wanyu/IteraTeR_v2 huggingface dataset:
Original:  Let%DIFDELCMD < {\Bbb %%% C be the complex field, and let%DIFDELCMD < {\Bbb %DIFDELCMD < %%% C ^X be the%DIFDELCMD < {\Bbb %%% C G-module of complex functions on X . By decomposing%DIFDELCMD < {\Bbb %%% C ^X into the direct sum of G-linear components, we obtain an orthogonal G-linear basis%DIFDELCMD < \widehat %%% X of%DIFDELCMD < {\Bbb %%% C ^X.
Revised:  Let%DIFDELCMD < {\Bbb %%% C be the complex field, and let%DIFDELCMD < {\Bbb %DIFDELCMD < %%% C ^X be the%DIFDELCMD < {\Bbb %%% C G-module of complex functions on X . By decomposing%DIFDELCMD < {\Bbb %%% C ^X into the direct sum of G-linear components, we obtain an orthogonal G-linear basis%DIFDELCMD < \widehat %%% X of%DIFDELCMD < {\Bbb %%% ^X.
Edit intent: fluency
Original:  As a generalization of the Fourier transforms of functions on finite abelian groups , for any f \in%DIFDELCMD < {\Bbb %%% C ^X we define the Fourier transform%DIFDELCMD < \widehat %%% f of f to be a function on%DIFDELCMD < \widehat %%% X . Then we define the bentness of f \in%DIFDELCMD < {\Bbb %%% C ^X in terms of  \widehat f(\lambda) for \lambda \in%DIFDELCMD < \widehat %%% X, and discuss its characterizations.
Revised:  As a generalization of the Fourier transforms of functions on finite abelian groups %DIFDELCMD < {\Bbb %%% C ^X we define the Fourier transform%DIFDELCMD < \widehat %%% f of f to be a function on%DIFDELCMD < \widehat %%% X . Then we define the bentness of f \in%DIFDELCMD < {\Bbb %%% C ^X in terms of \widehat f(\lambda) for \lambda \in%DIFDELCMD < \widehat %%% X, and discuss its characterizations.
Edit intent: clarity
Original:  As a generalization of the Fourier transforms of functions on finite abelian groups , for any f \in%DIFDELCMD < {\Bbb %%% C ^X we define the Fourier transform%DIFDELCMD < \widehat %%% f of f to be a function on%DIFDELCMD < \widehat %%% X . Then we define the bentness of f \in%DIFDELCMD < {\Bbb %%% C ^X in terms of  \widehat f(\lambda) for \lambda \in%DIFDELCMD < \widehat %%% X, and discuss its characterizations.
Revised:  As a generalization of the Fourier transforms of functions on finite abelian groups , for any f \in%DIFDELCMD < {\Bbb %%% ^X we define the Fourier transform%DIFDELCMD < \widehat %%% f of f to be a function on%DIFDELCMD < \widehat %%% X . Then we define the bentness of f \in%DIFDELCMD < {\Bbb %%% C ^X in terms of \widehat f(\lambda) for \lambda \in%DIFDELCMD < \widehat %%% X, and discuss its characterizations.
Edit intent: fluency
Original:  As a generalization of the Fourier transforms of functions on finite abelian groups , for any f \in%DIFDELCMD < {\Bbb %%% C ^X we define the Fourier transform%DIFDELCMD < \widehat %%% f of f to be a function on%DIFDELCMD < \widehat %%% X . Then we define the bentness of f \in%DIFDELCMD < {\Bbb %%% C ^X in terms of  \widehat f(\lambda) for \lambda \in%DIFDELCMD < \widehat %%% X, and discuss its characterizations.
Revised:  As a generalization of the Fourier transforms of functions on finite abelian groups , for any f \in%DIFDELCMD < {\Bbb %%% C ^X we define the Fourier transform%DIFDELCMD < \widehat %%% f of f to be a function on%DIFDELCMD < \widehat %%% . Then we define the bentness of f \in%DIFDELCMD < {\Bbb %%% C ^X in terms of \widehat f(\lambda) for \lambda \in%DIFDELCMD < \widehat %%% X, and discuss its characterizations.
