Split (1)

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shuffled_text stringlengths 267 3.16k	A stringclasses 6 values	B stringclasses 6 values	C stringclasses 6 values	D stringclasses 6 values	label stringclasses 4 values
A: This research was supported by the grant ”Agreement on the provision of a grant in the form of subsidies from the federal budget for implementation of state support for the creation and development of world-class scientific centers, including international world-class centers and scientific centers, carrying out...	ABC	ABC	BCA	BAC	Selection 4
A: In particular, several recent research works present the benefit of utilizing the polarization domain in recently proposed communication schemes including, but not limited to, MIMO spatial multiplexing [1]; spatial modulation (SM) [2, 3, 4]; non-orthogonal multiple access (NOMA) [5]; and beamforming [6, 7]B...	CAB	CAB	ABC	ACB	Selection 4
A: They generalize MZV and MSZV in a natural way and also build a bridge to the theory of partitionsB: Related works can be found in [Ba18], [BY18], [BC20], and [BKSYY23]. One of the referees of our paper pointed out that it would be interesting to see if the RHS of (5.18) can be expressible in terms of SMZV. T...	BAC	ACB	BCA	CBA	Selection 3
A: As shown by Example 8.5, the non-toric generalization has to involve all valuations instead of just one.Lastly, let us mention that our generalization of Boucksom–Chen theorem has important consequences in Archimedean pluripotential theory as wellB: It is of interest to generalize D to the non-toric setting ...	BAC	ABC	CAB	BCA	Selection 1
A: Notably, our result may extend to non-monotone profilesB: Additionally, the result depends only on the front solution, not on the initial condition. C: Therefore, for a broad class of nonlinear diffusive-dispersive equations of Burgers type, a traveling front solution is asymptotically, nonlinearly, and ...	BAC	CBA	BCA	CBA	Selection 3
A: Later, Takahashi and Nunokawa [15], studied the class 𝒮α,βsubscript𝒮𝛼𝛽\mathcal{S}_{\alpha,\beta}caligraphic_S start_POSTSUBSCRIPT italic_α , italic_β end_POSTSUBSCRIPT for the range α,β∈(0,1]𝛼𝛽01\alpha,\beta\in(0,1]italic_α , italic_β ∈ ( 0 , 1 ] and denoted it as 𝒮⁢𝒮∗⁢(α,β).𝒮superscript𝒮𝛼𝛽\mathcal{S...	CAB	ACB	ABC	BAC	Selection 1
A: Our contribution consists in providing a unified version of Biswas-Gómez-Muñoz proof, which holds for any almost-simple group G𝐺Gitalic_GB: Instead of using the theory of spectral covers, we will use the theory of cameral covers. C: The main difference between our paper and the other ones consists in th...	CBA	ACB	CAB	CBA	Selection 2
A: In Section 3 we define strongly Lech-independence and expansion property and prove some equivalent conditions. There are also some examples showing the relation between strongly Lech-independence and other notions. In Section 4 we use strongly Lech-independence to analyze the colength of powers of ideals and der...	BCA	BAC	BAC	CAB	Selection 4
A: However, this requires additional caveats when discussing the factorizationB: Therefore, we prefer not to consider them as snakes. C: If we consider meanders of total order one as snakes, the generating function and the corresponding numerical sequences take a somewhat more natural form	CBA	CBA	CBA	BCA	Selection 4
A: In addition, the isomorphism (46) is provided in [DHSSM22, Thm.14.10]B: At this stage, all that remains, amongst the assumptions of Theorem B is to prove that this “Betti χχ\upchiroman_χ-independence” isomorphism preserves mixed Hodge structures. Moreover, the original P=W conjecture has been proven [MS22, H...	BCA	BAC	BAC	CBA	Selection 1
A: However, there are numerous approaches to community detection [12, 33], and it is worthwhile to adapt other approaches, such as modularity maximization [29] and statistical influence using stochastic block models [31], to account for node-absorption ratesB: Absorption-scaled graphs provide a useful way to ad...	ACB	BAC	CAB	BCA	Selection 4
A: However, their methods can not cover the regime when the expected degree is Ω⁢(1)Ω1\Omega(1)roman_Ω ( 1 ) due to the lack of concentrationB: In subsequent works [25, 71] we proposed algorithms to achieve weak consistencyC: Additionally, [72] proposed Projected Tensor Power Method as the refinement stage ...	BAC	BCA	CAB	BCA	Selection 1
