Split (1)

train · 34 rows

shuffled_text stringlengths 286 1.81k	A stringclasses 6 values	B stringclasses 6 values	C stringclasses 6 values	D stringclasses 6 values	label stringclasses 4 values
A: \prime}}(y_{1})\,)\,,\,(y_{2}).\,\mathsf{inr}(\,\tau_{D}^{D^{\prime}}(y_{2})\,% )\,)italic_τ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ( italic_y ) = start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_El start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_y...	BAC	ABC	ACB	BAC	Selection 2
A: Here we use a somewhat different type of limit object, namely an ultralimit. We are certainly not the first to employ ultralimits (a.k.a. nonstandard analysis) in additive number theory; see for example [40]. B: However there is a very notable exception, namely the portion of the literature that exploits the...	ABC	CBA	CBA	CAB	Selection 4
A: These have been studied at least since the 1970s since they appear as leaves of foliations on compact 3-manifolds (see e.g. [25, 9, 11].) They are also studied from the point of view of Teichmüller theory (see e.g. [14, 19] and the references therein) and translation surfaces [26].B: For us, the interesting ...	CBA	ABC	ABC	CAB	Selection 1
A: Generalizations and alternative methods to solve the problem have been studied in [3, 17, 42, 47, 49], and the related determination of smooth structure was recently studied in [22, 23]B: [1, 13, 50]C: On a given domain of the Euclidean space, the problem can be reduced to inverse coefficient problems fo...	BCA	BAC	BCA	ABC	Selection 2
A: Note that row and column operations are effected by left- and right multiplications by transvectionsB: Thus recording the row and and column operations required to transform a diagonal matrix into the identity, allows us to write the input matrix as a product of transvections. C: The key idea is to tran...	CBA	CBA	BCA	CAB	Selection 3
A: propertiesB: Among others, the Jacobian [P,Q]∉K×𝑃𝑄superscript𝐾[P,Q]\notin K^{\times}[ italic_P , italic_Q ] ∉ italic_K start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPTC: Due to this fact we must use a slight variation of the system St⁢(n,m,(λi),F1−n)subscript𝑆𝑡𝑛𝑚subscript𝜆𝑖subscript𝐹1𝑛S_{t}(n,m,(\l...	ACB	CBA	ABC	ACB	Selection 3
A: The regular homomorphism in this paper is a generalization of what Samuel called a rational homomorphism in the context of Chow groups ([Sam58, Section 2.5]; see also [Mur85, Definition 1.6.1])B: Although the interest of this paper is motivic cohomology, the formalism of algebraic part and regular homomorphi...	BAC	ABC	ABC	ACB	Selection 1
A: In these proofs, some pre-models, and usually, a single or a family of Blaschke products are neededB: By surgery, the regularity and the existence of critical points on the boundaries of Siegel disks were proved at the same time. C: The proofs of the regularity results for the bounded type and PZ type Si...	BCA	ABC	BAC	ACB	Selection 1
A: Section 2 contains the necessary background materialB: In Section 3 we prove the local index theorem for families of ∂¯¯\bar{\partial}over¯ start_ARG ∂ end_ARG-operators on Riemann orbisurfaces that are factors of the hyperbolic plane by the action of finitely generated cofinite Fuchsian groups. Specifically...	BAC	BAC	BCA	ABC	Selection 3
A: A straightforward calculation shows that the relations in (5.1) holdB: A simple computation shows thatC: Since we have four families of cyclic vectors (indexed by i𝑖iitalic_i and j𝑗jitalic_j), we have four sets of coherent states (indexed by i𝑖iitalic_i and j𝑗jitalic_j)	BCA	ACB	CBA	BCA	Selection 2
A: The binomial coefficient (14) accounts for the remaining factors in the formula. The factors involving l𝑙litalic_l and n𝑛nitalic_n reduce to 1111 if the corresponding eigenspace exists but is unmarked (l𝑙litalic_l or n𝑛nitalic_n =0absent0=0= 0). ∎B: For counting purposes, we view them as ‘balls’ placed a...	BCA	BCA	BCA	CBA	Selection 4
