wanyu/IteraTeR_full_sent huggingface dataset:
Original:  We prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover, even on bipartite graphs, unless%DIFDELCMD < {\sf %%% NP \subseteq%DIFDELCMD < {\sf %%% coNP /%DIFDELCMD < {\sf %%% poly .
Revised:  We prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover, even on bipartite graphs, unless%DIFDELCMD < {\sf %%% NP \subseteq%DIFDELCMD < {\sf %%% /%DIFDELCMD < {\sf %%% poly .
Edit intent: fluency
Original:  We prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover, even on bipartite graphs, unless%DIFDELCMD < {\sf %%% NP \subseteq%DIFDELCMD < {\sf %%% coNP /%DIFDELCMD < {\sf %%% poly .
Revised:  We prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover, even on bipartite graphs, unless%DIFDELCMD < {\sf %%% NP \subseteq%DIFDELCMD < {\sf %%% coNP %DIFDELCMD < {\sf %%% poly .
Edit intent: coherence
Original:  We prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover, even on bipartite graphs, unless%DIFDELCMD < {\sf %%% NP \subseteq%DIFDELCMD < {\sf %%% coNP /%DIFDELCMD < {\sf %%% poly .
Revised:  We prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover, even on bipartite graphs, unless%DIFDELCMD < {\sf %%% NP \subseteq%DIFDELCMD < {\sf %%% coNP /%DIFDELCMD < {\sf %%% .
Edit intent: fluency
Original:  We prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover, even on bipartite graphs, unless%DIFDELCMD < {\sf %%% NP \subseteq%DIFDELCMD < {\sf %%% coNP /%DIFDELCMD < {\sf %%% poly . Motivated by a question of Boria et al.
Revised:  We prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover, even on bipartite graphs, unless%DIFDELCMD < {\sf %%% NP \subseteq%DIFDELCMD < {\sf %%% coNP /%DIFDELCMD < {\sf %%% poly question of Boria et al.
Edit intent: coherence
Original:  Math. 2015] about the existence of subquadratic kernels for MMVC parameterized by k, we rule out their existence unless P=NP, if we restrict the kernelization algorithms to apply only a type of natural reduction rules that we call "large optimal preserving rules". In particular, these rules contain the typical reduction rules to obtain linear kernels for Vertex Cover .
Revised:  Math. 2015] , and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal .
Edit intent: meaning-changed
Original:  On the positive side,  we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf  {\sf  {\sf   .
Revised:  On the positive side, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf {\sf {\sf .
Edit intent: meaning-changed
Original:  On the positive side,  we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf  {\sf  {\sf   .
Revised:  On the positive side, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition . Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless{\sf {\sf {\sf .
Edit intent: meaning-changed
Original:  On the positive side,  we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf  {\sf  {\sf   .
Revised:  On the positive side, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf NP {\sf {\sf .
Edit intent: fluency
Original:  On the positive side,  we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf  {\sf  {\sf   .
Revised:  On the positive side, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf \subseteq{\sf {\sf .
Edit intent: meaning-changed
Original:  On the positive side,  we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf  {\sf  {\sf   .
Revised:  On the positive side, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf {\sf coNP {\sf .
Edit intent: meaning-changed
Original:  On the positive side,  we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf  {\sf  {\sf   .
Revised:  On the positive side, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf {\sf /{\sf .
Edit intent: meaning-changed
Original:  On the positive side,  we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf  {\sf  {\sf   .
Revised:  On the positive side, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf {\sf {\sf poly .
Edit intent: fluency
Original:  On the positive side,  we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf  {\sf  {\sf   .
Revised:  On the positive side, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf {\sf {\sf . This indicates that parameters smaller than the solution size are unlike to yield polynomial kernels for MMVC .
Edit intent: meaning-changed
Original: In this paper, we extend the additive average Schwarz method to solve second order elliptic boundary value problems with heterogeneous coefficients inside the subdomains and across subdomain interfaces by the mortar technique, where the mortar finite element discretization is on nonmatching meshes.
Revised: In this paper, we extend the additive average Schwarz method to solve second order elliptic boundary value problems with heterogeneous coefficients inside the subdomains and across their interfaces by the mortar technique, where the mortar finite element discretization is on nonmatching meshes.
