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**A**: A special thanks is owed to the referee whose exceptionally detailed and patient reading resulted in many corrections and improvements to the article. The author is indebted to Peter Ozsváth for his consistent encouragement and guidance during this project.
**B**: The author would like to thank David Gabai and S... | ABC | CAB | ACB | CBA | Selection 2 |
**A**: I am very grateful to him for many helpful comments and for suggesting the definition of the size of a loose chart in Remark 1.13 (personal communication, December 3, 2020)**B**: I am also grateful to the referee for pointing out gaps in the proofs of the main result and Theorem 1.4 in the original version of th... | CAB | ACB | BAC | BCA | Selection 4 |
**A**:
By [1]delimited-[]1[1][ 1 ] we denote the poset 0≤1010\leq 10 ≤ 1**B**: We are aware that early in this section we also denoted the shift functor on graded objects**C**: Both are standard notation, and from our use of the poset 0≤1010\leq 10 ≤ 1 there is vanishingly little danger of confusing those two. | BCA | CAB | ABC | BAC | Selection 3 |
**A**: The proof relies on the expansion property of expanding Thurston maps**B**: We first demonstrate in the following lemma a mechanism to produce a periodic point**C**: Note that the closure of a flower may not necessarily be simply connected. As a result, it takes extra care to locate periodic points from the comb... | ABC | BAC | CBA | CAB | Selection 2 |
**A**: We start with preliminaries about QEEP, sparse Fourier transformation and compressed sensing in Sec. 2**B**: The rest of the paper is organized as follows**C**: We then introduce our QPE algorithm based on compressed sensing in Sec. 3 and prove several analytical results, including its Heisenberg-limit scaling. ... | CBA | CBA | BAC | ACB | Selection 3 |
**A**: A minimizer of the problem (1.5) exists under mild hypotheses on g𝑔gitalic_g, as we recalled in Lemma 2.2 (and this can be also found in [4, 10]), but, in general, this problem does not necessarily admit a minimizer if considered among the restricted class of the characteristic functions (see e.g. [2, 17] and [... | ACB | CBA | ABC | CAB | Selection 3 |
**A**:
Acknowledgments**B**: We would like to thank Oli Jones, Alice Kerr, Patrick Nairne and our supervisor Alessandro Sisto for helpful discussions and feedback**C**: We also thank the anonymous referee for very helpful comments which greatly improved the readability of the paper. | BAC | ABC | BCA | CAB | Selection 2 |
**A**: Specifically, we use Fourier series to derive inequalities and establish rigidity results. In particular, our method can recover their results.
**B**: Our approach is partly inspired by Shen and Wang [MR4308060], though there are several key differences. Shen and Wang use conformal transformations and compare th... | BAC | CAB | CBA | BCA | Selection 2 |
**A**: Step 3**B**: Conclusion and approximation.
We are in a position to leverage the dissipation and the regularity of the limiting equation to draw our conclusion**C**: Recall the parabolic estimate that for any ρ𝜌\rhoitalic_ρ solving (1.2) and any t>0𝑡0t>0italic_t > 0, | BCA | ABC | CAB | BAC | Selection 2 |
**A**: Theorem 3.4 can also be proved by either appropriately modifying the proof of Proposition 2.6.12 in [2] or by the methods described in Section 6 of [19]**B**: We are grateful to both David Blecher and Hannes Thiel for pointing these references to us**C**: An alternative way is to use
Corollary 4.1.7 in [7]. This... | CAB | BCA | ACB | ABC | Selection 4 |
**A**: Then the proof of Theorem 1.2 is given in Section 5, while the proof of Theorem 1.3 is given in Section 6.**B**: We end the introduction by giving a brief overview of the structure of the rest of the paper**C**: In Section 2, we review some basic facts about representations over local nonarchimedean field, and s... | BAC | BAC | ACB | CAB | Selection 4 |
**A**: There is no active deformation along each link, neither in length nor in curvature, and elongation is produced by new material points appearing at the tips of the filament.
Notice that this does not imply that a new part of the body is generated: for instance, one can consider, as in vine robots, a thin tubular ... | CAB | CAB | CAB | BAC | Selection 4 |
**A**: The inefficiency of the canonicalization-based algorithm is due to the fact that there exists some partition embedded in one vertex of the MPS/comb whose contraction yields a large-sized tensor**B**: As can be seen, the density matrix algorithm outperforms the canonicalization-based algorithm when the number of ... | BAC | CAB | BCA | ACB | Selection 1 |
**A**: P. Neff acknowledges support in the framework of the DFG-Priority Programme 2256 “Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials”, Neff 902/10-1, Project-No**B**:
Patrizio Neff is grateful for the helpful discussions with Michael A. Slawinski (Memorial University of... | BAC | BCA | CAB | CAB | Selection 1 |
**A**: The grading of all of them is supported in C𝐶Citalic_C, thus to understand the braiding, it is enough to consider the C𝐶Citalic_C-action. This way, we can regard them as Yetter-Drinfeld modules over C𝐶Citalic_C.