A: The paper is structured as followsB: The first syzygy of Hibi rings is discussed in Section 3. Explicit expression for the first Betti number for planar distributive lattices has been discussed in Section 4. C: In Section 2, we collect basic notations, terminology, and results that will be used in the pa...	ABC	BAC	BCA	ACB	Selection 4
A: The spines of an ordered abelian group, in the terminology of [28], are (interpretable) coloured linear orders determining the first order theory of the groupB: To the best of our knowledge, no systematic study of ordered abelian groups with finite spines has been carried out beforeC: In Section 2, we c...	ACB	ACB	BAC	ABC	Selection 4
A: For G𝐺Gitalic_G compact, a proof of this can be found in [Mil84, p. 1046]B: It is also true in the non-compact case as a special version of [Sch15, Section 5.4] (in particular [Sch15, Proof of Theorem 5.4.11]) where the case of non-compact orbifolds is treated.888To our knowledge [Sch15] is the only source...	CAB	BCA	CAB	ABC	Selection 4
A: The precise formulation of this fact is due to Roe [Roe16] and leads to the notion of the coarse index class of a Dirac operator with support.B: If this term is positive on some subset of the manifold only, then the index information is supported on the complement of that subsetC: One possible reason fo...	CBA	ACB	ACB	ABC	Selection 1
A: Most convergent proofs either assume extra regularity or special properties of the coefficients [AHPV, MR3050916, MR2306414, MR1286212, babuos85, MR1979846, MR2058933, HMV, MR1642758, MR3584539, MR2030161, MR2383203, vs1, vs2, MR2740478]B: Some methods work even considering that the solution has low regulari...	BAC	CAB	CAB	BCA	Selection 4
A: Consider again any finiteB: canonical embedding Ωc2⁢(M;ℂ)⊂C⁢(M)subscriptsuperscriptΩ2𝑐𝑀ℂ𝐶𝑀\Omega^{2}_{c}(M;{\mathbb{C}})\subset C(M)roman_Ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_M ; blackboard_C ) ⊂ italic_C ( italic_M )C: In addition to t...	BAC	CBA	CAB	ABC	Selection 3
A: I would also like to thank the anonymous referees for their many corrections and suggestionsB: In particular for a suggested simplification of the proof of Lemma 4.2.C: I would like to thank Mohammed Abouzaid, Marcel Bökstedt, Sylvain Courte, Tobias Ekholm, Yasha Eliashberg, Søren Galatius, and John Rog...	BAC	ACB	BCA	ACB	Selection 3
A: The author does not try to complete this point as he believes it not essential. B: Finally, a word of caution is in orderC: The equivariant K𝐾Kitalic_K-groups dealt in this paper are not identical to these dealt in [47] and [37] in the sense that both groups are just dense subset (or intersects with a d...	ACB	ACB	BAC	CAB	Selection 4
A: For full details, and proofs of the statements made here we refer the reader to [25, Sections 2-4]B: For additional general background on algebraic topology, and topological methods in group theory, we refer the reader to [48] and [24]. C: In this section we recall some of the relevant background from [2...	ABC	ABC	BCA	ACB	Selection 3
A: Additionally, we delved into the investigation of the optimized certainty equivalent on variable exponent Bochner–Lebesgue spaces as an illustrative example. The study of this paper gives a new set of risk measures to capture the fluctuation of volatility of financial markets and further investigation on how the...	ACB	CBA	BCA	BAC	Selection 2
A: We also prove that the Sugawara operators acting on the integrable highest weight representations of twisted affine Kac-Moody algebras are independent of the parameters up to scalarsB: This section is preparatory for Section 7.C: In Section 6, we prove the independence of parameters for integrable highe...	CBA	BCA	ABC	CBA	Selection 2
A: 8 and the punctured sphere picture in Appendix B) and use the topological gluing to obtain the bases on any Riemann surfaces.B: The topological operations also work in the cases with Chiral topological orders with gapless boundariesC: In such cases, one can start with the basis of real anyons on a spher...	ACB	CAB	BCA	ABC	Selection 2