A: Instead of passing a critical state of high energy, the process has to cross a valley of negative fitness through a sequence of deleterious mutationsB: Similarly to the fast dynamics after passing a high energy state, the adaptive dynamics system quickly attains a new metastable equilibrium once a fit mutant...	BCA	CBA	BAC	CAB	Selection 1
A: Though we have stated the modularity score of a partition for binary edge weights it is simple to take the weight of edges inside each part (instead of the number of edges) and to take the degree of a vertex v𝑣vitalic_v to be the sum of the weights of the edges incident to v𝑣vitalic_v (instead of the number of...	ACB	ACB	BAC	BCA	Selection 4
A: Then α⁢(x)𝛼𝑥\alpha(x)italic_α ( italic_x ) is a subshift of X𝑋Xitalic_X and α⁢(x)=α⁢(x′)𝛼𝑥𝛼superscript𝑥′\alpha(x)=\alpha(x^{\prime})italic_α ( italic_x ) = italic_α ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) impliesB: Conversely, for every x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X, let α⁢(x)={y∈A...	CBA	CAB	BCA	BAC	Selection 1
A: Further, sj−si∈Δupsubscript𝑠𝑗subscript𝑠𝑖superscriptΔups_{j}-s_{i}\in\Delta^{\mbox{\tiny up}}italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ roman_Δ start_POSTSUPERSCRIPT up end_POSTSUPERSCRIPTB: On the other hand, the induction hypothesi...	CAB	CBA	BAC	ACB	Selection 2
A: More precisely, these authors established the third term on the right-hand side inB: (2017)C: The result in Theorem 4 for s≥1/2𝑠12s\geq 1/2italic_s ≥ 1 / 2 (that is, 2⁢k+2≥d2𝑘2𝑑2k+2\geq d2 italic_k + 2 ≥ italic_d) was already derived in Sadhanala et al	BAC	CBA	CAB	ABC	Selection 2
A: In this study, we explore a nonlinear stochastic COVID-19 system, incorporating the influence of non-Gaussian noiseB: This consideration is crucial for a comprehensive understanding of the system’s behavior and its response to unpredictable environmental factors. C: The presence of non-Gaussian noise add...	BAC	BCA	CBA	ACB	Selection 4
A: Recent studies have illustrated the versatility of persistent homology in analyzing complex networks, including brain networksB: (2019); Xing et al. (2022) highlighted the application of persistent homology in evaluating temporal changes in topological network features.C: Sizemore et al	CAB	ABC	ACB	CBA	Selection 3
A: Thanks are also due to Colin Guillarmou for pointing out to us that we could use Fried’s result for the proof of Theorem 3B: Finally, we would like to thank the anonymous referee for his comments and suggestions that helped to improve the manuscript. The first author is supported from the European Research C...	CBA	ABC	BCA	ABC	Selection 3
A: Next, in Section 4, we show that the pISSf conditions on control gains additionally guarantee ISSt for the system in the sense of (10). B: First, in Section 3, we find the conditions on control gains that satisfy the pISSf criterion in (9)C: In the subsequent sections, our approach of finding the control...	ACB	CBA	BCA	CAB	Selection 2
A: Financial support was provided by the POSCO Science Fellowship, as well as Korea NRF grants (nos. 2020R1F1A1A01048645 and RS-2023-00237811). B: The author expresses gratitude to both institutions for providing conducive environments for focused research and generous supportC: This work was undertaken dur...	BAC	CBA	ACB	CAB	Selection 2
A: The development of the following lemma is motivated by the convenience and simplicity of considering points on interior edges of a given origin face and destination face of an arbitrary landscape in any convex unit polyhedronB: By doing so, we eliminate the need for four variables when constructing arguments...	BCA	BAC	CAB	ABC	Selection 4
A: More recently, Ding, Fukushima, Sun and Xu [17, 18, 19] gave detailed results in the case of Bernoulli hard obstacles in dimension d≥2𝑑2d\geq 2italic_d ≥ 2. We will give a more complete description of the results for Bernoulli (soft or hard) obstacles, as well as a comparison with our own work and techniques, i...	ABC	ABC	CBA	CAB	Selection 4
A: This paper is organized as followsB: In Section 3 we discuss the Gaussian modified torsional rigidity. In Section 4 we define the Gaussian version of the Kohler-Jobin rearrangement, study its properties, and prove Theorem 1.1.C: In Section 2 we discuss some preliminaries	CAB	BCA	ACB	BCA	Selection 3
