robench-2024b
Collection
48 items • Updated
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**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 transvections**B**: 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**: properties**B**: Among others, the Jacobian [P,Q]∉K×𝑃𝑄superscript𝐾[P,Q]\notin K^{\times}[ italic_P , italic_Q ] ∉ italic_K start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT**C**: 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 needed**B**: 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 material**B**: 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) hold**B**: A simple computation shows that**C**: 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 follows**B**: 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 group**B**: To the best of our knowledge, no systematic study of ordered abelian groups with finite spines has been carried out before**C**: 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 subset**C**:
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 finite**B**: 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 suggestions**B**: 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 order**C**: 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 scalars**B**: 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 boundaries**C**: 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 field**C**: 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_β**B**: We only need to argue for the conclusion in the case that β𝛽\betaitalic_β is a nonzero limit ordinal**C**: 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_S**B**: 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 1.1**B**: 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 nilpotent**C**: 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 4**B**: 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 dimensions**B**: 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 predictable**B**: 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 if**C**: 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 cobordisms**B**: As in the case of closed curves in closed symplectic manifolds, we must add nodal configurations à la Kontsevich’s stable map compactification**C**: 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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