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<|MaskedSetence|> Without her far reaching vision this could not have come to being. <|MaskedSetence|> We have also benefited from discussions with Pavel Safronov. <|MaskedSetence|> was partially funded by ANR grants ENUMGEOM 18-CE40-0009 and COSY 21-CE40-0002, and by a Fellowship of the University of Strasbourg Institute for Advanced Study within the French national programme “Investment for the future” (IdEx-Unistra).
. | **A**: We would also like to thank the Institute for Advanced Study, Princeton, for its support in the academic year 2021/22, during which much of this work was completed.
A.O.
**B**: We would like to thank Nathalie Wahl and Jonathan Laurent Clivio for their explanations on signs in TQFT.
**C**:
Acknowledgements.
This paper is a split-off from our collaboration with Nancy Hingston on Poincaré duality.
| CBA | CBA | CBA | CBA | Selection 1 |
<|MaskedSetence|> By Proposition 4.12, λ𝜆\lambdaitalic_λ is continuous at x𝑥xitalic_x. If λ(x)>α𝜆𝑥𝛼\lambda(x)>\alphaitalic_λ ( italic_x ) > italic_α, there exists δ1>0subscript𝛿10\delta_{1}>0italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > 0, such that for any y∈(x−δ1,x+δ1)𝑦𝑥subscript𝛿1𝑥subscript𝛿1y\in(x-\delta_{1},x+\delta_{1})italic_y ∈ ( italic_x - italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x + italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ), λ(y)>α𝜆𝑦𝛼\lambda(y)>\alphaitalic_λ ( italic_y ) > italic_α, which implies 𝟙Eα(y)=𝟙Eα(x)=0subscript1subscript𝐸𝛼𝑦subscript1subscript𝐸𝛼𝑥0\mathds{1}_{E_{\alpha}}(y)=\mathds{1}_{E_{\alpha}}(x)=0blackboard_1 start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) = blackboard_1 start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) = 0. Similarly, If λ(x)<α𝜆𝑥𝛼\lambda(x)<\alphaitalic_λ ( italic_x ) < italic_α, there exists δ2>0subscript𝛿20\delta_{2}>0italic_δ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > 0, such that for any y∈(x−δ2,x+δ2)𝑦𝑥subscript𝛿2𝑥subscript𝛿2y\in(x-\delta_{2},x+\delta_{2})italic_y ∈ ( italic_x - italic_δ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x + italic_δ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ), 𝟙Eα(y)=𝟙Eα(x)=1subscript1subscript𝐸𝛼𝑦subscript1subscript𝐸𝛼𝑥1\mathds{1}_{E_{\alpha}}(y)=\mathds{1}_{E_{\alpha}}(x)=1blackboard_1 start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_y ) = blackboard_1 start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x ) = 1. <|MaskedSetence|> <|MaskedSetence|> | **A**: Therefore,
.
**B**: Thus, 𝟙Eαsubscript1subscript𝐸𝛼\mathds{1}_{E_{\alpha}}blackboard_1 start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_POSTSUBSCRIPT is continuous at x𝑥xitalic_x.
**C**:
Let x∈(s−1,1)𝑥superscript𝑠11x\in(s^{-1},1)italic_x ∈ ( italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , 1 ) be an s𝑠sitalic_s-ary irrational number with λ(x)≠α𝜆𝑥𝛼\lambda(x)\neq\alphaitalic_λ ( italic_x ) ≠ italic_α.
| CBA | CBA | CBA | ACB | Selection 2 |
The conclusions of Theorem 1.2 and Theorem 1.1 rule out non-simple blowup phenomenon in several applicable situations. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> For example, in the blowup analysis of Toda systems, which has ties with conformal geometry, algebraic geometry, integrable system and complex analysis [28, 36, 37, 42, 46], one always needs to compare blowup speeds of different components. If different components all tend to infinity in a neighborhood of one blowup point, the behavior of the “fast” component is similar to a quantized singular source to “slow” components. Therefore the asymptotic behavior of blowup solutions of Liouville equation with quantized singular source provides crucial information for systems.
. | **A**: Even though we study only one equation in this article, it represents certain situations in systems.
**B**: The proofs of the main results should lead to advances in multiple related problems.
**C**: They seem to suggest that the only case that non-simple blowup solutions occur is when the profile of blowup solutions is very close to global solutions in the classification theorem of Prajapat-Tarantello [35].
| BCA | CBA | CBA | CBA | Selection 4 |
<|MaskedSetence|> Perhaps the most important example of this and the inspiration for a lot of what has followed, is the seminal result of Spielman and Teng [39] on the performance of the simplex algorithm, see also Vershynin [41] and Dadush and Huiberts [12].
Spielman and Teng [39] inspired the following model of Bohman, Frieze and Martin [8]. <|MaskedSetence|> Here α𝛼\alphaitalic_α is a positive constant and 𝒢(α)𝒢𝛼\mathcal{G}(\alpha)caligraphic_G ( italic_α ) is the set of graphs with vertex set [n]delimited-[]𝑛[n][ italic_n ] and minimum degree at least αn𝛼𝑛\alpha nitalic_α italic_n. They show that adding O(n)𝑂𝑛O(n)italic_O ( italic_n ) random edges to G𝐺Gitalic_G is enough to create a Hamilton cycle w.h.p. <|MaskedSetence|> Research on this model and its variations has been quite substantial, see for example [4], [5], [6], [7], [9], [10], [13], [16], [23], [30], [31], [32], [36], [37], [38], [40].. | **A**: They consider adding random edges to an arbitrary member G𝐺Gitalic_G of 𝒢(α)𝒢𝛼\mathcal{G}(\alpha)caligraphic_G ( italic_α ).
**B**:
It is often the case that adding some randomness to a combinatorial structure can lead to significant positive change.