A: The transpose network in (5) is defined as the conceptually dual dynamical model of (1), and it has the same set of vertices with every edge reversed compared to the orientation of the corresponding edge in the original network (1)B: The relation between the identifiability properties in the original and tr...	BCA	BCA	CBA	ACB	Selection 4
A: We have established the asymptotic theory in a class of directed random graph model parameterized by the differentially private bi-sequence and illustrated application to the Probit model. The result shows that statistical inference can be made using the noisy bi-sequenceB: However, the asymptotic normality...	BCA	ACB	BAC	CBA	Selection 2
A: At the time when the project was performed, Jose Agudelo was an undergraduate student at North Dakota State University, Brooke Dippold was an undergraduate student at Longwood University, Ian Klein was an undergraduate student at Carleton College, Alex Kokot was an undergraduate student at the University of Notr...	ACB	BCA	CAB	BAC	Selection 2
A: Obviously, this is not guaranteed for each instance of the algorithm. The Gauss-Southwell method leads to faster convergence at the cost of extra computations and evaluations of gradients during the selection of coordinates which can be an issue in large-scale problems [25].B: On the other hand, the use of ...	CBA	ACB	ABC	ABC	Selection 1
A: Gicquaud and T. Marsh, we give in this paper a geometric characterization of complete non-compact Kähler manifolds admitting a compactification by a strictly pseudoconvex CR structure. The study is more intricate in the complex hyperbolic setting than in the real hyperbolic one, due to the anisotropic nature of ...	ACB	CAB	ACB	BCA	Selection 2
A: In this section we discuss the general geometric structure with which ℋℋ{\mathscr{H}}script_H and ℒℒ{\mathscr{L}}script_L are eventually endowedB: For example, in [LS-2014] the introduction of a ∂¯¯\bar{\partial}over¯ start_ARG ∂ end_ARG-operator to define the holomorphic structure of a smooth (quasi)-Hilbe...	BAC	ACB	BCA	CBA	Selection 2
A: We would like to express our sincere gratitude to our scientific adviser Nikolai Vavilov for formulating the problem and for a constant support, without which this paper would never have been writtenB: AcknowledgmentC: The authors are grateful to Alexei Stepanov for carefully reading our original manusc...	ACB	BCA	BAC	CAB	Selection 3
A: For this we mainly follow [UN]. Then we recall the classification of T-invariant Cartier divisors in terms of Cartier divisorial support functions, following the work in [AIPSV] and [PS].B: In Section 3 we restrict to the complexity one case and we describe the invariant subvarieties induced by the polyhedra...	CBA	CAB	ACB	BAC	Selection 1
A: Then there exists two prime numbers ℓ1,ℓ2≠psubscriptnormal-ℓ1subscriptnormal-ℓ2𝑝\ell_{1},\ell_{2}\neq proman_ℓ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_ℓ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≠ italic_p, ℓ1≠ℓ2subscriptnormal-ℓ1subscriptnormal-ℓ2\ell_{1}\neq\ell_{2}roman_ℓ start_POSTSUBSCRIPT 1 end_POST...	CAB	BCA	CBA	ABC	Selection 3
A: On the contrary, we show that the Bayes optimal algorithm performs sub-optimally with some of the worst model parameters, which implies that maximizing the Bayesian objective differs substantially from maximizing the frequentist objectiveB: The fact that the Bayes optimal algorithm is suboptimal means that ...	CBA	CAB	ACB	CAB	Selection 3
A: For many of our results, we will be operating in a regime in which k𝑘kitalic_k is fixed, m𝑚mitalic_m is large (as a function of k𝑘kitalic_k) and λ𝜆\lambdaitalic_λ is arbitraryB: We will employ asymptotic notation with respect to this regime.C: Over the course of this section and the next one, we wil...	CBA	CAB	ABC	BCA	Selection 4
A: Acknowledgement: The authors would like to thank the anonymous referees for their many useful comments and suggestionsB: The second author is supported by CSIR (India). The third author is supported by the Prime Minister’s Research Fellows (PMRF) Scheme.C: The first author is supported by the Science an...	BAC	CAB	ACB	ABC	Selection 3
