the three above classes of problems, on graphs of bounded cyclomatic number.
**C... | CAB | BCA | BCA | BCA | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> The quantization conditions derived from finite polynomial solutions of these equations allowed us to obtain discrete energy spectra for each field configuration considered.
Across the different scenarios studied—from constant and exponential profiles to hyperbolic field configur... | **A**: The resulting equations, although not adopting the standard eigenvalue form, exhibit deep connections with Heun equations.
**B**: However, in hyperbolic configurations, the slope depends on the quantum number nn, and only the ground state retains a slope equal to −vd-v_{\mathrm{d}}.
**C**: In this work, we ana... | CAB | CAB | ACB | CAB | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> On the other hand, [6] gives an interpretation of the fixed point iteration for ψα\psi_{\alpha} as a Newton-Raphson method under additional assumptions.
To the best of our knowledge, there is no general convergence proof yet for the fixed point iteration via (2.6) although the al... | **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**: This was probably not realized in [6] since it was not stated in the description/manual of [10].
**C**: 2.2, Cor.
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<|MaskedSetence|> Normal Loading occurs when all cargo types are placed under lower density ones and are lighter than water; thus the constraint matrix is positive definite by the Proposition. In Reverse Loading heavier cargoes are placed over lighter ones. <|MaskedSetence|> <|MaskedSetence|> In the cases where the ... | **A**: As to stability requirements, we examine a Relaxed Stability Margin with metacentric height at least μ=4\mu=4 and a Strict Stability Margin with μ=6\mu=6.
**B**: Table 2: Cargo types
We consider two cases for the loading order.
**C**: We then solved the resulting four cases using the GRG Excel Solver, and th... | BAC | BAC | BAC | ACB | Selection 3 |
The rest of the paper is organized as follows. 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} (fir... | **A**: Discussion and unsolved problems are addressed in the concluding section.
.
**B**: In section 3, we give a simple bijection linking caterpillar tree parking distributions and 𝐮\mathbf{u}-parking distributions.
**C**: In section 4, we introduce the first-return decomposition, which is our main tool for the ma... | ACB | BCA | BCA | BCA | Selection 2 |
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**: Moreover, we report on the results of a small Monte Carlo simulation study.
**B**: In the last section, the basic idea is carried over to the two-sided alternative H1∗:F≠GH_{1}^{*}:F\neq G.
**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**: A key innovation lies in their algebraic reformulation of the curvature term into specialized representations of the Lie algebras, which provides crucial structural insights while eliminating classical singularities associated with curvature-dependent operators.
**B**:
The curvature term Θ\Theta in the Weitze... | BCA | BCA | BCA | BCA | Selection 3 |
as k→∞.k\to\infty.
This contradicts (5.4). Thus, the claim is proved.
Now choose An:={u∈Xn:‖u‖=ϱn}A_{n}:=\{u\in X_{n}\;:\;\|u\|=\varrho_{n}\}. <|MaskedSetence|> <|MaskedSetence|> Thus, ℰ^α,λ\hat{\mathcal{E}}_{\alpha,\lambda} admits a sequence of critical points {un}\{u_{n}\} in D2,p(ℝN)D^{2,p}(\mathbb{R}^{N}) such... | **A**: Therefore, ℰ^α,λ\hat{\mathcal{E}}_{\alpha,\lambda} satisfies all the assumption in Theorem 5.2.
**B**: Clearly γ(An)=n\gamma(A_{n})=n and AnA_{n} is closed and symmetric, and hence An∈ΣnA_{n}\in\Sigma_{n} and also from (5.3), supu∈Anℰ^α,λ(u)<0.\sup_{u\in A_{n}}\hat{\mathcal{E}}_{\alpha,\lambda}(u)<0.
**C**: ... | BAC | CBA | BAC | BAC | Selection 3 |
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**: A key advantage of PDOM is that the quadratic nature of the smooth term keeps the Hessian constant, allowing its inverse to be precomputed and stored, thus eliminating the need for costly iterative matrix inversions typical of quasi-Newton methods.
**B**: While PDOM provides theoretical guarantees for convergen... | BAC | CAB | CAB | CAB | Selection 4 |
<|MaskedSetence|> Most notably, several authors have suggested applying the row-sum technique in a Petrov-Galerkin framework by choosing (approximate) dual functions as test functions. <|MaskedSetence|> Since then, there has been a surge of interest in approximate dual functions [29, 30, 31, 32]. Unfortunately, such ... | **A**: Although the idea, whose origins date back to Cottrell [4], initially did not attract much attention, it was taken up again recently in [28] with promising results.
**B**: Intensive research is currently underway to improve its accuracy and several paths are being explored.
**C**: Additionally, quadrature rule... | BAC | BAC | BAC | BAC | Selection 1 |
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