**C**: This is in contrast to the approximately 12nlogn12𝑛𝑛\frac{1}{2}n\log ndivide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_n roman_log italic_n edges needed if we rely only on the random edges.
| BCA | BAC | BAC | BAC | Selection 2 |
<|MaskedSetence|> The first breakthrough in the case of L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT supercritical was made by Jeanjean [10], where a mountain-pass type argument for the scaled functional J~(u,t):=J(t⋆u)assign~𝐽𝑢𝑡𝐽⋆𝑡𝑢\tilde{J}(u,t):=J(t\star u)over~ start_ARG italic_J end_ARG ( italic_u , italic_t ) := italic_J ( italic_t ⋆ italic_u ) with t⋆u(⋅):=tN2u(t⋅)t\star u(\cdot):=t^{\frac{N}{2}}u(t\cdot)italic_t ⋆ italic_u ( ⋅ ) := italic_t start_POSTSUPERSCRIPT divide start_ARG italic_N end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT italic_u ( italic_t ⋅ ) was introduced. <|MaskedSetence|> Ikoma and Tanaka [9] established a deformation result on 𝒮csubscript𝒮𝑐\mathcal{S}_{c}caligraphic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, and gave an alternative proof of the results in [2, 10].
Bartsch and Soave [3, 4] demonstrated that the set 𝒮c∩ℳsubscript𝒮𝑐ℳ\mathcal{S}_{c}\cap\mathcal{M}caligraphic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∩ caligraphic_M is a C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT manifold and constitutes a natural constraint. They developed a minimax approach on 𝒮c∩ℳsubscript𝒮𝑐ℳ\mathcal{S}_{c}\cap\mathcal{M}caligraphic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∩ caligraphic_M based on the σ𝜎\sigmaitalic_σ-homotopy stable family of compact subsets of 𝒮c∩ℳsubscript𝒮𝑐ℳ\mathcal{S}_{c}\cap\mathcal{M}caligraphic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∩ caligraphic_M to investigate the existence and multiplicity of normalized solutions of (1.1). <|MaskedSetence|> | **A**:
If f𝑓fitalic_f admits a L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT supercritical growth at infinity, i.e., p>2+4N𝑝24𝑁p>2+\frac{4}{N}italic_p > 2 + divide start_ARG 4 end_ARG start_ARG italic_N end_ARG, then J|𝒮cevaluated-at𝐽subscript𝒮𝑐J|_{\mathcal{S}_{c}}italic_J | start_POSTSUBSCRIPT caligraphic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT is unbounded below and the direct minimization does not work.
**B**: Subsequently, Bartsch and de Valerioda [2] applied the genus theory to obtain infinitely many normalized solutions of (1.1).
**C**: In these studies, the following Ambrosetti-Rabinowitz (AR) condition:
.
| ABC | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> The research for this paper was conducted while the second author was a J. L. <|MaskedSetence|> <|MaskedSetence|> Bañuelos acknowledges the countless conversations he had for almost 40 years with the late Richard Gundy on topics related to those of this paper.. | **A**: Doob research assistant professor at the University of Illinois at Urbana-Champaign.
**B**:
Acknowledgments
We express our thanks to Renming Song for a helpful conversation on positive-definite functions and reference [Jacob1], and to Tomasz Szarek for reference [Kov].
We are grateful to Mark Ashbaugh for useful conversations on special functions.
**C**: With deep appreciation and respect, R.
| BAC | BAC | BAC | CAB | Selection 1 |
The state-of-the-art method for the analysis of these data, DESeq2 (Love et al., 2014), uses a negative binomial model with a dispersion parameter that is allowed to differ between genes, but does not depend on covariates. In particular, it does not depend on pathological stage. <|MaskedSetence|> <|MaskedSetence|> Since we would expect approximately 1,000 rejections if the DESeq2 model fits well, this gives clear indication of lack of fit of that model, at least in some of the genes. <|MaskedSetence|> | **A**: We tested this assumption for each gene using a GAMLSS model.
**B**: The null hypothesis that dispersion did not depend on tumor stage was rejected for 5,967 out of 20,119 genes at the unadjusted 5% level.
**C**: Based on the simulations in Section 8 we would, therefore, expect DESeq2 to be anti-conservative for these data.
.
| CBA | ABC | ABC | ABC | Selection 4 |
Nonetheless, in the last years, some proofs that avoid these tools have appeared. For example, in 2002, such a proof was developed by Elliot in [2]. <|MaskedSetence|> In turn, this served as the stepping stone for another new proof of Linnik’s theorem which circumvented the combination of the three aforementioned principles. <|MaskedSetence|> <|MaskedSetence|> | **A**: Another pretentious proof of Linnik’s theorem is presented in [7, Chapter 27] and a basic element of the proof is a flexible variant of (1.1) where every prime is weighted with 1/p1𝑝1/p1 / italic_p instead of logp𝑝\log proman_log italic_p.
**B**: Even though the alternative approaches recover Linnik’s theorem, they do not provide a pretentious proof for an estimate of the same quantitative strength as (1.1).
.
**C**: Later, in 2016, Granville, Harper and Soundararajan [3] studied pretentiously the distribution of multiplicative functions on arithmetic progressions and as a consequence of their general results they were able to show a weaker form of (1.1).
| CAB | CAB | CAB | ACB | Selection 2 |
Concisely speaking, hard thresholding hyperinterpolation is the unique solution to an ℓ0subscriptℓ0\ell_{0}roman_ℓ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-regularized weighted discrete least square problem, and has been proved to be an effective tool in denoising from the numerical examples. Hard thresholding hyperinterpolation satisfies the Pythagorean theorem with respect to the discrete (semi) inner product, which is an important geometric property that Lasso hyperinterpolation [2] and hybrid hyperinterpolation [1] do not possess. <|MaskedSetence|> What’s more, a practical criterion, using the sum of the difference between the regularization parameter and the product of noise coefficients and signs of hyperinterpolation coefficients, is established to judge the denoising abilities of hard thresholding hyperinterpolation and Lasso hyperinterpolation.
With the aid of the Marcinkiewicz-Zygmund property [3], one can bypass the quadrature exactness as in [4], which can break the restriction of the application of hard thresholding hyperinterpolation. <|MaskedSetence|> Furthermore, one may combine the springback penalty [6] with the weighted least squares problem (2.7) to obtain a more stable and effective approximation scheme. <|MaskedSetence|> | **A**: Then we use the reciprocal of Christoffel function to prove that the upper bound of the uniform norm of hard thresholding hyperinterpolation operator is not greater than that of hyperinterpolation operator.
**B**: In addition, it seems promising to discuss the relation between different types of noise and denoising ability of hard thresholding hyperinterpolation..