A: Also, it can be extended to give individual non-trivial eigenvalues with non-trivial coboundaries to general S𝑆Sitalic_S-adic sequences, even when they are neither finitary, nor constant-length, nor otherwise, recognizableB: Finally we remark that this kind of construction is essentially the only kind that...	ABC	BAC	BCA	BCA	Selection 2
A: quickly found four other infinite families by looking at the eigenmatrices of various primitive rank 3333 graphs but did not publish the results. In Winter 2021, C.-Y.LB: and W.-H.Y. joined the project of finding all nontrivial real ETFs obtained as spherical embeddings of primitive rank 3333 graphs.C: T...	CBA	BCA	CAB	ACB	Selection 2
A: Lastly, dash-dotted curves take into account both the corralated interference and the correlated MRC.B: The dashed lines depict the scenario with finite pool of Q=24𝑄24Q=24italic_Q = 24 pilot sequences and the solid horizontal line is the corresponding lower bound as given by (29)C: The markers represen...	CBA	CAB	CAB	ABC	Selection 1
A: In fact, in our algorithm, the yielded path has length at most (300)9/2⁢log⁡300superscript30092300(300)^{9/2}\log{300}( 300 ) start_POSTSUPERSCRIPT 9 / 2 end_POSTSUPERSCRIPT roman_log 300 times the length of the smallest spanning treeB: Moreover, the exponent 3333 in our theorem is sharp and can not be lower...	CAB	ACB	ABC	BCA	Selection 4
A: In the case of a minimizing T𝑇Titalic_T around a spherical inclusion, T𝑇Titalic_T will approach the particle surface since the nematic and magnetic exchange length become small w.r.tB: 6]). In the case of a peanut-shaped particle aligned with the magnetic field we expect one of three different minimizing c...	CBA	CAB	ABC	ACB	Selection 4
A: is an ∞\infty∞-stackB: In order to be able to compute this, it would be convenient to know that W¯⁢A¯𝑊𝐴\overline{W}Aover¯ start_ARG italic_W end_ARG italic_A is fibrant in the Čech model structure, i.eC: This follows thanks to the following wonderful theorem.	BAC	BCA	ABC	ACB	Selection 1
A: The polymatroids of this form are now called Chow polymatroids,333This naming is unfortunate for us because we will later associate a polymatroid to the formats of non-degenerate Chow forms of a variety, which are not themselves Chow polymatroidsB: an interesting (and proper) subclass of polymatroids. C:...	BAC	BAC	BCA	ACB	Selection 3
A: This suggests that geometric semilattices should be an “affine” counterpart to geometric lattices (among whose main examples we find posets of intersections of arrangements of linear hyperplanes). The following structure theorem is an abstract counterpart to this linear-affine relationship. B: The poset of i...	CBA	BCA	BCA	ACB	Selection 1
A: We first show a suitable mapping class exists, and then relate the parity to a certain determinant of the mapping class group element acting on a symmetrically self-dual representationB: Because the determinant, which is always ±1plus-or-minus1\pm 1± 1, is preserved by reduction mod p𝑝pitalic_p for odd p𝑝p...	ACB	BCA	ABC	ACB	Selection 2
A: I would also like to thank the anonymous referee for carefully reading the paper and making a number of helpful suggestions. B: I am also very grateful to Alessio Martini for bringing an error in Lemma 2.1 of an earlier version of this article to my attentionC: I am deeply grateful to my advisor Detlef M...	BAC	ABC	ACB	CBA	Selection 4
A: The second named author was supported by National Key R&D Program of China 2021YFA1003100, NSFC-11825101, NSFC-11522101 and NSFC-11431013B: The third named author was supported by China Postdoctoral Science Foundation BX20230402 and 2023M743719. C: Acknowledgements	ACB	CBA	BCA	CBA	Selection 3
A: Section 2 contains all necessary preliminaries on integral, superintegral and Kleinian sphere packingsB: Sections 3.2, 3.3 and 3.4 contain the proofs of our main theorems. Finally, in Section 3.5 we provide an example of a properly integral packing and study its properties.C: There we also recall Vinber...	CBA	CAB	ACB	CAB	Selection 3
A: 2022YFA1004900, the National Natural Science Foundation of China under Grant NoB: The work is supported by the National Key Research and Development Program of China under Grant NoC: 12201637 and No. 62202475, the Natural Science Foundation of Hunan Province of China under Grant No.2021JJ40701, and the ...	BCA	BAC	CAB	CBA	Selection 2
