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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 , ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) **B**: sansserif_inl ( italic_τ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) , ( italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) **C**: sansserif_inr ( italic_τ start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) ) holds under the appropriate context | 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 Furstenberg correspondence principle between combinatorial problems and ergodic theory**C**: See [12] for the original application to Szemerédi’s theorem, or [56] for a more recent application to Gowers norms over finite fields | 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 case is that of surfaces of infinite type**C**:
In this paper we are concerned with n=2𝑛2n=2italic_n = 2, i.e. surfaces | 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 for elliptic equations which were solved in [62].
We are concerned with the stability of the inverse problem. | 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 transform the diagonal matrix with the help of row and column operations into the identity matrix in a way similar to an algorithm to compute the elementary divisors of an integer matrix, as described for example in [23, Chapter 7, Section 3] | 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,(\lambda_{i}),F_{1-n})italic_S start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_n , italic_m , ( italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , italic_F start_POSTSUBSCRIPT 1 - italic_n end_POSTSUBSCRIPT ). Our computations provide an | 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 homomorphisms can be developed in a categorical setting**C**: To motivate our discussion, let us review the classical case of Chow groups.
| 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 Siegel disks stated previously are all based on surgery: either quasiconformal or trans-quasiconformal | 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,**C**:
The paper is organized as follows | 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 arbitrarily into ‘boxes’ labelled with the available markings**C**: Within a row, tiles are unordered | 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 paper | 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 collect a few useful facts about ordered abelian groups. In Section 3 we apply Schmitt’s characterization of lexicographic sums of ordered archimedian groups to characterize groups with finite spines. | 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 currently in print where a full proof of the regularity of Diff(G)Diff𝐺\operatorname{Diff}(G)roman_Diff ( italic_G ) for the non-compact G𝐺Gitalic_G in our setting can be found**C**: See however [Nee06, Theorem III.4.1.] and [Glö15, Corollary 13.7] where an unpublished preprint by H. Glöckner which paved the way for the treatment in [Sch15] is referenced. Further, we refer to [Mic83, 4.6] for related considerations. | 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 for the invertibility of a Dirac operator and hence the triviality of the index is the positivity of the zero-order term in the Weizenboeck formula, e.g., the scalar curvature of the manifold in the case of the spin Dirac operator | 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 regularity [MR2801210, MR2753343, MR3225627, MR3177856, MR2861254]
but are based on ideas that differ considerably from what we advocate here**C**: It is hard to approximate such problem in its full generality using numerical methods, in particular because of the low regularity of the solution and its multiscale behavior | 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 this let us find a Clifford module for C(M)𝐶𝑀C(M)italic_C ( italic_M ) | 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 Rognes for conversations surrounding the material in this paper | 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 dense subset) in the original K𝐾Kitalic_K-groups (the both groups are suitably topologized) | 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 [25] needed for the rest of the article | 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 these risk measures can be applied to financial markets can be carried on.**B**: Through further refinement of the axioms associated with these risk measures, we derived their dual representations**C**:
In this paper, we introduced risk measures on a unique variable exponent Bochner–Lebesgue space denoted as Lp(⋅)superscript𝐿𝑝⋅L^{p(\cdot)}italic_L start_POSTSUPERSCRIPT italic_p ( ⋅ ) end_POSTSUPERSCRIPT and then explored dynamic and cash sub-additive risk measures in this space | 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 highest weight representations of twisted affine Kac-Moody algebras over a base | 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 sphere (recall Fig | 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 representation category to the category of coherent sheaves.
| 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 out within the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ’5-100’**C**: We thank the referees for the careful reading of the first version of the text and for many helpful remarks | 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 have disjoint tubes**C**: In particular, it suffices to change tubes one at a time. | 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 Theorem 2.14 the family version of Theorem 2.12, which did not appear in [26]. | 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_POSTSUPERSCRIPT entropy solutions is given in B, and further comments on our duality results are postponed to C and D.**C**:
The rest of this paper is organized as follows | 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 variable and constant coefficient settings in n≥3𝑛3n\geq 3italic_n ≥ 3, see [1, 2, 9] for more details. This work provides additional methods and techniques toward handling the variable coefficient case which may help advance the research in this direction.
**C**: As a result, the corresponding local smoothing conjecture is resolved | 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.4.9] gives a different definition of a predictable stopping time | 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 pseudoholomorphic “buildings”, consisting of various curves in different symplectic cobordisms whose asymptotic Reeb orbits agree end-to-end to form a chain.
| 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 generators of random walks on graphs and their corresponding Laplace-Beltrami operator have been utilized in Machine Learning, as evidenced by belkin2003laplacian , roweis2000nonlinear , tenenbaum2000global , and van2008visualizing **C**: In joharinad2023mathematical , Jost and Joharinad demonstrated that the convergence properties of random walks enable the investigation of approximating a Riemannian manifold by graphs | 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 end_ID start_ID over¯ start_ARG italic_B end_ARG end_ID ) = italic_b roman_N ( italic_B italic_u ) = italic_b italic_B start_ID over¯ start_ARG italic_B end_ARG end_ID , end_CELL end_ROW start_ROW start_CELL italic_c italic_C start_ID over¯ start_ARG italic_C end_ARG end_ID end_CELL start_CELL ↦ end_CELL start_CELL italic_c ( start_ID over¯ start_ARG italic_u end_ARG end_ID italic_C start_ID over¯ start_ARG italic_u end_ARG end_ID ) ( italic_u start_ID over¯ start_ARG italic_C end_ARG end_ID italic_u ) = italic_c roman_N ( start_ID over¯ start_ARG italic_u end_ARG end_ID italic_C start_ID over¯ start_ARG italic_u end_ARG end_ID ) = italic_c roman_N ( italic_C ) **B**: }\,\overline{\!u\!\!\;}\!\>}\mbox{})=c\mathord{\mathrm{N}}(C).\mathord{\mathrm%
{N}}(u)^{2}=cC\mathord{\mbox{}\,\overline{\!C\!\!\;}\!\>}\mbox{},\end{array}start_ARRAY start_ROW start_CELL italic_a italic_b italic_c end_CELL start_CELL ↦ end_CELL start_CELL italic_a italic_b italic_c , end_CELL end_ROW start_ROW start_CELL italic_a italic_A start_ID over¯ start_ARG italic_A end_ARG end_ID end_CELL start_CELL ↦ end_CELL start_CELL italic_a ( italic_u italic_A ) ( start_ID over¯ start_ARG italic_A end_ARG end_ID start_ID over¯ start_ARG italic_u end_ARG end_ID ) = italic_a italic_N ( italic_u italic_A ) = italic_a roman_N ( italic_A ) **C**: roman_N ( italic_u ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_c italic_C start_ID over¯ start_ARG italic_C end_ARG end_ID , end_CELL end_ROW end_ARRAY | ABC | CBA | BAC | CBA | Selection 3 |
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