**C**: Once the quadrature exactness is not required, there are many quadrature rules that we can take, such as (Quasi) Monte-Carlo rules [15].
| ABC | ACB | ACB | ACB | Selection 3 |
Additional Experiment
To further compare with state-of-the-art models, we used the newly developed Python framework for spatiotemporal predictive learning (OpenSTL)444https://github.com/chengtan9907/OpenSTL [18]. The subset of the temperate dataset from the WeatherBench [81] was used for experiments. The subset dataset contains temperature values from 2016 to 2018 with a spatial resolution of 5.625° over a grid of 32 × 64 points. <|MaskedSetence|> The models were trained on 85% of the data, validated on 7%, and tested on the remaining samples. <|MaskedSetence|> The models from OpenSTL were trained using default parameters. <|MaskedSetence|> We used the Adam optimizer with a learning rate 1e-3 and batch size equal to 8. Note that the proposed model does not require training, as it is an algebraic method and, therefore, utilizes only standard linear algebra operations to make predictions. The TT-DMD method only takes 51 seconds to reconstruct the data from DMD modes and make predictions. The RMSE and MAE values have been recorded and are summarized in Table 4. The sample experiments are available555https://github.com/ShakirSofi/TensorizingDMD/tree/main/Tensorizing_DMD. | **A**: The dataset consists of 17520 (1-hour temporal resolution) temperature distribution maps (32 x 64).
**B**: In this experiment, we trained the following models: ConvLSTM [13], PhyDNet [20], PredRNN [19], TAU [22], SimVP [21], MAR [53] and compared with the proposed TT-DMD model.
**C**: However, the input sequence length (lookback) was changed to 12, and the output sequence length (prediction horizon) was also set to 12.
| ABC | BAC | ABC | ABC | Selection 1 |
<|MaskedSetence|> To solve the IE we implement a solver that performs an iterative procedure to obtain a solution, see Appendix B.3 and Appendix D. <|MaskedSetence|> This procedure allows our deep learning model to be independent of the temporal grid points, therefore resulting in a continuous model, since the model internally uses randomly sampled points to generate the successive iterations, as opposed to using fixed grid points. <|MaskedSetence|> See also Figure 2 for a visualization of the general solving procedure.
Algorithm 1 NIE method training step. Integration is performed using the module torch.quad, with the Monte Carlo method.. | **A**: NIEs in this form comprise two neural networks, namely K𝐾Kitalic_K and F𝐹Fitalic_F.
We observe that in IEs, the initial condition is embedded in the equation itself, and it is not an arbitrary value to be specified as an extra condition.
**B**: The general algorithm for training NIE is given in Algorithm 1, and a diagrammatic overview of it is shown in Figure 1.
**C**: During the iterations, Monte Carlo sampling is performed to evaluate the integrals.
| ACB | ABC | ACB | ACB | Selection 1 |
One drawback to the SINDy technique is that the time derivative of system states is needed to build the linear system. <|MaskedSetence|> A common approach to address this issue is to apply low-pass filters to reduce the noise. However, it is known [4] that low-pass filtering does not apply in certain situations such as handling EEG data [16] in neuroscience or turbulence data [24] in computational fluid dynamics, where high-frequency data is present.
Hence, a technique which is robust to noise is preferable in these situations. <|MaskedSetence|> Weak-SINDy utilizes test functions to access the derivative data via integration by parts. <|MaskedSetence|> | **A**: Similarly, the Occupation Kernel technique uses test functions to access the derivative data via the fundamental theorem of calculus.
.
**B**: Integral formulations of SINDy, such as the weak-SINDy technique developed by Messenger and Bortz [15, 14] and the Occupation Kernel techniques developed by Rosenfeld et al. [21, 20], are naturally robust to noise as the derivative data is accessed in a circuitous fashion via integral equations.
**C**: These time derivatives often need to be estimated from the system states, which is untenable in the presence of noise.
| CBA | CBA | CBA | BAC | Selection 1 |
In an attempt to move towards constructing an analytical solution, we propose a new solution scheme, which is based on the matrix sparsification technique developed to solve tropical optimization problems and two-sided inequalities in [20, 21, 22, 23, 26, 24]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Finally, a computational scheme based on the approach to derive all solutions of the two-sided equation is discussed.
The paper is organized as follows. In Section 2, we give an overview of basic definitions and preliminary results of tropical algebra, which underlie subsequent developments. Section 3 offers a solution of a system of vector inequalities, which is a key ingredient in the solution of the two-sided equation. In Section 4, we present our main results, which include a necessary and sufficient condition for solutions of the equation and represent a direct solution of the equation in a compact vector form. Numerical examples are given in Section 5. Section 6 presents some concluding remarks.. | **A**: To illustrate the approach and to compare the result with those of existing solution procedures, we apply our solution technique to handle two-sided equations known in the literature.
**B**: We use this technique to reduce the two-sided equation to a set of vector inequalities that involve row-monomial matrices obtained from the given matrices.
**C**: An existence condition of solutions for the inequalities is established, and a direct representation of the solutions is derived in a compact vector form.
| CBA | BCA | BCA | BCA | Selection 2 |
λ=(λ1,..,λk)\lambda=(\lambda_{1},..,\lambda_{k})italic_λ = ( italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , . . , italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) of non-negative integers that
sum to m𝑚mitalic_m. <|MaskedSetence|> Two partitions
are equivalent if they differ only by a string of 00’s at the end.
A partition λ=(λ1,..,λk)\lambda=(\lambda_{1},..,\lambda_{k})italic_λ = ( italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , . <|MaskedSetence|> <|MaskedSetence|> | **A**: In this case, we write λ⊢mproves𝜆𝑚\lambda\vdash mitalic_λ ⊢ italic_m.
**B**: .
**C**: , italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ), with λk>0subscript𝜆𝑘0\lambda_{k}>0italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT > 0,
is graphically encoded by a Young diagram, which is a finite.
| ABC | ABC | ABC | ABC | Selection 1 |
<|MaskedSetence|> It was first constructed by Matetski, Quastel and Remenik in [MQR21] recently, as a Markov process with explicit transition probability by analyzing the totally asymmetric simple exclusion process (TASEP). However, the derivation of explicit formulas for the multi-point distribution of H(x,τ)H𝑥𝜏\mathrm{H}(x,\tau)roman_H ( italic_x , italic_τ ) is a challenging task due to the complexity of the transition probability formula. Pioneering efforts in this field have unfolded over the past two decades. <|MaskedSetence|> Meanwhile, the multi-point distribution along the spatial direction, (H(x1,τ),⋯,H(xn,τ))Hsubscript𝑥1𝜏⋯Hsubscript𝑥𝑛𝜏\left(\mathrm{H}(x_{1},\tau),\cdots,\mathrm{H}(x_{n},\tau)\right)( roman_H ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ ) , ⋯ , roman_H ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_τ ) ) for a fixed τ𝜏\tauitalic_τ, is characterized by the finite-dimensional distribution of the Airy process and its analogues, as explored in works by [PS02, Joh03, IS04, BFM07, BFPS07, BFS08, MQR21]. <|MaskedSetence|> These breakthroughs were achieved by Johansson and Raham [JR21] as well as independently by Liu [Liu22a].