**B**: The following irreducible Yetter-Drinfeld modules over W𝑊Witalic_W have a finite-dimension... | ABC | CBA | BCA | CAB | Selection 4 |
**A**: One can follow in a similar way as in the proof of Theorem 3.5**B**: Using the convergences given in (3.15), we take limit supremum in (3.12) to find
**C**: In the critical case r=3𝑟3r=3italic_r = 3, note that the operator G(⋅)G⋅\mathrm{G}(\cdot)roman_G ( ⋅ ) is monotone (see (2.34)), and hence we provide a sh... | ACB | BAC | BCA | BCA | Selection 1 |
**A**: This is due to the fact that the DMD produces equation-free solutions, whereas for the POD, for an increasing number of modes larger reduced-order Galerkin projected ordinary differential equations have to be solved.
**B**: The speed-up factors of both DMD algorithms increase more rapidly than of the POD with an... | CAB | ACB | CBA | BCA | Selection 3 |
**A**: Therefore, by dimensionality arguments in this linear setting of matrices**B**: there must exist solutions to this system of equations, if fact there will be a unique solution for every x∈L𝑥𝐿x\in Litalic_x ∈ italic_L.
**C**: It can be easily verified that these two groups of equations are independent and do no... | ABC | BCA | ACB | ACB | Selection 2 |
**A**: Particularized to an unbiased coin as input and a biased coin with parameter τ𝜏\tauitalic_τ as output, the referred algorithm iteratively replaces an interval, initialised as [0,1)01[0,1)[ 0 , 1 ), by either its lower or upper half, as indicated by a random input, until the interval no longer contains τ𝜏\tauit... | BCA | CAB | BCA | BAC | Selection 4 |
**A**: I am grateful to Alessandra Sarti for her constant support, for the supervision of this work and for many useful discussions.
I would like to thank Giovanni Mongardi, Enrico Fatighenti and Simone Novario for many interesting discussions**B**: Acknowledgments: Most of the results of this paper are part of my PhD ... | ABC | CBA | ABC | BAC | Selection 4 |
**A**:
This important section can be interpreted as providing population level versions of our main convergence results for the proposed algorithms presented above**B**: We will discuss the precise meaning of this in what follows, and we also indicate how this correspondence is being used to prove the convergence resu... | ACB | BAC | BAC | BAC | Selection 1 |
**A**: However, to the best of our knowledge, it is not known how to extend those general conjectures to the “quantum” (i.e. general q𝑞qitalic_q) case**B**: Let us note that the q=1𝑞1q=1italic_q = 1 case discussed above is a special case of a very general set of conjectures due to D. Ben-Zvi, Y. Sakellaridis and A. V... | ACB | BAC | CAB | CAB | Selection 2 |
**A**: We will take advantage of this in Section 5 to skip a direct proof of connectedness for our combinatorial HHS.
**B**: In the proof of Theorem 2.18, it is shown that the definition of a combinatorial HHS implies W𝑊Witalic_W is connected; see Sections 1 and 5.2 of [BHMS20]**C**: We emphasis that there is no a pri... | CBA | CAB | BCA | ACB | Selection 1 |
**A**: This action is difficult to describe, but we give an example to demonstrate that it is not the same as our action which is given in Figure 3**B**: First, we review the definitions and known results about parking functions.**C**:
There is another action of the braid group on rooted labeled forests coming from th... | ABC | ACB | ACB | BCA | Selection 4 |
**A**: But in general, given a projection, it may be difficult to determine if a singular curve exists**B**:
If there exists a projection P𝑃Pitalic_P of a virtual knot K𝐾Kitalic_K onto a projection surface S𝑆Sitalic_S that intersects a nontrivial curve exactly once, that is a singular curve**C**: Figure 3 gives an ... | BCA | ACB | ABC | BAC | Selection 4 |
**A**: In their model of noise, every bit in every exchanged message is flipped independently with some probability smaller than 1/2121/21 / 2. As a result, the authors give natural protocols that solve the aforementioned problems efficiently. The work [26] generalizes the above study to opinion sets of any cardinality... | BCA | ACB | CBA | CAB | Selection 4 |
**A**: These spaces are very natural since they appear in the description of boundary strata of the usual spaces of abelian differentials (see e.g**B**:
A central ingredient in our study will be spaces of meromorphic differentials on curves satisfying residue conditions at some of their poles and the intersection numb... | CBA | BAC | ABC | ACB | Selection 2 |