Edit intent: fluency
Original:  As a generalization of the Fourier transforms of functions on finite abelian groups , for any f \in%DIFDELCMD < {\Bbb %%% C ^X we define the Fourier transform%DIFDELCMD < \widehat %%% f of f to be a function on%DIFDELCMD < \widehat %%% X . Then we define the bentness of f \in%DIFDELCMD < {\Bbb %%% C ^X in terms of  \widehat f(\lambda) for \lambda \in%DIFDELCMD < \widehat %%% X, and discuss its characterizations.
Revised:  As a generalization of the Fourier transforms of functions on finite abelian groups , for any f \in%DIFDELCMD < {\Bbb %%% C ^X we define the Fourier transform%DIFDELCMD < \widehat %%% f of f to be a function on%DIFDELCMD < \widehat %%% X . Then we define the bentness of f \in%DIFDELCMD < {\Bbb %%% ^X in terms of \widehat f(\lambda) for \lambda \in%DIFDELCMD < \widehat %%% X, and discuss its characterizations.
Edit intent: fluency
Original:  In order to explain our main results clearly, examples are also presented.
Revised:  In order to explain the main results clearly, examples are also presented.
Edit intent: fluency
Original:  The performance of a statistical machine translation system is empirically found to improve by using the conditional probabilities of phrase pairs computed by the RNN Encoder--Decoder as an additional feature in the existing log-linear model.
Revised:  The performance of a statistical machine translation system is empirically found to improve by using the conditional probabilities of phrase pairs computed by the RNN Encoder--Decoder as an additional feature in the existing linear model.
Edit intent: clarity
Original: In this paper, we propose a novel neural network model called RNN Encoder--Decoder that consists of two recurrent neural networks (RNN).
Revised: In this paper, we propose a novel neural network model called RNN Encoder-Decoder that consists of two recurrent neural networks (RNN).
Edit intent: fluency
Original:  The performance of a statistical machine translation system is empirically found to improve by using the conditional probabilities of phrase pairs computed by the RNN Encoder--Decoder as an additional feature in the existing linear model.
Revised:  The performance of a statistical machine translation system is empirically found to improve by using the conditional probabilities of phrase pairs computed by the RNN Encoder-Decoder as an additional feature in the existing linear model.
Edit intent: fluency
Original:  The performance of a statistical machine translation system is empirically found to improve by using the conditional probabilities of phrase pairs computed by the RNN Encoder--Decoder as an additional feature in the existing linear model.
Revised:  The performance of a statistical machine translation system is empirically found to improve by using the conditional probabilities of phrase pairs computed by the RNN Encoder--Decoder as an additional feature in the existing log-linear model.
Edit intent: clarity
Original: Stochastic gradient algorithms compute the gradient based on only one sample (or just a few samples ) and enjoy low computational cost per iteration.
Revised: Stochastic gradient algorithms estimate the gradient based on only one sample (or just a few samples ) and enjoy low computational cost per iteration.
Edit intent: clarity
Original: Stochastic gradient algorithms compute the gradient based on only one sample (or just a few samples ) and enjoy low computational cost per iteration.
Revised: Stochastic gradient algorithms compute the gradient based on only one or a few samples ) and enjoy low computational cost per iteration.
Edit intent: clarity
Original: Stochastic gradient algorithms compute the gradient based on only one sample (or just a few samples ) and enjoy low computational cost per iteration.
Revised: Stochastic gradient algorithms compute the gradient based on only one sample (or just a few samples and enjoy low computational cost per iteration.
Edit intent: fluency
Original:  They are widely used in large-scale optimization problems.
Revised:  They have been widely used in large-scale optimization problems.
Edit intent: clarity
Original:  Under the strongly convex condition , these variance-reduced stochastic gradient algorithms achieve a linear convergence rate.
Revised:  Under the strongly convex condition , these variance-reduced stochastic gradient algorithms achieve a linear convergence rate.
Edit intent: fluency
Original:  However, in many machine learning problems , the objective function to be minimized is convex butnot strongly convex.
Revised:  However, many machine learning problems , the objective function to be minimized is convex butnot strongly convex.
Edit intent: fluency
Original:  However, in many machine learning problems , the objective function to be minimized is convex butnot strongly convex.
Revised:  However, in many machine learning problems not strongly convex.