The multi-point distribution formula enables us to delve into the finer details of the KPZ fixed point. For an inhomogeneous Markov process, we naturally turn our attention to the probabilistic properties of the conditional field {H(x,τ)∣H(x′,τ′)=h′}conditional-setH𝑥𝜏Hsuperscript𝑥′superscript𝜏′superscripth′\{\mathrm{H}(x,\tau)\mid\mathrm{H}(x^{\prime},\tau^{\prime})=\mathrm{h}^{%. | **A**:
The limit space-time field H(x,τ)H𝑥𝜏\mathrm{H}(x,\tau)roman_H ( italic_x , italic_τ ), where x∈ℝ,t≥0formulae-sequence𝑥ℝ𝑡0x\in\mathbb{R},t\geq 0italic_x ∈ blackboard_R , italic_t ≥ 0, of the KPZ universality class, is called the KPZ fixed point, which depends on the initial condition H(x,0)=h0(x)H𝑥0subscriptℎ0𝑥\mathrm{H}(x,0)=h_{0}(x)roman_H ( italic_x , 0 ) = italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) for a function h0subscriptℎ0h_{0}italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT in the space of upper-semi-continuous functions.
**B**: Recently, significant progress has been made in obtaining general explicit formulas for joint distributions, even when considering different temporal points (H(x1,τ1),⋯,H(xn,τn))Hsubscript𝑥1subscript𝜏1⋯Hsubscript𝑥𝑛subscript𝜏𝑛\left(\mathrm{H}(x_{1},\tau_{1}),\cdots,\mathrm{H}(x_{n},\tau_{n})\right)( roman_H ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , ⋯ , roman_H ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ).
**C**: Notably, [BDJ99, Joh00] demonstrated that the marginal one-point distribution of H(x,τ)H𝑥𝜏\mathrm{H}(x,\tau)roman_H ( italic_x , italic_τ ) follows the Tracy-Widom distribution.
| ACB | CAB | ACB | ACB | Selection 1 |
<|MaskedSetence|> This is IL. <|MaskedSetence|> This law is equivalent to the principle that in order to prove a proposition it suffices to show that its negation is contradictory. In IL, such an argument does not constitute sufficient evidence for its conclusion.
Heyting [27] and Kolmogorov [31] provided a semantics for intuitionistic proof that captures the evidential character of intuitionism, called the Brouwer-Heyting-Kolmogorov (BHK) interpretation of IL. <|MaskedSetence|> | **A**: Famously, as a consequence, IL rejects the law of the excluded middle — that is, the meta-theoretic statement that either a statement or its negation is valid.
**B**: It is now the standard explanation of the logic..
**C**:
Intuitionism, as defined by Brouwer [6], is the view that an argument is valid when it provides sufficient evidence for its conclusion.
| CAB | CAB | CAB | CBA | Selection 1 |
<|MaskedSetence|> Firstly we find a positive integer K𝐾Kitalic_K such that ∑i=0kCiki!xi−(x+1)Ksuperscriptsubscript𝑖0𝑘subscriptsubscript𝐶𝑖𝑘𝑖superscript𝑥𝑖superscript𝑥1𝐾\displaystyle\sum_{i=0}^{k}\displaystyle\frac{{}_{k}C_{i}}{i!}x^{i}-(x+1)^{K}∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT divide start_ARG start_FLOATSUBSCRIPT italic_k end_FLOATSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_i ! end_ARG italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT - ( italic_x + 1 ) start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT is greater than or equal to 0 for x>0𝑥0x>0italic_x > 0. <|MaskedSetence|> In other words we find K𝐾Kitalic_K such that k≥K𝑘𝐾k\geq Kitalic_k ≥ italic_K and Ciki!≥CiKsubscriptsubscript𝐶𝑖𝑘𝑖subscriptsubscript𝐶𝑖𝐾\displaystyle\frac{{}_{k}C_{i}}{i!}\geq{}_{K}C_{i}divide start_ARG start_FLOATSUBSCRIPT italic_k end_FLOATSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG italic_i ! end_ARG ≥ start_FLOATSUBSCRIPT italic_K end_FLOATSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for 0≤i≤K0𝑖𝐾0\leq i\leq K0 ≤ italic_i ≤ italic_K. If i=0𝑖0i=0italic_i = 0, it is trivial. <|MaskedSetence|> Multiplying both sides of the inequality by (i!)2superscript𝑖2(i!)^{2}( italic_i ! ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, then we have that
. | **A**: So we consider cases of 1≤i≤K1𝑖𝐾1\leq i\leq K1 ≤ italic_i ≤ italic_K.
**B**:
Proof.
We fix k≥1𝑘1k\geq 1italic_k ≥ 1.
**C**: It is enough that all coefficients of this polynomial are non-negative.
| BCA | ACB | BCA | BCA | Selection 4 |
<|MaskedSetence|> To see this, we note that c⊗ztensor-product𝑐𝑧c\otimes zitalic_c ⊗ italic_z has bidegree (0,0)00(0,0)( 0 , 0 ) and a1⊗v2tensor-productsubscript𝑎1subscript𝑣2a_{1}\otimes v_{2}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊗ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT has bidegree (1,1)11(1,1)( 1 , 1 ). It follows from the immersed curve computation, which we omit for simplicity, that there exists a unique hat-flavored homology classes of bidegrees (0,0)00(0,0)( 0 , 0 ) (which is clearly ω𝜔\omegaitalic_ω) and (1,1)11(1,1)( 1 , 1 ) (denoted as ζ𝜁\zetaitalic_ζ in Figure 5.3. <|MaskedSetence|> We will also note that b1⊗v2tensor-productsubscript𝑏1subscript𝑣2b_{1}\otimes v_{2}italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊗ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is identified with α𝛼\alphaitalic_α in the figure. <|MaskedSetence|> | **A**: Hence we see that c⊗ztensor-product𝑐𝑧c\otimes zitalic_c ⊗ italic_z and a1⊗v2tensor-productsubscript𝑎1subscript𝑣2a_{1}\otimes v_{2}italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊗ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are identified with ω𝜔\omegaitalic_ω and ζ𝜁\zetaitalic_ζ.