**A**: Another strand of this literature incorporates instrumental variables to deal with unobserved confounding [Carrasco et al., 2007, Singh et al., 2019]**B**: We study a different problem that combines short term experimental data with long term observational data.**C**: We complement a literature that incorporates... | ABC | BCA | BAC | CAB | Selection 2 |
**A**: In this section, we briefly recall the mirror constructions of Gross-Siebert and Keel-Yu in [GS19] and [KY23]**B**: As was mentioned in Remark 3.1, the logarthimic construction given in [GS19] applies to a larger class of log Calabi-Yau targets than the non-archimedean construction given in [KY23]**C**: However,... | CBA | ABC | BCA | BCA | Selection 2 |
**A**: Indeed, since it holds for any metric d∈𝒥(∂X)𝑑𝒥𝑋d\in\mathcal{J}(\partial X)italic_d ∈ caligraphic_J ( ∂ italic_X ) and for any ε>0𝜀0\varepsilon>0italic_ε > 0, we obtain the lower bound of the theorem**B**:
The theorem follows from this claim**C**: We now prove separately each of the two inequalities in th... | ACB | ACB | BAC | CAB | Selection 3 |
**A**: of composable morphisms**B**: of S𝑆Sitalic_S-letters, e.g. (s1,…,sn)subscript𝑠1…subscript𝑠𝑛\left(s_{1},\ldots,s_{n}\right)( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )**C**: The prefix
S𝑆Sitalic_S- is sometimes left out when the considered... | ABC | CAB | CAB | CBA | Selection 1 |
**A**:
First, Theorem 1 applied to SAA establishes an upper bound which is at least as tight as the previous one present in the literature (Mohri and Muñoz Medina, 2012) which involves the uniform deviation (see (7))**B**: While this was sufficient to derive tight guarantees on the rate of the worst-case regret, a nat... | ACB | BAC | BAC | BAC | Selection 1 |
**A**: The heuristics can be further categorised as approximation algorithms, meta-heuristics and hyper-heuristics. The approximation algorithms, like first fit, next fit and worst fit, do not provide optimal solutions but provide guaranteed bounds on the solution**B**:
For solving the BPPs, a variety of algorithms ha... | CAB | BAC | CBA | CAB | Selection 2 |
**A**: This gives rise to a Jacobi-type Newton method, globalized by an Armijo-type linesearch, which guarantees that at the new iterate both players will decrease their predicted objectives; this avoids, as much as possible, convergence to undesired stationary points. On the other hand, the method can also be seen as ... | ACB | ABC | CBA | CAB | Selection 3 |
**A**: The rest of the squares are cartesian either by construction or by the two out of three property for cartesian squares**B**: Finally, we show the commuting square of interest is cartesian by considering the following commuting diagram.**C**:
The top right square is the fine and saturated cartesian square of Dia... | ABC | BCA | CBA | BAC | Selection 2 |
**A**: It would be very interesting to close the gap. In particular, it would be good to know whether there is a simple exponential upper bound, or whether the numbers grow more quickly.
**B**: In Theorem 1.1, we have given a doubly exponential upper bound on the d𝑑ditalic_d-dimensional Ramsey numbers**C**: From below... | ABC | CAB | BAC | ACB | Selection 2 |
**A**: (2022); Rather et al**B**: (2022).
However, multiparticle entanglement offers a complex and rich structure resulting in the impossibility of quantification by means of a single number.**C**: Still, the analysis of AME states is important for understanding quantum error correction and regarded as one of the centr... | ABC | ACB | BCA | CAB | Selection 3 |
**A**: Using this method they constructed an essential, natural, regular uniform algebra on the closed unit disc D¯¯𝐷\overline{D}over¯ start_ARG italic_D end_ARG [12, Theorem 1.2]**B**: Repeating the proof of [12, Theorem 1.2] with the strongly regular uniform algebra of Theorem 1.2 above in place of McKissick’s norma... | ACB | BAC | ACB | BCA | Selection 4 |
**A**: This is in contrast to the RCDT-POD, which retains more of the shape of the original when only five modes are used for reconstruction, although affected by the intrinsic RCDT error. The result improves slightly for ten modes, but most of the shape is captured in the first few modes, unlike in the case of physica... | CAB | ACB | CBA | BCA | Selection 3 |
**A**: The key idea of this generalization is to select columns from two matrices that share the same column indices in order to extract the features of one dataset that are most pertinent to the other**B**: The MCUR approximation of triplet matrices was also presented in [3] , with the goal of identifying pertinent ch... | CAB | CAB | ACB | BCA | Selection 3 |
**A**: On the other hand, out-splits for topological graphs correspond to a factorisation of the source map, which defines the right-module structure of the graph correspondence. This makes the noncommutative analogy for out-splits more difficult to pin down than in the case of in-splits.
**B**: In-splits for topologic... | ABC | CAB | ABC | BCA | Selection 2 |
**A**: (See, for example, [GoldbringHart23, Section 4].) Thus, if all free group factors have the same first-order theory, then this might serve as evidence as to why the free group factor isomorphism problem is so difficult. Of course, if some free group factors have distinct first-order theories, then this would prov... | ACB | BAC | BAC | CBA | Selection 4 |
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