A: Almost surely either a player was chosen on Step 1 or 2 or the sum of just the odd-numbered terms (given by B’s moves) of expression (2) diverges to ∞\infty∞, by Lemma 3.6B: Thus player B is chosen on this step with probability 00, finishing the proof. ∎C: In the latter case, the sum of the even-numbered...	BCA	BCA	CAB	ACB	Selection 4
A: We refer to Section S13 of the Supplementary Materials [Balocchi et al., 2024+] for more details.B: For all these SSP, the prediction is somewhat accurate, and most often the credible intervals contain the “true” value of the functional of interest. When the data was generated from a power-law Zipf distribut...	ABC	BCA	CBA	ABC	Selection 3
A: It also contains a Red join gadget in [1.5,13.5]1.513.5[1.5,13.5][ 1.5 , 13.5 ]. Its long intervals terminate at 1.5,7.51.57.51.5,7.51.5 , 7.5 and start from 10.5,13.510.513.510.5,13.510.5 , 13.5. B: It also contains the second part of the third switch gadget in [14.5,17.5]14.517.5[14.5,17.5][ 14.5 , 17.5 ]....	ABC	BAC	CBA	ACB	Selection 3
A: ∎ B: Next consider the case where G𝐺Gitalic_G is abelian-by-compactC: Then exactly by the same argument as above, G𝐺Gitalic_G has the same asymptotic dimension and Hirsch length as its cocompact abelian subgroup (which could clearly assumed to be closed and so locally compact) and the result follow fro...	CAB	BAC	ACB	BAC	Selection 1
A: We give a further simplification of this latter property into a condition of bifilteredness thanks to our key observation, which harmonizes this result with our definitions of bi-accessibility. We then prove the 2-categories of flat pseudofunctors to be themselves bi-accessible and bipresentable whenever their d...	CBA	BCA	BAC	BCA	Selection 1
A: The third author was supported by EPSRC grant EP/W522338/1 and London Mathematical Society grant ECF-2021-27B: The authors have no relevant financial or non-financial interests to disclose. There are no datasets associated with the article. The authors thank the two reviewers for their helpful comments and G...	BCA	CBA	CBA	BAC	Selection 1
A: The main goal of this paper is to study finite and infinite horizon risk averse MDPs in a similar framework as above, but with latent costs and randomized actions, under a weakly continuous transition mechanism, subject to state-dependent DRMs. Below we provide a high level discussion on some of the open proble...	ABC	CAB	BCA	ACB	Selection 4
A: Motivated by applications in fluid dynamics, we suppose that u𝑢uitalic_u is divergence-free, B: In spite of its mathematical simplicity, advection equations are of fundamental importance in a variety of models in physicsC: By considering this linear model it is supposed that the velocity field has no or...	BCA	CBA	ACB	ABC	Selection 2
A: All the symmetry calculation results of this paper can immediately be obtained from our general result for the given coefficients of PDE (3.37)B: All that is needed is the calculation of the semi-invariant K⁢(x)𝐾𝑥K(x)italic_K ( italic_x ) defined by (2.5)C: Then, the condition for the equation to have...	BCA	ABC	CBA	CAB	Selection 2
A: The proofs of these results are given in the subsequent sections.B: The main results obtained in the present paper are Theorems 3.1 and 4.1 from sections 3 and 4 respectivelyC: The next section briefly recalls only the essential part of [8] used in the sequel	ABC	BCA	CBA	BCA	Selection 3
A: The second is to explore more complicated Turan-type properties, in the spirit of the paper by Keller and Lifshitz [33]B: Below an example of such a statement for several families.C: The first direction is to broaden the class of potential ‘ambient’ families	ABC	BCA	ABC	ACB	Selection 2
A: Two curves are stably homeomorphic if there is a homeomorphism of their regular neighborhoods in the ambient surfaces mapping the first curve onto the second one preserving the orientations of the curve and the surface.B: A curve is the image of a generic immersion of oriented circles into an oriented (close...	BAC	CAB	CBA	BAC	Selection 2