**B**:
It is natural to ask how the pairing formula identifies the homology class of HFK^^𝐻𝐹𝐾\widehat{HFK}over^ start_ARG italic_H italic_F italic_K end_ARG as shown above with elements of CFA^(T∞,P3,−1)⊠M⊠^𝐶𝐹𝐴subscript𝑇subscript𝑃31𝑀\widehat{CFA}(T_{\infty},P_{3,-1})\boxtimes Mover^ start_ARG italic_C italic_F italic_A end_ARG ( italic_T start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 3 , - 1 end_POSTSUBSCRIPT ) ⊠ italic_M.
**C**: We do not have to know about how other homology classes are identified.
.
| CAB | BAC | BAC | BAC | Selection 3 |
Let f1,⋯,fk:M→ℝ:subscript𝑓1⋯subscript𝑓𝑘→𝑀ℝf_{1},\cdots,f_{k}:M\to\mathbb{R}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_f start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT : italic_M → blackboard_R be smooth functions such that for all x∈M𝑥𝑀x\in Mitalic_x ∈ italic_M, df1,x,⋯,dfk,x𝑑subscript𝑓1𝑥⋯𝑑subscript𝑓𝑘𝑥df_{1,x},\cdots,df_{k,x}italic_d italic_f start_POSTSUBSCRIPT 1 , italic_x end_POSTSUBSCRIPT , ⋯ , italic_d italic_f start_POSTSUBSCRIPT italic_k , italic_x end_POSTSUBSCRIPT span Tx∗Msuperscriptsubscript𝑇𝑥𝑀T_{x}^{*}Mitalic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_M. <|MaskedSetence|> The forms dfi𝑑subscript𝑓𝑖df_{i}italic_d italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT define a surjective bundle map θ→T∗M→𝜃superscript𝑇𝑀\theta\to T^{*}Mitalic_θ → italic_T start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_M. <|MaskedSetence|> <|MaskedSetence|> The map p𝑝pitalic_p determines the vector fields X1,⋯,Xksubscript𝑋1⋯subscript𝑋𝑘X_{1},\cdots,X_{k}italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_X start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT.
∎
. | **A**: Let θ=M×ℝk𝜃𝑀superscriptℝ𝑘\theta=M\times\mathbb{R}^{k}italic_θ = italic_M × blackboard_R start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT be the trivial vector bundle on M𝑀Mitalic_M of rank k𝑘kitalic_k.
**B**: Its dual is an injective bundle map i:TM→θ:𝑖→𝑇𝑀𝜃i:TM\to\thetaitalic_i : italic_T italic_M → italic_θ.
**C**: Let p:θ→TM:𝑝→𝜃𝑇𝑀p:\theta\to TMitalic_p : italic_θ → italic_T italic_M be any bundle map such that p∘i=Id𝑝𝑖Idp\circ i=\mathrm{Id}italic_p ∘ italic_i = roman_Id.
| ABC | ABC | CBA | ABC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> Assuming to the contrary that a solitary wave is stable, on one hand, we show the upper bound for this functional. On the other hand, its time derivative is lower bounded, which implies growth in time, contradicting the first fact, the boundedness. <|MaskedSetence|> The two-dimensional version of the virial-type functional was introduced by the second author and her collaborators in [29, 30].
In our approach it is sufficient to use a truncation of this functional, avoiding. | **A**: 4.2.
**B**: Virial-type estimates
In this part, we introduce a virial-type functional, which is used to show instability of solitary waves in the supercritical case.
**C**: This type of functional was used in the 1d context of the critical gKdV equation in [71]
(see also the instability argument reviewed in the supercritical gKdV case in [24] and [28]).
| ABC | ABC | ABC | ABC | Selection 4 |
Proof.
We take each of the single edge and two edges from each triangle. Then in the n𝑛nitalic_n-vertex graph, we have at least 2⋅(α−ξ)n+(1−α−ξ)n=(1+α−3ξ)n⋅2𝛼𝜉𝑛1𝛼𝜉𝑛1𝛼3𝜉𝑛2\cdot(\alpha-\xi)n+(1-\alpha-\xi)n=(1+\alpha-3\xi)n2 ⋅ ( italic_α - italic_ξ ) italic_n + ( 1 - italic_α - italic_ξ ) italic_n = ( 1 + italic_α - 3 italic_ξ ) italic_n
edges. <|MaskedSetence|> <|MaskedSetence|> Do it repeatedly until we obtain a rainbow cycle, which is of length at most Clogn𝐶𝑛C\log nitalic_C roman_log italic_n. <|MaskedSetence|> | **A**: This completes the proof..
**B**: If this cycle is not rainbow, we can replace two edges of the same color, which must come from the same triangle, by the other edge in the triangle to get a shorter cycle.
**C**: Since 1+α−3ξ>11𝛼3𝜉11+\alpha-3\xi>11 + italic_α - 3 italic_ξ > 1, Theorem 2 implies that there is a cycle of length at most Clogn𝐶𝑛C\log nitalic_C roman_log italic_n for some constant C(α,ξ)>0𝐶𝛼𝜉0C(\alpha,\xi)>0italic_C ( italic_α , italic_ξ ) > 0.
| CBA | CAB | CBA | CBA | Selection 3 |
6 Discussion
This paper proposes and analyzes a stochastic version of the recent proximal distance algorithm. The algorithm allows for a wide range of constraints to be considered in a projection-based framework. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> By leveraging tools from the analysis of stochastic and incremental schemes, we are able to derive new guarantees even when the MM geometry no longer holds due to subsampling.
. | **A**: Such results were previously unavailable even in the non-stochastic, “batch” version of the method.
**B**: In particular, we establish convergence guarantees when the parameter ρ𝜌\rhoitalic_ρ—and in turn the sequence of objectives—change over iterations.
**C**: Our contributions now extend these merits to large-scale settings, and provide new theoretical insights.
| ACB | CBA | CBA | CBA | Selection 4 |
<|MaskedSetence|> Cheng and S.T. Yau [CY80]. <|MaskedSetence|> This metric is also unique up to scaling by a constant. <|MaskedSetence|> This is still an open conjecture in complex geometry.