A: These are analog and enhancement of the known results in [DM82, EHS08]B: In this section we shall present some applications of Tannakian duality to study homomorphisms between group schemes over a fieldC: At the end, we recall the construction of the torsor associated to a tensor functor from a represent...	BCA	CBA	CBA	BAC	Selection 4
A: The proof is by induction on β𝛽\betaitalic_βB: We only need to argue for the conclusion in the case that β𝛽\betaitalic_β is a nonzero limit ordinalC: In that case the conclusion follows easily from the induction hypothesis and the fact that for every α<β𝛼𝛽\alpha<\betaitalic_α < italic_β,	ABC	BAC	CAB	BAC	Selection 1
A: The results of Section 4 has been obtained under support of the RSF grant 19-11-00056. The work of both authors has also been supported in part by the Simons Foundation. The first author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury. B: This research was carried ou...	CBA	ABC	CAB	BAC	Selection 1
A: Any tubing of S𝑆Sitalic_S is isotopic to a tubing where each tube has one foot on the component of S𝑆Sitalic_S containing K𝐾Kitalic_K, and one foot on a closed component of S𝑆Sitalic_SB: Since tubes are boundaries of 3-dimensional 1-handles, we may assume that, after an isotopy, any two such tubings hav...	CAB	ABC	CAB	CBA	Selection 2
A: After our paper appeared on the arXiv, similar results as in Theorem 2.12 were also obtained in [26], see Theorem 1.1B: In [26] the authors also considered the generic fiber of so-called weak parabolic fibrations, in which the residue of the Higgs field is not required to be nilpotentC: We will prove Th...	BAC	ABC	CAB	BCA	Selection 2
A: We recall basic facts in Section 2, we state our main results in Section 3, and prove them in Section 4B: For completeness, some results for minimal discontinuous viscosity solutions are proved in A, a complete proof of well-posedness for L∞superscript𝐿L^{\infty}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPE...	CBA	ACB	BCA	ABC	Selection 3
A: Very recently, Guth-Wang-Zhang [10] established the sharp square function estimate in the Euclidean case in 2+1212+12 + 1 dimensionsB: For the variable coefficient setting, the Kakeya compression phenomena will happen which leads to the difference in the numerology of local smoothing conjecture between the v...	CAB	ABC	ACB	CBA	Selection 3
A: Let us point out that in [66, Definition IV.5.4] a process which we call accessible is called predictableB: Fortunately, according to [49, Theorem I.6.6], τ𝜏\tauitalic_τ is a predictable stopping time according to [49, Definition I.4.9] if and only ifC: On the other hand, Metivier in [49, Definition I....	ABC	CBA	ACB	BAC	Selection 3
A: The SFT compactness theorem extends Gromov’s compactness theorem to punctured curves in symplectic cobordismsB: As in the case of closed curves in closed symplectic manifolds, we must add nodal configurations à la Kontsevich’s stable map compactificationC: However, we must additionally allow multi-level ...	ABC	BCA	BCA	BCA	Selection 1
A: Additionally, chung2007four employed random walks to establish Cheeger’s inequality, which was further utilized in shi2000normalized for studies in Image Segmentation. It remains an open and unexplored question whether the techniques developed herein can be applied in a similar direction. B: The discrete g...	BCA	CBA	ABC	CAB	Selection 2
A: roman_N ( italic_u ) = italic_a italic_A start_ID over¯ start_ARG italic_A end_ARG end_ID , end_CELL end_ROW start_ROW start_CELL italic_b italic_B start_ID over¯ start_ARG italic_B end_ARG end_ID end_CELL start_CELL ↦ end_CELL start_CELL italic_b ( italic_B italic_u ) ( start_ID over¯ start_ARG italic_u end_ARG...	ABC	CBA	BAC	CBA	Selection 3

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