A: This does not correspond to a valid move, since the support of ν𝜈\nuitalic_ν would be {−10,−9,9,10}109910\{-10,-9,9,10\}{ - 10 , - 9 , 9 , 10 }. Intuitively, such moves should not ultimately help push mass far away, and indeed in Lemma 4.2 we show for a different relaxation of highest Elo that they don’t help.*...	BAC	CAB	BCA	ABC	Selection 2
A: We will show in this subsection that such kind of bijection does not exist; see Corollary 5.5.B: One may think that it might work with a different choice of bijection in Theorem 3.8C: We shall show that the Prasad conjecture does not hold for ℓℓ\ellroman_ℓ-modular representations with the V𝑉Vitalic_V c...	CBA	ACB	CAB	BCA	Selection 1
A: Finally, we prove structure results for primal and dual solutions. B: Then, we obtain a Strassen-type result for a variant of the convex order involving positively 1111-homogenous functionsC: We first obtain an improved duality result showing that under mild conditions there is dual attainment	CBA	BCA	BAC	BCA	Selection 1
A: [duc, 1.3.1]), so it makes sense to speak of branches issued from a point of C𝐶Citalic_CB: See [duc, 1.7] for some elements from the branch language which we use here. See also [duc, pg. 210] for the notion of a virtual disc. C: A Berkovich curve has the structure of a real graph (cf	BAC	ACB	ACB	BCA	Selection 4
A: We will need that the following constructions yield algebraic representations, all clear from the definitions: B: Equivalently, the adjoint maps G×V→V→𝐺𝑉𝑉G\times V\to Vitalic_G × italic_V → italic_V satisfy the usual equations for an action and are morphisms of varietiesC: We will also refer to this a...	ACB	CAB	ACB	CBA	Selection 4
A: We also flag up the very recent [spiliopolus], which treats the case of SDEs which are non-linear in the sense of McKean.B: The literature on multiscale methods is extremely vast so in the above we have mentioned only the works which are most relevant to our discussionC: Other related works (without any...	ABC	CAB	BCA	CBA	Selection 2
A: L𝐿Litalic_L) is finitely generatedB: In most of the applications, we assume F𝐹Fitalic_F (respC: Since the ring is local, any finitely generated flat module is free.***for the case F𝐹Fitalic_F (resp. L𝐿Litalic_L) is not-necessarily finitely generated, see Remark 3.32 (resp. Proposition 3.20).	ABC	CAB	BAC	ACB	Selection 3
A: Choose the singular cycle with the smallest initial point (in the standard representation)B: Choose a complex linear subdigraphC: Find its smallest point of singularity. Either it is an enclosed point or a corner point of an IDI path, as described earlier.	BCA	BCA	BCA	BAC	Selection 4
A: Despite many works, complete classifications of real and strongly real elements are known only for a very few families of Lie groupsB: Mostly, the equivalence between real and strongly real classes has been understood for certain groups, e.g., [Wo, ST]. A complete classification of such elements is known onl...	BCA	CBA	CAB	CAB	Selection 1
A: After this paper was written, we realized that the analogue of γ𝛾\gammaitalic_γ-coisotropic- with γ𝛾\gammaitalic_γ replaced by the Hofer distance (and under the name of “locally rigid”) had already been defined in [Ush19]B: Using such a notion, the property that for a submanifold, the equivalence between l...	ABC	ABC	BCA	CAB	Selection 3
A: Section 2 details the mathematical set-up of translationally invariant stabilizer codes in terms of commutative algebra. Rudiments of symplectic geometry over group rings of ΛΛ\Lambdaroman_Λ are laid out here. Section 3 makes the connection between topological excitations and the functor ExtExt\mathrm{Ext}roman...	ABC	ACB	CBA	BAC	Selection 1
A: This section is organized as follows. In Subsection 5.1, we establish terminology and notationB: In Subsection 5.3 we prove that for connected polyhedral complexes imbedded in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, all cells are pointed or all cells are...	ABC	BAC	ACB	ABC	Selection 3
A: For a syntomic k𝑘kitalic_k-scheme X𝑋Xitalic_X, we refer to Corollary 2.1.6, [BL22, Warning 4.6.2, Prop. 5.1.1]111Strictly speaking, [BL22, Prop. 5.1.1] constructs the Nygaard filtration only in the affine caseB: for the definition of the Nygaard filtrationC: However, similarly to [BL22, Not. 5.5.23], ...	CBA	BCA	CAB	ACB	Selection 4