. | **A**: Yau conjectured that the Cheng-Yau metric of a bounded pseudoconvex domain coincides with its Bergman metric if and only if the domain is homogeneous [Yau82]; recall that a domain in ℂnsuperscriptℂ𝑛\mathbb{C}^{n}blackboard_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is called homogeneous if its automorphism group acts transitively on it.
**B**:
For any bounded strongly pseudoconvex domain there exists a natural complete Kähler-Einstein metric, whose existence was proved by S.Y.
**C**: The Cheng-Yau metric can be constructed explicitly by solving a Monge-Ampere type equation.
| BCA | BCA | BAC | BCA | Selection 1 |
However, some data requires filtration along multiple parameters to fully capture its information: this is the role of multi-parameter persistent homology [10, 9]. <|MaskedSetence|> Additionally, single parameter persistent homology is not robust to outliers in a point cloud; these outliers can lead to a misinterpretation of the persistent homology, with the unnecessary creation or destruction of homology classes. Multi-parameter persistence can be a natural fix to this problem by adding a dimension depending on the density of the samples (see, e.g., [15, 6]). <|MaskedSetence|> <|MaskedSetence|> | **A**: In some contexts, it can be helpful to use multiple parameters to capture the details of the data [11, 19, 26, 17, 29].
**B**: Unfortunately, understanding, visualizing, and computing invariants in multi-parameter persistent homology remains a difficult task both mathematically and computationally.
**C**: This difficulty holds as well when it comes to computing distances between such invariants.
.
| ABC | ABC | ABC | ABC | Selection 2 |
In the stochastic approximation literature, similar techniques have been successfully applied to
study optimization and games in various settings such as on Riemannian or primal-dual spaces
[26, 28, 37]. <|MaskedSetence|> [12], Bubeck et al. <|MaskedSetence|> <|MaskedSetence|> Moreover, the integration of the Picard process with the theory of WAPT plays a pivotal role in our analysis, and both of these aspects present original contributions.
. | **A**: The application to sampling has also been previously explored by Chau et al.
**B**: [11] in different contexts.
**C**: What distinguishes our work from the existing literature is the advantage of generalizing the Picard process to encompass a vastly wider class of algorithms, specifically the LRM schemes.
| ABC | ABC | ACB | ABC | Selection 4 |
<|MaskedSetence|> Uniform rate of expansion and non-degeneracy
Now we are ready to show that the support of our solution strictly expands with respect to streamlines. <|MaskedSetence|> The construction of the barrier function in a thin strip domain was enough in Section 6, since there we showed the propagation of cone monotonicity over time, which came from the interior of the support, εκsuperscript𝜀𝜅\varepsilon^{\kappa}italic_ε start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT-away from the boundary. <|MaskedSetence|> This necessitates our construction of the barrier different from the previous section.
. | **A**: To show this we apply sup-convolutions as in Section 5 to construct perturbed subsolutions, but our domain is no longer a thin strip near the free boundary.
**B**: 7.2.
**C**: Here we will show propagation of the interior non-degeneracy, which only holds unit distance away from the boundary.
| CBA | BAC | BAC | BAC | Selection 2 |
<|MaskedSetence|> In Section 2, after establishing notation (Subsection 2.1), market dynamics, and admissible strategies (Subsection 2.2), we introduce alternative descriptions of the law of one price by means of (i) the price process S𝑆Sitalic_S; (ii) pricing functionals; and (iii) state price densities (Subsections 2.3–2.5). We conclude Section 2 with a review of ℰℰ\mathscr{E}script_E-densities and ℰℰ\mathscr{E}script_E-martingales (Subsection 2.6). Section 3 contains the main results of the paper. In Subsection 3.1, we pull the various notions of the law of one price together to demonstrate their equivalence (Theorem 3.2). Subsection 3.2 illustrates several phenomena that arise when the law of one price fails. <|MaskedSetence|> We next examine the consequences of LOP for quadratic hedging (Subsection 3.4), extend its applicability to the conditional framework of Hansen and Richard [19] (Subsection 3.5), and study the resulting mean–variance portfolio selection in the presence of a contingent claim (Subsection 3.6). <|MaskedSetence|> | **A**: In Subsection 3.3 we offer some intuition for the law of one price and interpret the main result (Theorem 3.2) as a market extension theorem.
**B**: Section 4 contains proofs of the main theorem presented via several partial statements of independent interest.
2 Problem formulation.
**C**: The paper is organised as follows.
| CAB | CAB | CAB | ACB | Selection 1 |
<|MaskedSetence|> We also remark that we crucially use the spectral decomposition for a specific cuspidal datum. This is presumably different than a usual proof of, e.g., Weyl law (see e.g. [20], [41]) where one does another sum over all cuspidal data in addition to the averages in Theorem 3. <|MaskedSetence|> <|MaskedSetence|> This way we may only achieve a polynomial strength dependence on level(K)level𝐾\mathrm{level}(K)roman_level ( italic_K ) in Theorem 1 (and Remark 1.2) as opposed to a poly-logarithmic strength.
. | **A**: This extra average will not give us the result of the strength as in, e.g., Theorem 1.
**B**:
It is worth noting that we do not use the geometric side of the trace formula, rather only use the two different expressions of the spectral side given in 6.1 (the spectral side of the pre-trace formula) and 6.2 (the spectral side of the trace formula); also see 6.3.
**C**: The primary reason is that the contribution from the cusp forms will dominate.
| BAC | BAC | ACB | BAC | Selection 1 |
First examples of Lie superalgebras, actually Lie super rings over ℤℤ{\mathbb{Z}}blackboard_Z, appeared in 1941, in topology as the sets of homotopy groups with the Whitehead product, see [Wh]. Associated with these examples were modular Lie superalgebras over finite fields. <|MaskedSetence|> An observation of Emmy Noether in 1925 shifted the attention to the “homology groups” of a space, and algebraic techniques were developed for computational purposes in the 1930’s. <|MaskedSetence|> <|MaskedSetence|> | **A**: Yet, homology remained a part of the realm of topology until about 1945.”, see [We, pp.
**B**: Similarity of Lie superalgebras and modular Lie algebras is so striking that sometimes one hears and reads that “when p=2𝑝2p=2italic_p = 2, there is no difference between Lie algebras and Lie superalgebras”, which is not true as is lucidly explained in [BGLLS].
