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<|MaskedSetence|> Techniques for deciding (2) are harder to come by. <|MaskedSetence|> On the other hand, left-orderability is not known to be decidable for three-manifold groups. <|MaskedSetence|> Here are a few practical methods for proving the left-orderability of a 3-manifold group:
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Technology for deciding conditions (1) and (3) is well developed; for example, Dunfield verified the equivalence of (1) and (3) for 99.8% of the manifolds in his census of ∼300,000similar-toabsent300000{\sim}300,000∼ 300 , 000 small hyperbolic rational homology spheres [Dun19].
**B**: See Calegari and Dunfiel... | ACB | ABC | ACB | ACB | Selection 3 |
Considering the recent breakthroughs on maximizing measures and ground states in uniformly expanding and uniformly hyperbolic settings mentioned above, it is natural to consider the properties of these measures for generic potentials in non-uniformly expanding settings.
One primary source of non-uniform expansion in ... | **A**: To limit the complexity, we assume that all critical points are preperiodic (i.e., have finite forward orbits) in this paper.
**B**: For recent works related to Lattès maps in the context of Thurston maps in particular, see [LZ18, BM20, BHI21].
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**C**: These rational maps are called the postcritically-finite ... | ACB | ACB | ACB | ABC | Selection 3 |
The goal of this paper is to study the question of existence of optimal sets, that is, minimizers of (1.4). <|MaskedSetence|> Indeed, as said above, relaxation arguments allow to deduce that a minimizing set exists if and only if there is a function minimizing (1.5) which is a characteristic function, and this is, o... | **A**: First of all, we underline that existence should not be expected in general.
**B**: As a matter of fact, in all the results where existence of optimal sets is established, as in the ones described above, the optimal sets are actually balls, and there is not really an argument which provides existence, but rathe... | ABC | ABC | ABC | ABC | Selection 2 |
<|MaskedSetence|> Shen and Wang use conformal transformations and compare the boundary integral (1.4) of the underlying domain with that of the punctured disk. <|MaskedSetence|> <|MaskedSetence|> Specifically, we use Fourier series to derive inequalities and establish rigidity results. In particular, our method can ... | **A**:
For CCE manifolds in dimension 4444, there is also some interest in studying specific compactifications; refer to [MR1909634, MR3886176, MR4130465, MR4727591] for an incomplete list.
Our approach is partly inspired by Shen and Wang [MR4308060], though there are several key differences.
**B**: Subsequently, t... | ABC | ABC | ABC | CBA | Selection 1 |
<|MaskedSetence|> It is complemented with the proper boundary conditions corresponding to the behaviour of particles at the boundary. The boundary condition in (1.1) is prescribed as zero influx when α=β=0𝛼𝛽0\alpha=\beta=0italic_α = italic_β = 0, and it is generally governed by the balance relations between the dist... | **A**: The operator ℒ1subscriptℒ1\mathcal{L}_{1}caligraphic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT serves as a prototype for describing collision processes of linear relaxation type, including the case of neutron transport.
**B**: This regime with ε→0→𝜀0\varepsilon\rightarrow 0italic_ε → 0 is often referred to as ... | CAB | CAB | CAB | BAC | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> In particular we will work with G=ℤ/3ℤ𝐺ℤ3ℤG=\mathbb{Z}/3\mathbb{Z}italic_G = blackboard_Z / 3 blackboard_Z. The augmentation ideal of a group ring is a well known algebraic object (see [20, Section 11.2], [8, Example 7, p. 245] for instance) and is easily generalized to operator ... | **A**: We include details here for completeness.
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**B**: We will now construct, for certain values of p𝑝pitalic_p, a class of Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-operator algebras
that have no cai, cannot be nondegenerately represented (on any Banach space),
and whose mu... | ABC | BCA | BCA | BCA | Selection 2 |
be the spherical Hecke algebra away from 𝚃𝚃\mathtt{T}typewriter_T, where 𝔬F¯vsubscript𝔬subscript¯𝐹𝑣\mathfrak{o}_{\underline{F}_{v}}fraktur_o start_POSTSUBSCRIPT under¯ start_ARG italic_F end_ARG start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_POSTSUBSCRIPT is the ring of integers of F¯vsubscript¯𝐹𝑣\underlin... | **A**: Put ℳv′+=ℳv+⊗ℳv+superscriptsubscriptℳ𝑣′tensor-productsuperscriptsubscriptℳ𝑣superscriptsubscriptℳ𝑣\mathcal{M}_{v}^{\prime+}=\mathcal{M}_{v}^{+}\otimes\mathcal{M}_{v}^{+}caligraphic_M start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ + end_POSTSUPERSCRIPT = caligraphic_M start_POSTSUBSCRIP... | ABC | BAC | BAC | BAC | Selection 4 |
In this paper, we propose four models for an enhanced two-link swimmer to achieve controllability, thus overcoming the Scallop Theorem in the context of microswimming.