Edit intent: clarity
Original:  In this paper, we propose a Variance-Reduced Projected Stochastic Gradient (VRPSG) algorithm, which can efficiently solve a class of constrained optimization problems .
Revised:  In this paper, we propose a Variance-Reduced Projected Stochastic Gradient (VRPSG) to solve a class of constrained optimization problems .
Edit intent: clarity
Original:  As the main technical contribution of this paper, we show that the proposed VRPSG algorithm achieves a linear convergence rate without the strong convexityassumption.
Revised:  As the main technical contribution of this paper, we show that the proposed VRPSG algorithm achieves a linear convergence rate the strong convexityassumption.
Edit intent: clarity
Original:  To the best of our knowledge, this is the first work that establishes the linear convergence rate for the variance-reduced stochastic gradient algorithm\emph{without strong convexity.
Revised:  To the best of our knowledge, this is the first work that establishes the linear convergence rate for the variance-reduced stochastic gradient \emph{without strong convexity.
Edit intent: clarity
Original: In this paper we consider large state space continuous time Markov chains  arising in the field of systems biology.
Revised: In this paper we consider large state space continuous time Markov chains (MCs) arising in the field of systems biology.
Edit intent: clarity
Original:  For a class of such models, namely, for density dependent families  , Kurtz has proposed two kinds of approximations.
Revised:  For density dependent families , Kurtz has proposed two kinds of approximations.
Edit intent: clarity
Original:  One is based on ordinary differential equations and provides a deterministic approximation while the other uses a diffusion process with which the resulting approximation is stochastic .
Revised:  One is based on ordinary differential equations , while the other uses a diffusion process with which the resulting approximation is stochastic .
Edit intent: clarity
Original:  One is based on ordinary differential equations and provides a deterministic approximation while the other uses a diffusion process with which the resulting approximation is stochastic .
Revised:  One is based on ordinary differential equations and provides a deterministic approximation while the other uses a diffusion process .
Edit intent: clarity
Original:  In  this paper we propose to apply the jump-diffusion approximation only to the density dependent components associated with high population levels.
Revised:  In this paper we propose to apply the jump-diffusion approximation only to those components of the model that are in density dependent form and are associated with high population levels.
Edit intent: clarity
Original:  We show that the stochastic differential equations that characterize this process can be derived automatically both from the description of the original Markov chain or starting from the correspondent Petri net .
Revised:  We show that the stochastic differential equations that characterize this process can be derived automatically both from the description of the original Markov chains or starting from the correspondent Petri net .
Edit intent: fluency
Original:  We give a randomized near-linear-time (1-\varepsilon)-approximation algorithm for this problem: in O(  ( n /\varepsilon ^4) \log ^2 n \log( 1/ \delta )) time we find a convex polygon contained in P that, with probability at least 1-\delta , has area at least (1-\varepsilon) times the area of an optimal solution  .
Revised:  We give a randomized near-linear-time (1-\varepsilon)-approximation algorithm for this problem: in O( ( 1 /\varepsilon ^4) \log ^2 n \log( 1/ \delta )) time we find a convex polygon contained in P that, with probability at least 1-\delta , has area at least (1-\varepsilon) times the area of an optimal solution .
Edit intent: fluency
Original:  We give a randomized near-linear-time (1-\varepsilon)-approximation algorithm for this problem: in O(  ( n /\varepsilon ^4) \log ^2 n \log( 1/ \delta )) time we find a convex polygon contained in P that, with probability at least 1-\delta , has area at least (1-\varepsilon) times the area of an optimal solution  .
Revised:  We give a randomized near-linear-time (1-\varepsilon)-approximation algorithm for this problem: in O( ( n /\varepsilon ^4) \log ^2 n \log( 1/ \varepsilon^4 )) time we find a convex polygon contained in P that, with probability at least 1-\delta , has area at least (1-\varepsilon) times the area of an optimal solution .
Edit intent: clarity
Original:  However, the ex-sheep-farmer, artist, and satirist, Samuel Butler , was able to address, in informational terms depleted of theological trappings, both organic evolution and mind in the universe.
Revised:  However, Samuel Butler was able to address, in informational terms depleted of theological trappings, both organic evolution and mind in the universe.