Weibel writes: “Homological algebra had its origins in the 19-th century, via the work of Riemann (1857) and Betti (1871) on “homology numbers”, and the rigorous development of the notion of homology numbers by Poincaré in 1895.
**C**: 797–836]..
| BAC | BAC | ABC | BAC | Selection 2 |
Outline. <|MaskedSetence|> <|MaskedSetence|> We combine the two approaches in Section 4, proving Theorem 1.5. Finally, we analyse non-abelian bases in Section 5, proving Theorems 1.12 and 1.13.
Acknowledgements. <|MaskedSetence|> | **A**: We start with some group-theoretic results on wreath products and their epimorphisms in Section 2.
**B**: The authors are indebted to Giles Gardam, Anthony Genevois, Peter Kropholler, Markus Steenbock and John Wilson for useful conversations.
They also wish to thank the organisers of the conference YGGT X - Newcastle (Online), where this work was started..
**C**: Then we move on to stable finiteness in Section 3, proving Proposition 1.7 and its consequences.
| BAC | ACB | ACB | ACB | Selection 4 |
<|MaskedSetence|> How can a lamplighter turn all the lights off using x𝑥xitalic_x, y𝑦yitalic_y, σ𝜎\sigmaitalic_σ, and τ𝜏\tauitalic_τ? He has four types of moves at his disposal: he can navigate the Cayley graph of K𝐾Kitalic_K (by using x𝑥xitalic_x and y𝑦yitalic_y); because σ=[x,a]a=x−1a−1xa2𝜎𝑥𝑎𝑎superscript𝑥1superscript𝑎1𝑥superscript𝑎2\sigma=[x,a]a=x^{-1}a^{-1}xa^{2}italic_σ = [ italic_x , italic_a ] italic_a = italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, he can decrement by 1111 the lamp one step away in the x−1superscript𝑥1x^{-1}italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT-direction at the expense of incrementing the lamp where he stands by 2222; and likewise in the y−1superscript𝑦1y^{-1}italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT-direction using τ𝜏\tauitalic_τ. The answer is he sets the lamp at e𝑒eitalic_e to 00 at the expense of setting the lamp at x𝑥xitalic_x to 2222. Then he sets the lamp at x𝑥xitalic_x to 00 at the expense of setting the lamp at x2superscript𝑥2x^{2}italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT to 4444. And so on, until the lamp at xnsuperscript𝑥𝑛x^{n}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is set to 2nsuperscript2𝑛2^{n}2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. He then sets that to 00 and, proceeding in the y−1superscript𝑦1y^{-1}italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT direction, sets the lamp at xny−1superscript𝑥𝑛superscript𝑦1x^{n}y^{-1}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT to 2n−1superscript2𝑛12^{n-1}2 start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT. Continuing likewise in the y−1superscript𝑦1y^{-1}italic_y start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT-direction he sets the lamp at xny−(n−1)superscript𝑥𝑛superscript𝑦𝑛1x^{n}y^{-(n-1)}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - ( italic_n - 1 ) end_POSTSUPERSCRIPT to 2222. <|MaskedSetence|> <|MaskedSetence|> | **A**: Any path from e𝑒eitalic_e to xny−nsuperscript𝑥𝑛superscript𝑦𝑛x^{n}y^{-n}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT in the Cayley graph must rise to height n𝑛nitalic_n to escape Pnsubscript𝑃𝑛P_{n}italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT and the settings of the lights must be incrementally adjusted on the way up so that the number of σ𝜎\sigmaitalic_σ- and τ𝜏\tauitalic_τ-moves grows exponentially with the height.
.
**B**: Suppose the lights at the elements e𝑒eitalic_e and xny−nsuperscript𝑥𝑛superscript𝑦𝑛x^{n}y^{-n}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT of K𝐾Kitalic_K are set to 1111 and −11-1- 1, respectively, and all other lights are off (set to 00).
**C**: Finally, he adjusts the lamp at xny−(n−1)superscript𝑥𝑛superscript𝑦𝑛1x^{n}y^{-(n-1)}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - ( italic_n - 1 ) end_POSTSUPERSCRIPT to zero at the expense of changing the lamp at xny−nsuperscript𝑥𝑛superscript𝑦𝑛x^{n}y^{-n}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT, but as that was initially set to −11-1- 1, this results in all lights being off, as required.
The above method takes at least 2nsuperscript2𝑛2^{n}2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT moves, but could it have been accomplished with fewer? The hypothesis involving Pnsubscript𝑃𝑛P_{n}italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, xny−nsuperscript𝑥𝑛superscript𝑦𝑛x^{n}y^{-n}italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT, and the epimorphism K→ℤ→𝐾ℤK\to\mathbb{Z}italic_K → blackboard_Z ensures it cannot.
| BCA | BCA | BCA | BCA | Selection 3 |
The structure of the paper is as follows. <|MaskedSetence|> In Section 3 we prove Theorem 1.6 using case-by-case arguments. <|MaskedSetence|> We also determine when a more refined decomposition exists in terms of reflection subgroups minimally containing Sylow ℓℓ\ellroman_ℓ-subgroups (see Theorem 4.2). In Section 5 we prove Theorem 1.5 using case-by-case arguments and MAGMA [BCP97] computations. <|MaskedSetence|> | **A**: In Section 4 we prove Theorem 1.4.
**B**: In Section 6 we generalize the notion of Coxeter diagram automorphism to the complex setting and prove an analogue to Theorem 1.6 (see Theorem 6.4).
.
**C**: In Section 2, we introduce definitions and prove some preliminary results.
| CAB | CAB | CAB | ABC | Selection 2 |
(Integrability of imaginary geometry coupled with LQG.) The aforementioned integrability of quantum triangles, and the welding results in this paper, and the mating of trees theory [DMS21] can together be used to study the integrablity of imaginary geometry coupled with LQG. For example, a class of permutons (i.e. scaling limit of permutation) called the skew Brownian permutons were recently introduced in [Bor21], with the Baxter permuton [BM22] as a special case. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: As shown in [BHSY22, Proposition 1.14], the expected portion of inversions for these permutons is related to a natural quantity in imaginary geometry coupled with LQG.
**B**: See [BGS22] for other applications of SLE/LQG to permutons.
•.
