text_with_holes stringlengths 124 5.48k | text_candidates stringlengths 49 3.74k | A stringclasses 6
values | B stringclasses 6
values | C stringclasses 6
values | D stringclasses 6
values | label stringclasses 4
values |
|---|---|---|---|---|---|---|
<|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 Ins... | **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.
Th... | 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... | **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... | 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... | **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 so... | 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 Bo... | **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 12nl... | 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... | **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_PO... | 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 use... | 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|> ... | **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 f... | 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 pr... | **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 theore... | 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 hyperint... | **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 denoi... | 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 data... | **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 sequen... | 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 ... | **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, a... | 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 ... | **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 e... | 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|> F... | **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... | 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 t... | **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\m... | **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)H𝑥0subsc... | 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 semanti... | **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... | 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_POSTS... | **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... | **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 natu... | 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... | **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 →... | 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-dimens... | **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 t... | 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 +... | **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 i... | 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 stochas... | **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 lar... | 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 automor... | 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 misinterpre... | **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... | 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... | **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 co... | **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... | **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 orga... | 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|> <|Mas... | **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 ... | 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 N... | **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... | 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 - New... | 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−1xa2𝜎𝑥𝑎𝑎superscr... | **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 it... | 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 ... | **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. scal... | **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 thi... | 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... | **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 Rie... | 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... | **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_POS... | 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 ... | **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... | **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 woul... | **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 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.