A: We note here that the model above does not include a latent period, however since the results in this paper relate to the final outcome of an SIR epidemic they are insensitive to quite general assumptions concerning a latent period (see e.gB: Moreover, the results of the present paper (suitably modified) ca...	BCA	BCA	ACB	ABC	Selection 3
A: Besides, when the interval of mesh becomes larger, the helicity fails to conserve as well as the refining one. B: Experiments show that our scheme preserves the helicity orders of magnitude better with a simple modification in the definition of the vorticityC: Figure 8 show that the helicity of our model...	ABC	CAB	ABC	CBA	Selection 2
A: Persistent homology, an important tool in TDA, captures the evolution of the homology of a family of topological spaces associated to the given datasetB: Traditionally, large topological features are emphasized over small onesC: In the traditional setup, homology classes represented by geometrically lar...	BAC	CBA	BCA	ABC	Selection 1
A: In contrast, the upper bound confidence (UCB) method is used in Xiong et al. (2021) for adaptive exploration. However, they require strictly positive state transition and observation emission kernels to ensure fast convergence to the stationary distribution. The more related work is Jin et al. (2020a), which con...	CBA	ABC	BCA	ACB	Selection 1
A: As we have seen before the papers on perfect edge domination are less frequentB: There is some more bibliography to add to the already vast literature [3, 10, 23, 31, 33, 35] on dominating induced matchingsC: There is a paper [16] where the authors describe ILP formulations for the PED problem, together ...	BCA	ABC	BCA	BAC	Selection 4
A: Therefore the proof of Theorem 1.1 cannot be directly appliedB: Nevertheless, as announced in the introduction, the strong Liouville property still holds, see the statement of Theorem 1.2. C: As for discrete ∞\infty∞-harmonic functions, recall that F∞subscript𝐹F_{\infty}italic_F start_POSTSUBSCRIPT ∞ en...	BAC	BCA	CAB	CBA	Selection 2
A: We will give two proofs for our main theorem, each based on these two definitions. Hence we repeat here both definitions.B: One important invariant of a greedoid is the greedoid polynomial. It was defined by Björner, Korte and Lovász, and they give two equivalent definitions (greedoid, , Theorem 6.1)C: ...	BCA	BAC	BCA	CAB	Selection 4
A: The typically nonlinear and large-scale structure characterizing NNs, however, generally complicate their analysis 26, 21. Therefore, available approaches, despite commonly working well in practice, generally come without formal certificates of closed-loop stability and performance. B: More recently, instead...	CAB	CAB	ABC	CBA	Selection 4
A: The existence of solutions for arbitrary large initial data is an open problemB: This uniqueness result seems new for chemotaxis problems in the context of critical spaces, including the classical Keller-Segel system (1.1). This issue has been raised in the context of Navier-Stokes equations (see [20] and so...	ACB	BCA	CAB	CBA	Selection 2
A: Then, using the existence of nondegenerate pairings between these quantum objects, we prove in Subsection 6.3 isomorphisms between the Hopf-cyclic (co)homology of the coordinate algebra of a quantum linear group, and the relative Hopf-cyclic (co)homology of the extended quantum enveloping algebra of the correspo...	BCA	BAC	ABC	CBA	Selection 4
A: The alternative strategy will be followed up elsewhere, both in the present context and in the setting of [17].B: There is a possibility to modify the approach by invoking Neumann-to-Dirichlet maps instead, which would have two advantages: one could consider all rates of vertex volume decay in (iii), and cer...	CAB	BAC	ACB	CBA	Selection 4
A: F] (numerical characterization of k𝑘kitalic_k-Du Bois) and [MP22a, Cor. 3.40] (bounds on dimΣdimensionΣ\dim\Sigmaroman_dim roman_Σ in terms of the relevant numerical invariant). ∎B: The normality claim is [MP22a, Cor. 5.6]C: For hypersurface singularities, various dimension bounds (covering the claim of...	ACB	CAB	CBA	BCA	Selection 2
A: Case 2: large profitB: In this case, agents of type θ=0𝜃0\theta=0italic_θ = 0 would choose to run trials: their expected profit from seeking approval is $40 million. On average, 5% of such agents would receive approval, so many of the approved drugs may actually be ineffective. For example, if there were 20...	ACB	CAB	BAC	CBA	Selection 1