To present our results, we discuss different concepts of controllability, such as total controllability (to prescribe the initial and final positions an... | **A**: Moreover, on a more technical perspective, gait controllability is usually the gateway through which the stronger total controllability is proven, making an independent notion even more useful.
Classical models of slender microswimmers are based on the capability to control curvature or angles along the swimme... | ACB | CAB | CAB | CAB | Selection 2 |
Note that the generation of the embedding tree only depends on the tensor network graph structure, rather than the actual tensor data.
The relationship between the embedding tree and the orderings of the edges is further explained in Section 4.3.
While our method introduces a novel perspective on the construction of ... | **A**: For instance, Seitz et al.
**B**: [54], Szalay et al.
**C**: Other studies, such as those by Nakatani and Chan [55], Murg et al.
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<|MaskedSetence|> Let us consider any F1,F2∈ℱ|U|∩ConFrLsubscript𝐹1subscript𝐹2subscriptℱ𝑈ConFr𝐿F_{1},\,F_{2}\in\mathcal{F}_{|U|}\cap\operatorname{Con}\operatorname{Fr}Litalic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ caligraphic_F start_POSTSUBSCRIPT | italic_U... | **A**: Moreover, G⊧Lmodels𝐺𝐿G\models Litalic_G ⊧ italic_L implies that G↑([n]∖U)⊧L,↑𝐺delimited-[]𝑛𝑈models𝐿G{\uparrow}([n]\setminus U)\models L,italic_G ↑ ( [ italic_n ] ∖ italic_U ) ⊧ italic_L , so G′⊧Lmodelssuperscript𝐺′𝐿G^{\prime}\models Litalic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊧ italic_L since G... | CBA | CBA | CBA | CBA | Selection 2 |
The rest of the paper is organized as follows. In the next section, we discuss the functional setting of the problem described in (1.1). After defining necessary function spaces, we define the linear and nonlinear operators and show that these operators satisfy a monotonicity property for r≥3𝑟3r\geq 3italic_r ≥ 3 (see... | **A**: After providing an abstract formulation of the system (1.1), the existence and uniqueness of a global weak solution in the Leray-Hopf sense is examined in Section 3.
**B**: Furthermore, we show that the sum of linear and nonlinear operators satisfy demicontinuous property also (Lemma 2.8).
**C**: Due to techni... | BAC | CAB | BAC | BAC | Selection 3 |
<|MaskedSetence|> There is no such criteria for ranking the contributions of the different DMD modes. <|MaskedSetence|> <|MaskedSetence|> The amplitude criterion is also not perfect because there exist modes with very high amplitudes but which are very fast damped. The selection based on frequency/growth rate has al... | **A**:
In POD the modes are ranked by energy level through the POD singular values.
**B**: Different criteria are developed depending on what can be considered important for the models used.
**C**: The DMD modes can then be selected based on their amplitude or based on their frequency/growth rate.
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In his paper [4], F. Forstneric showed that every closed oriented 3-manifold can be embedded totally real into ℂ3superscriptℂ3\mathbb{C}^{3}blackboard_C start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. <|MaskedSetence|> <|MaskedSetence|> In this situation, all complex tangents are necessarily of dimension 1111 and gener... | **A**: In 2018, Kasuya and Takase proved that given any smooth link L𝐿Litalic_L in a closed 3-manifold M𝑀Mitalic_M with trivial fundamental class [L]∈H1(M)delimited-[]𝐿subscript𝐻1𝑀[L]\in H_{1}(M)[ italic_L ] ∈ italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_M ), there exists a smooth embedding M↪ℂ3↪𝑀su... | CBA | CBA | CBA | CBA | Selection 1 |
This paper describes a general algorithm that solves the above problem when there exists a representation of τ𝜏\tauitalic_τ as a series with rational terms. The algorithm is described in §2, and its basic properties are addressed. <|MaskedSetence|> Application to specific values, including Euler’s constant γ𝛾\gamm... | **A**: Conclusions and future work are presented in §5.