Edit intent: clarity
Original:  We prove that the condition number for the system of algebraic equations resulting from the extended additive average Schwarz method, corresponding to both coarse spaces, is of the order O(H/h) and independent of jumps of the coefficients, where H and h are the mesh parameters.
Revised:  We prove that the condition numbers of the systems of algebraic equations resulting from the extended additive average Schwarz method, corresponding to both coarse spaces, is of the order O(H/h) and independent of jumps of the coefficients, where H and h are the mesh parameters.
Edit intent: fluency
Original:  We prove that the condition number for the system of algebraic equations resulting from the extended additive average Schwarz method, corresponding to both coarse spaces, is of the order O(H/h) and independent of jumps of the coefficients, where H and h are the mesh parameters.
Revised:  We prove that the condition number for the system of algebraic equations resulting from the extended additive average Schwarz method, corresponding to both coarse spaces, are of the order O(H/h) and independent of jumps of the coefficients, where H and h are the mesh parameters.
Edit intent: fluency
Original:  We prove that the condition number for the system of algebraic equations resulting from the extended additive average Schwarz method, corresponding to both coarse spaces, is of the order O(H/h) and independent of jumps of the coefficients, where H and h are the mesh parameters.
Revised:  We prove that the condition number for the system of algebraic equations resulting from the extended additive average Schwarz method, corresponding to both coarse spaces, is of the order O(H/h) and independent of jumps in the coefficients, where H and h are the mesh parameters.
Edit intent: clarity
Original: Continuity of the value of the martingale optimal transport problem on the real line w.r.t. its marginals was recently established in Backhoff-Veraguas and Pammer [2] and Wiesel [ 19 ].
Revised: Continuity of the value of the martingale optimal transport problem on the real line w.r.t. its marginals was recently established in Backhoff-Veraguas and Pammer [2] and Wiesel [ 21 ].
Edit intent: meaning-changed
Original:  In particular, using results from Beiglb\"ock, Jourdain, Margheriti, and Pammer [ 4 ], we show that the set of martingale measures with fixed marginals is continuous, i.e., lower- and upper hemicontinuous, w.r.t. its marginals.
Revised:  In particular, using results from Beiglb\"ock, Jourdain, Margheriti, and Pammer [ 5 ], we show that the set of martingale measures with fixed marginals is continuous, i.e., lower- and upper hemicontinuous, w.r.t. its marginals.
Edit intent: meaning-changed
Original:  For a given set P of n strings, the first algorithm to compute HOG was given by Cazaux and Rivals [IPL20] requiring O(||P||+n^2) time using superlinear space, where ||P|| is the cummulative sum of the lengths of strings in P.
Revised:  For a given set P of n strings, the first algorithm to compute HOG was given by Cazaux and Rivals [IPL20] requiring O(||P||+n^2) time using superlinear space, where ||P|| is the cumulative sum of the lengths of strings in P.
Edit intent: fluency
Original:  In this paper, we achieve the desired optimal bounds by presenting a simple algorithm that does not use any complex data structures  ]  .
Revised:  In this paper, we achieve the desired optimal bounds by presenting a simple algorithm that does not use any complex data structures ] for a special case of the All Pairs Suffix Prefix (APSP) problem from O(||P||+n^2) time to optimal O(||P||) time, which may be of independent interest .
Edit intent: meaning-changed
Original:  We welcome developers to join the Ivy community by writing their own functions, layers and libraries in Ivy, maximizing their audience and helping to accelerate DL research through the creation of lifelong inter-framework codebases.
Revised:  We welcome developers to join the Ivy community by writing their own functions, layers and libraries in Ivy, maximizing their audience and helping to accelerate DL research through inter-framework codebases.
Edit intent: clarity
Original: The random-cluster model is a unifying framework for studying random graphs, spin systems and random networks . The model is closely related to the classical ferromagnetic Ising and Potts models and is often viewed as a generalization of these models .
Revised: The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models and is often viewed as a generalization of these models .