**C**: In a subsequent work we will derive an exact expression for this quantity.
| ACB | ACB | BAC | ACB | Selection 2 |
<|MaskedSetence|> In Section 2 we first recall some basic facts about Riemannian geometry of surfaces and about sprays. <|MaskedSetence|> Finally, we give the necessary background and relevant results on needle decomposition. <|MaskedSetence|> We also mention an analogue of the curvature-dimension condition from the theory of metric measure spaces [20, 12] in the setting of sprays on surfaces. In Section 4 we prove the equivalence between the nonnegative curvature condition and the Brunn-Minkowski inequality in the case of simple, proper metric sprays (and, more generally, in the case of projectively Finsler-metrizable sprays on surfaces). In Section 5 we provide some examples of weighted spray spaces satisfying the Brunn-Minkowski inequality.
. | **A**:
The paper is organized as follows.
**B**: We then prove Proposition 2.15 regarding projective Finsler-metrizability of magnetic sprays.
**C**: In Section 3 we introduce the notion of a nonnegatively curved weighted spray space, and give a characterization of such spaces in the case of a metric spray on a Riemannian surface.
| ABC | ABC | ABC | ABC | Selection 1 |
<|MaskedSetence|> We will identify Selϕ^(E^n)subscriptSel^italic-ϕsubscript^𝐸𝑛\operatorname{Sel}_{\hat{\phi}}(\hat{E}_{n})roman_Sel start_POSTSUBSCRIPT over^ start_ARG italic_ϕ end_ARG end_POSTSUBSCRIPT ( over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) with the kernel of some ω1(n)+rsubscript𝜔1𝑛𝑟\omega_{1}(n)+ritalic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_n ) + italic_r by ω(n)+c𝜔𝑛𝑐\omega(n)+citalic_ω ( italic_n ) + italic_c matrix, where r,c∈{−1,0,1}𝑟𝑐101r,c\in\{-1,0,1\}italic_r , italic_c ∈ { - 1 , 0 , 1 }. <|MaskedSetence|> Furthermore, as n𝑛nitalic_n grows, we expect that usually ω1(n)≈12loglognsubscript𝜔1𝑛12𝑛\omega_{1}(n)\approx\frac{1}{2}\log\log nitalic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_n ) ≈ divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_log roman_log italic_n and ω(n)≈loglogn𝜔𝑛𝑛\omega(n)\approx\log\log nitalic_ω ( italic_n ) ≈ roman_log roman_log italic_n in the sense of the Erdős–Kac theorem. Heuristically, viewing the entries of the matrix as random elements in 𝔽3subscript𝔽3\mathbb{F}_{3}blackboard_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT picked uniformly, the probability that the matrix takes the maximum possible rank tends to 1111 as n𝑛nitalic_n tends to infinity.
This observation is similar to that in work of Fouvry, Koymans, and Pagano on the 4444-rank of class groups of biquadratic fields [9]. <|MaskedSetence|> | **A**: Then the dimensions of the matrix guarantees the lower bound (1.6).
**B**:
Furthermore, since rankSelϕ(En)ranksubscriptSelitalic-ϕsubscript𝐸𝑛\operatorname{rank}\operatorname{Sel}_{\phi}(E_{n})roman_rank roman_Sel start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is almost always 00 by (1.5), putting this into (1.4) shows that the rank of Selϕ^(E^n)subscriptSel^italic-ϕsubscript^𝐸𝑛\operatorname{Sel}_{\hat{\phi}}(\hat{E}_{n})roman_Sel start_POSTSUBSCRIPT over^ start_ARG italic_ϕ end_ARG end_POSTSUBSCRIPT ( over^ start_ARG italic_E end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) almost always takes the lower bound in (1.6).
In Section 2.2, we show how the lower bound (1.6) and the relation (1.4) can be visualized more explicitly by studying matrices of cubic residue symbols
formed from packaging the local solvability conditions of the cubic equations from (1.1) and (1.2).
**C**: There they show that for almost all odd positive squarefree integers n𝑛nitalic_n, the 4444-rank of the class group of ℚ(n,−1)ℚ𝑛1\mathbb{Q}(\sqrt{n},\sqrt{-1})blackboard_Q ( square-root start_ARG italic_n end_ARG , square-root start_ARG - 1 end_ARG ) is 1111 less than the number of primes dividing n𝑛nitalic_n that are congruent to 3mod4modulo343\bmod 43 roman_mod 4..
| ACB | BAC | BAC | BAC | Selection 4 |
3.1.1 δ−limit-from𝛿\delta-italic_δ -stability in layerwise training Algorithm 1
In this section, we investigate the relevance of manifold regularization in our framework. The main motivation of manifold regularization is to promote δ−limit-from𝛿\delta-italic_δ -stability. <|MaskedSetence|> Thus, stability in this sense is closely related to continuity. <|MaskedSetence|> <|MaskedSetence|> | **A**:
.
**B**: Here, stability means if two data points are “similar” to each other in some sense, then the network predictions on the two data points must be close to each other.
**C**: We now provide the details.
| BCA | BCA | BCA | BCA | Selection 1 |
The CLT used to be the central part of mathematics before 1940s. <|MaskedSetence|> However, the Lindenberg-Lévy CLT is the most well-known and widely-used theorem among statisticians and practitioners. Other versions of CLT such as De Moivre-Laplace CLT and Hajék-Sidak CLT are also used in literature. <|MaskedSetence|> <|MaskedSetence|> | **A**: They have paramount influence in both theory and practice.
**B**: We provide an example below.
Example 3.1 (Kernel density estimator [15])..
**C**: The definitive answer to CLT is given by William Feller in 1940s.
| CAB | CAB | CAB | CAB | Selection 1 |
This work was initiated during the “Big mapping class groups” reading seminar at Bielefeld in Spring 2021. <|MaskedSetence|> MP was partially supported by a grant of the Romanian Ministry of Education and Research, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2020-2798, within PNCDI III. <|MaskedSetence|> He would like to thank Kai-Uwe Bux for his support when he was a postdoc at Bielefeld and many helpful discussions. <|MaskedSetence|> After informing Lvzhou Chen of our results, he informed us recently of his on-going joint work with Danny Calegari and Nathalie Wahl on closely related topics.
. | **A**: XW is currently a member of LMNS and supported by a starter grant at Fudan University.
**B**: He also thanks Javier Aramayona, Lvzhou Chen and Jonas Fleisig for discussions related to this project.
**C**: We thank the members of the reading group.
| CAB | CAB | CAB | ABC | Selection 3 |