**B**: Proofs to all results are given in §6.
.
**C**: The complexity of the algorithm is analysed in §3.
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The literature on the estimation of low-dimensional structures (or filaments) is rich, and different approaches use different geometric ideas. For example, the local principal curves (Einbeck et al., 2005, 2008) are formed by tracking the localized version of the first principal component directions, but the method re... | **A**: Genovese et al.
**B**: (2006) can be used to detect the presence of a single filament in a noise background.
**C**: The candy model (Stoica et al., 2007) uses possibly connected cylinders (in 3D) of a fixed radius and
height to represent the filaments.
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1.3. What is done in this paper?
In this paper we deal with the case M=N−1𝑀𝑁1M=N-1italic_M = italic_N - 1 for generic q𝑞qitalic_q. <|MaskedSetence|> There we consider (among other things) the case q=1𝑞1q=1italic_q = 1. <|MaskedSetence|> <|MaskedSetence|> | **A**: The advantage of the M=N−1𝑀𝑁1M=N-1italic_M = italic_N - 1 assumption is that in this case the group UM,Nsubscript𝑈𝑀𝑁U_{M,N}italic_U start_POSTSUBSCRIPT italic_M , italic_N end_POSTSUBSCRIPT is
trivial, so GL(M,𝐎)⋉UM,N(𝐅)GLleft-normal-factor-semidirect-product𝑀𝐎subscript𝑈𝑀𝑁𝐅{\mathop{\operatorname{\r... | ACB | BCA | ACB | ACB | Selection 1 |
At first, this seems to present an issue with composing genus one and genus two (or more) knots. <|MaskedSetence|> <|MaskedSetence|> Then we take half of this double to compose, and therefore everything being composed is boundary compatible in that both pieces include their boundary. Since we can also consider singul... | **A**: For any genus one knot in a nonsingular composition, we first remove its cork, and then double it over what remains of the cork boundary to obtain a genus two surface that includes its boundary.
**B**: This is why Theorem 4.2 is stated to exclude the composition of a genus one knot with a genus at least two kno... | CAB | BCA | CAB | CAB | Selection 1 |
In [49], the authors show that, in the voter model, the presence of stubborn agents with opposite opinions precludes the convergence to consensus. The work [42] studies the asynchronous voter rule and the asynchronous majority rule dynamics with Poisson clocks when the opinion set is binary.
The authors use mean-field ... | **A**: Otherwise, either no agreement is possible, or the process converges to an agreement towards a single opinion, which is that of the largest stubborn community.
**B**: If the two sizes are close to each other and not too large, then agreement on both opinions is possible.
**C**: In the second, there are stubbor... | CBA | ACB | CBA | CBA | Selection 1 |
6.1. Top-ψ𝜓\psiitalic_ψ for strata with residue conditions
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 numbers of their fundamental classes with powers of ψ𝜓\psiitalic_ψ-classes. <|MaskedSetence|>... | **A**: [BCG+19b]).
**B**: Moreover, their fundamental classes have many nice properties and explicit descriptions.
**C**: These spaces are very natural since they appear in the description of boundary strata of the usual spaces of abelian differentials (see e.g.
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(3)
The eigenvalues decay at least polynomially. <|MaskedSetence|> A higher value of b𝑏bitalic_b corresponds to a lower effective dimension, better control of the variance of our estimator, and hence a faster rate. The limit b→∞→𝑏b\rightarrow\inftyitalic_b → ∞ gives an RKHS with finite dimension [Caponnetto and De... | **A**: Figure 2 verifies polynomial decay of the empirical eigenvalues in the real world application of Section 6; the Project STAR data have a low effective dimension as required by Assumptions 5.2 and 5.3.222Specifically, we divide each empirical eigenvalue by the trace of the corresponding matrix, to convey the frac... | BCA | BCA | ACB | BCA | Selection 2 |
<|MaskedSetence|> Related Work
Mandel in [Man21] has previously proven the full Frobenius structure conjecture for all cluster varieties. <|MaskedSetence|> <|MaskedSetence|> Instead of assuming a toric model, we use the maximal degeneracy assumption on the boundary to facilitate the connection with tropical geometr... | **A**:
1.1.
**B**: This allows for tropical curve counting techniques to be utilized in the study of the enumerative geometry of cluster varieties.