Edit intent: meaning-changed
Original:  The random-cluster model is parametrized by an  edge probability p and a cluster weight q. Our focus is on the critical regime: p = p_c(q) and q \in (1,2), where p_c(q) is the threshold corresponding to the order-disorder phase transition of the model.
Revised:  The random-cluster model is parametrized by an it edge probability p and a cluster weight q. Our focus is on the critical regime: p = p_c(q) and q \in (1,2), where p_c(q) is the threshold corresponding to the order-disorder phase transition of the model.
Edit intent: fluency
Original:  This  provides a nearly optimal bound (up to the \log\log n factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no previous bound was known.
Revised:  This also provides a nearly optimal bound (up to the \log\log n factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no previous bound was known.
Edit intent: coherence
Original:  This  provides a nearly optimal bound (up to the \log\log n factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no previous bound was known.
Revised:  This provides a nearly optimal bound (up to the \log\log n factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no non-trivial bound was previously known.
Edit intent: meaning-changed
Original:  Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes, and tailored estimates for critical random graphs  .
Revised:  Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes, and tailored estimates for critical random graphs . In addition, we derive an improved comparison inequality between the mixing time of the Chayes-Machta dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting .
Edit intent: meaning-changed
Original:  Additionally, in the context of multi-product bundling pricing, we use our model and provide profit guarantees for simple strategies such as bundle-size pricing and component-pricing with respect to the optimal personalized mixed bundling solution .
Revised:  Additionally, in the context of multi-product bundling pricing, we use our model and provide profit guarantees for simple strategies such as bundle-size pricing and component-pricing with respect to optimal personalized mixed bundling solution .
Edit intent: fluency
Original:  Additionally, in the context of multi-product bundling pricing, we use our model and provide profit guarantees for simple strategies such as bundle-size pricing and component-pricing with respect to the optimal personalized mixed bundling solution .
Revised:  Additionally, in the context of multi-product bundling pricing, we use our model and provide profit guarantees for simple strategies such as bundle-size pricing and component-pricing with respect to the optimal personalized mixed bundle pricing .
Edit intent: clarity
Original: During the special period of the COVID-19 outbreak, this project investigated the driving factors in different information diffusion modes (i.e. broadcasting mode, contagion mode) based on the nomination relations in a charitable social relay campaign on Sina Weibo.
Revised: During the special period of the COVID-19 outbreak, this project investigated the driving factors in different information diffusion modes (i.e. broadcasting mode, contagion mode) based on the nomination relations in a social welfare campaign on Weibo.
Edit intent: clarity
Original:  First, it allows us to show that a (hereditary) class of matrices on a finite alphabet either contains at least n! matrices of size n \times n, or at most c^n for some constant c.
Revised:  First, it allows us to show that a (hereditary) class of matrices over a finite alphabet either contains at least n! matrices of size n \times n, or at most c^n for some constant c.
Edit intent: fluency
Original:  This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class on a finite alphabet, answers our small conjecture [SODA '21] in the case of ordered graphs, and with more work, settles a question first asked by Balogh, Bollob\'as, and Morris [Eur.
Revised:  This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class over a finite alphabet, answers our small conjecture [SODA '21] in the case of ordered graphs, and with more work, settles a question first asked by Balogh, Bollob\'as, and Morris [Eur.
Edit intent: clarity
Original: In this paper, we combine a linearized iterative method with multi-scale deep neural network to compute oscillatory flows for stationary Navior-Stokes equation in complex domains. The multiscale neural network converts the high frequency components in target to low frequency components before training, thus accelerating the convergence of training of neural network for all frequencies .
Revised: In this paper, we present linearized learning methods to accelerate the convergence of training of neural network for all frequencies .
Edit intent: clarity
Original:  To solve the stationary nonlinear Navier-Stokes  equation, we introduce the idea of linearization of Navier-Stokes equation and iterative methods to treat the nonlinear convection term .
Revised:  To solve the stationary nonlinear Navier-Stokes (NS) equation, we introduce the idea of linearization of Navier-Stokes equation and iterative methods to treat the nonlinear convection term .
Edit intent: meaning-changed
Original:  To solve the stationary nonlinear Navier-Stokes  equation, we introduce the idea of linearization of Navier-Stokes equation and iterative methods to treat the nonlinear convection term .