A: Here, we discuss basic properties for the boundary of a gradient Einstein-type manifold by means of γ𝛾\gammaitalic_γB: Our proof follows as in the latter two papers. For the sake of clarity, we have adapted the proof to our case.C: We observe that such properties has been proved in particular cases by ...	CAB	BAC	CBA	ACB	Selection 4
A: In this case, the group operation is the vector additionB: Traditionally, the additive notation is used for the group operation in the theory of mixed lattice groups and semigroups,C: A mixed lattice vector space is defined similarly as a partially ordered vector space V𝑉Vitalic_V with two partial orde...	BCA	BAC	CBA	CAB	Selection 1
A: These results, that are here not mentioned, have a less prominent role in our treatment as they will be deduced from the main structure with which we shall workB: In summary, Chevallier provedC: Chevallier also obtains other results connected to the Lie algebra structure of 𝒮𝒮\mathcal{S}caligraphic_S ...	BCA	ABC	CAB	ABC	Selection 1
A: Both results in Table 1 and in Figure 2b suggest that neural operators perform mapping of finite-banded functions.888One may object that we considered only smooth dataB: This is, indeed, true, but as shown in [De ̵+22] both FNO and DeepONet produce Runge-like oscillations when applied to data with discontin...	BAC	ABC	ACB	BCA	Selection 2
A: Let x:=∑i=1kλi⁢δgi⊗asi∈ℓ1⁢(G)⊗𝒜assign𝑥superscriptsubscript𝑖1𝑘tensor-productsubscript𝜆𝑖subscript𝛿subscript𝑔𝑖subscript𝑎subscript𝑠𝑖tensor-productsuperscriptℓ1𝐺𝒜x:=\sum_{i=1}^{k}\lambda_{i}\delta_{g_{i}}\otimes a_{s_{i}}\in\ell^{1}(G)% \otimes\mathcal{A}italic_x := ∑ start_POSTSUBSCRIPT italic_i = 1 en...	BAC	CBA	BCA	BCA	Selection 1
A: Set m:=γ⁢a⁢p⁢(n−12)assign𝑚𝛾𝑎𝑝binomial𝑛12m:=\gamma ap\binom{n-1}{2}italic_m := italic_γ italic_a italic_p ( FRACOP start_ARG italic_n - 1 end_ARG start_ARG 2 end_ARG )B: Fix a≤λ⁢n𝑎𝜆𝑛a\leq\lambda nitalic_a ≤ italic_λ italic_n, b≤C⁢a𝑏𝐶𝑎b\leq Caitalic_b ≤ italic_C italic_a, and two disjoint sets A𝐴Ai...	CAB	CBA	ABC	BAC	Selection 3
A: A dimer covering or perfect matching is a collection of edges in the graph G𝐺Gitalic_G such that each vertex is covered in exactly one edge. The set of all dimer coverings of G𝐺Gitalic_G will be denoted as ℳ⁢(G)ℳ𝐺\mathcal{M}(G)caligraphic_M ( italic_G ).B: Kasteleyn has shown that every plane graph has a ...	ACB	BCA	BAC	CBA	Selection 4
A: In order to do this first we prove the existence of dlt models (local and global) and the ACC property for log canonical thresholdsB: Note that the ACC for for log canonical thresholds is also proved in [Fuj22a]. C: We will prove termination of flips for effective pairs as in [Bir07]	CAB	BAC	BCA	CAB	Selection 3
A: The first line of research studies offline RL in standard MDPs without any partial observabilityB: The second line of research studies online RL in POMDPs where the actions are specified by history-dependent policies. Thus, the actions does not directly depends on the latent states and thus these works do no...	CAB	BAC	ABC	BCA	Selection 4
A: Bercu2020Efficient designed an online Newton’s method for logistic regression, and Boyer2023asymptotic generalized that method to general regression problemsB: The asymptotics of second-order Newton’s methods for unconstrained problems have recently been investigatedC: Compared to first-order methods th...	CBA	CAB	CAB	BAC	Selection 4
A: Another open problem is the analysis of isoparametric generalized Taylor-Hood families in 2D and 3D to cope with curved boundaries. Perturbation arguments similar to those used in [6], [7] for isogeometric generalized Taylor-Hood families seem to be a promising approach for this open problem.B: The present ...	CBA	CAB	BAC	ABC	Selection 2
A: The detection of columns in B𝐵Bitalic_B at each step can be made with cost O⁢(ln⁡(s)⁢n)𝑂𝑠𝑛O(\ln(s)n)italic_O ( roman_ln ( italic_s ) italic_n )B: This provides a total cost O⁢(ln⁡(s)⁢s⁢n)𝑂𝑠𝑠𝑛O(\ln(s)sn)italic_O ( roman_ln ( italic_s ) italic_s italic_n ).C: O⁢(bi⁢n)𝑂subscript𝑏𝑖𝑛O(b_{i}n)itali...	ACB	BCA	ABC	ACB	Selection 2

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