**C**: In these situations, the target geometries can be tamed by considering toric models associated with various seeds of the cluster variety, corresponding to differe... | ACB | ACB | ACB | ACB | Selection 2 |
the notion of class, which measures the complexity of normalising
length-three words. <|MaskedSetence|> 21[m]21delimited-[]𝑚21\left[m\right]21 [ italic_m ]) for the alternating sequence 121…121…121\ldots121 …
(resp. <|MaskedSetence|> <|MaskedSetence|> right-class. | **A**: A quadratic normalisation (A,N)𝐴𝑁\left(A,N\right)( italic_A , italic_N )
is said to be of left-class m𝑚mitalic_m (resp.
**B**: 212…212…212\ldots212 …) of length m𝑚mitalic_m.
**C**: For a nonnegative integer m𝑚mitalic_m, we write 12[m]12delimited-[]𝑚12\left[m\right]12 [ italic_m ]
(resp.
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<|MaskedSetence|> <|MaskedSetence|> We discuss in more details connections between our work and previous contamination models in Section A.2.3.
Closest to us are Brustle et al. (2020); Guo et al. (2021) who study optimal auctions. <|MaskedSetence|> The framework we develop is anchored around a setting that allows ... | **A**: (Blanc et al., 2021) establishes that adaptive adversaries yield equivalent performance to oblivious ones in many settings.
We note that our framework is general enough to derive guarantees on the regret under the ϵitalic-ϵ\epsilonitalic_ϵ-contamination model and the strong contamination model.
**B**: They cons... | BAC | BAC | CAB | BAC | Selection 1 |
Due to a long manual setup time, the company lacks the flexibility to introduce new PCBs in the build process as well as in responding to unprecedented situations, like COVID-19, for which the company may have to reschedule the build process according to the changed demands. Moreover, the manual setup also leads to a s... | **A**: The components can be categorised into two categories, (i) the components which need one to five slots on a container for loading different PCB components onto the CAP machine, and are put on the trolleys, and (ii) the components needing more than five slots on the container and are loaded onto the stackers.
**... | ACB | ACB | ABC | ACB | Selection 4 |
We note that the Armijo-type conditions (13) and (16) can be guaranteed to be satisfied at a small enough t𝑡titalic_t without requiring local Lipschitz continuity of the gradients. <|MaskedSetence|> <|MaskedSetence|> This is analogous to what is known in traditional optimization. On the other hand, conditions (14),... | **A**: Therefore, the analysis is more complex, being necessary to test conditions (14), (17), (15), and (18) inside the algorithm, since they are only guaranteed to be fulfilled when t𝑡titalic_t is sufficiently small.
Next, we discuss the case when the iterates all lie on a bounded set and the problem is sufficient... | CBA | BAC | CBA | CBA | Selection 3 |
For many important applications entanglement has been proven to be a powerful resource. An example of resourceful states are the absolutely maximally entangled (AME) states which maximized the entanglement in the bipartitions, but are notoriously difficult to characterize Scott (2004); Facchi et al. <|MaskedSetence|... | **A**: (2022).
However, multiparticle entanglement offers a complex and rich structure resulting in the impossibility of quantification by means of a single number..
**B**: (2008); Reuvers (2018); Gour and Wallach (2010); Huber et al.
**C**: (2017, 2018); Contreras and Goyeneche (2022).
Still, the analysis of AME sta... | BAC | BCA | BCA | BCA | Selection 4 |
<|MaskedSetence|> To observe and test the errors described above and assess the capabilities of RCDT in the context of MOR, we consider a number of images and flow case studies in section 3.1 and section 4. Test cases to quantify specific errors can be found in section 3.1. <|MaskedSetence|> <|MaskedSetence|> Final ... | **A**: We then test our workflow for a multi-phase fluid wave and the flow around an airfoil using high-resolution CFD data.
**B**: These include discontinuous images of a unit circle and a circular ring and continuous Gaussian functions to observe the intrinsic error in the RCDT workflow, and the flow field given by ... | CBA | CBA | CBA | CBA | Selection 1 |
A deterministic or random selection of the columns and rows is possible, with the option of obtaining additive or relative approximation errors. <|MaskedSetence|> The DEIM is another deterministic way to select a matrix’s columns or rows and relies on the top singular vectors [6]. <|MaskedSetence|> For additional de... | **A**: The three probability distributions that are most typically used to choose columns and rows are uniform, length-squared, and leverage-score distributions.
**B**: It is generally known that the columns or rows with the highest volume might provide virtually optimal solutions in a deterministic scenario [29].
**... | BAC | ABC | BAC | BAC | Selection 3 |
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