Revised:  To solve the stationary nonlinear Navier-Stokes equation, we integrate the procedure of linearization of Navier-Stokes equation and iterative methods to treat the nonlinear convection term .
Edit intent: clarity
Original:  To solve the stationary nonlinear Navier-Stokes  equation, we introduce the idea of linearization of Navier-Stokes equation and iterative methods to treat the nonlinear convection term . Three forms of linearizations will be considered.
Revised:  To solve the stationary nonlinear Navier-Stokes equation, we introduce the idea of linearization of Navier-Stokes equation and iterative methods to treat the nonlinear convection term in the NS equation into the training process of multi-scale deep neural network approximation of the NS solution. Four forms of linearizations will be considered.
Edit intent: meaning-changed
Original:  First we will conduct a benchmark problem of the linearized schemes in comparison with schemes based directly on the nonlinear PDE. Then, a Navier Stokes flow with high frequency components in a 2-D domain with a hole are learned by the linearized multiscale deep multiscale neural network .
Revised:  First we will conduct a benchmark problem , we solve the highly oscillating stationary flows utilizing the proposed linearized learning with multi-scale neural network for complex domains .
Edit intent: meaning-changed
Original:  We design and analyze online algorithms with efficient tradeoffs between their consistency (i.e., the competitive ratio assuming no prediction error) and their robustness (i.e., the competitive ratio under adversarial error), and whose performance degrades gently as a function of the error.
Revised:  We design and analyze online algorithms with efficient tradeoffs between consistency (i.e., the competitive ratio assuming no prediction error) and their robustness (i.e., the competitive ratio under adversarial error), and whose performance degrades gently as a function of the error.
Edit intent: fluency
Original:  We design and analyze online algorithms with efficient tradeoffs between their consistency (i.e., the competitive ratio assuming no prediction error) and their robustness (i.e., the competitive ratio under adversarial error), and whose performance degrades gently as a function of the error.
Revised:  We design and analyze online algorithms with efficient tradeoffs between their consistency (i.e., the competitive ratio assuming no prediction error) and robustness (i.e., the competitive ratio under adversarial error), and whose performance degrades gently as a function of the error.
Edit intent: fluency
Original:  We design and analyze online algorithms with efficient tradeoffs between their consistency (i.e., the competitive ratio assuming no prediction error) and their robustness (i.e., the competitive ratio under adversarial error), and whose performance degrades gently as a function of the error.
Revised:  We design and analyze online algorithms with efficient tradeoffs between their consistency (i.e., the competitive ratio assuming no prediction error) and their robustness (i.e., the competitive ratio under adversarial error), and whose performance degrades gently as a function of the prediction error. This is the first theoretical study of online bin packing in the realistic setting of erroneous predictions, as well as the first experimental study in the setting in which the input is generated according to both static and evolving distributions.
Edit intent: meaning-changed
Original: Event cameras are a kind of bio-inspired sensors that generate data when the brightness changes . Because of the advantages of low-latency and high dynamic range (HDR) , they are widely used in the field of mobile robots .
Revised: Event cameras are a kind of bio-inspired sensors that generate data when the brightness changes , which are of low-latency and high dynamic range (HDR) , they are widely used in the field of mobile robots .
Edit intent: coherence
Original:  Our findings indicated that the audiovisual stimuli improved the performance of the RSMP BCI, and that the late positive potential (LPP) at Pz contributed to classification accuracy  .
Revised:  Our findings indicated that the audiovisual stimuli improved the performance of the RSMP BCI, and that the late positive potential (LPP) at Pz contributed to classification accuracy . Online accuracy of BCI reached 85.7+-11.5\% .
Edit intent: meaning-changed
Original:  We perform evaluations on multiple downstream tasks on the GLUE benchmark and IMDB reviews with standard sequence length, and find that our Nystr\"omformer performs comparably, or in a few cases, even slightly better, than standard Transformer.
Revised:  We perform evaluations on multiple downstream tasks on the GLUE benchmark and IMDB reviews with standard sequence length, and find that our Nystr\"{o performs comparably, or in a few cases, even slightly better, than standard Transformer.
Edit intent: fluency