Edit intent: clarity
Original: Understanding how epidemics spread in a system is a crucial step to prevent and control outbreaks, with broad implications on the system's functioning, health, and associated epidemic costs.
Revised: Understanding how epidemics spread in a system is a crucial step to prevent and control outbreaks, with broad implications on the system's functioning, health, and associated costs.
Edit intent: clarity
Original:  One important ingredient to consider is the pattern of disease-transmission contacts among the elements, however lack of data or possible delays in providing updated records may hinder its use, especially for time-varying patterns.
Revised:  One important ingredient to consider is the pattern of disease-transmission contacts among the elements, however lack of data or delays in providing updated records may hinder its use, especially for time-varying patterns.
Edit intent: clarity
Original:  We define the node's loyalty as a local measure of the tendency to maintain contacts with the same elements over time, and uncover important non-trivial correlations with the node's epidemic risk.
Revised:  We define the node's loyalty as a local measure of its tendency to maintain contacts with the same elements over time, and uncover important non-trivial correlations with the node's epidemic risk.
Edit intent: clarity
Original:  We show that a risk assessment analysis incorporating this knowledge and based on past structural and temporal pattern properties provide accurate predictions for both systems.
Revised:  We show that a risk assessment analysis incorporating this knowledge and based on past structural and temporal pattern properties provides accurate predictions for both systems.
Edit intent: fluency
Original:  High accuracy  is recovered across variations of the system's features, whereas the predictive power is found to be system-specific.
Revised:  High accuracy of our predictions is recovered across variations of the system's features, whereas the predictive power is found to be system-specific.
Edit intent: clarity
Original:  High accuracy  is recovered across variations of the system's features, whereas the predictive power is found to be system-specific.
Revised:  High accuracy is recovered across different settings, while the amount of possible predictions is system-specific.
Edit intent: clarity
Original:  The Joint Research Centre has developed an integrated model to provide a macro-scale set of indices to evaluate the connectivity of  Natura 2000 network, which forms the backbone of a Green Infrastructure for Europe.
Revised:  The Joint Research Centre has developed an integrated model to provide a macro-scale set of indices to evaluate the connectivity of the Natura 2000 network, which forms the backbone of a Green Infrastructure for Europe.
Edit intent: fluency
Original:  The model allows a wide assessment and comparison to be performed across country in terms of structural (spatially connected or isolated sites) and functional connectivity (least-cost distances between sites influenced by distribution, distance and land cover).
Revised:  The model allows a wide assessment and comparison to be performed across countries in terms of structural (spatially connected or isolated sites) and functional connectivity (least-cost distances between sites influenced by distribution, distance and land cover).
Edit intent: fluency
Original:  Main conclusion:  Natura 2000 network in Europe shows differences among countries in terms of the sizes and number of sites, their distribution as well as distances between sites.
Revised:  Main conclusion: The Natura 2000 network in Europe shows differences among countries in terms of the sizes and number of sites, their distribution as well as distances between sites.
Edit intent: fluency
Original:  Main conclusion:  Natura 2000 network in Europe shows differences among countries in terms of the sizes and number of sites, their distribution as well as distances between sites.
Revised:  Main conclusion: Natura 2000 network in Europe shows differences among countries in terms of the sizes and numbers of sites, their distribution as well as distances between sites.
Edit intent: fluency
Original:  Connectivity has been assessed on the basis of  500 m average inter-site distance, roads and intensive land use as barrier effect as well as the presence of "green" corridors.
Revised:  Connectivity has been assessed on the basis of a 500 m average inter-site distance, roads and intensive land use as barrier effect as well as the presence of "green" corridors.
Edit intent: fluency
Original:  Connectivity has been assessed on the basis of  500 m average inter-site distance, roads and intensive land use as barrier effect as well as the presence of "green" corridors.
Revised:  Connectivity has been assessed on the basis of 500 m average inter-site distance, roads and intensive land use as barrier effects as well as the presence of "green" corridors.
Edit intent: fluency
Original:  Yet previous work addressing the impacts of partial network data is surprisingly limited, focuses primarily on missing nodes, and suggests that network statistics derived from subsampled data are not suitable estimators for the same network statistics describing the overall network topology.
Revised:  Previous work addressing the impacts of partial network data is surprisingly limited, focuses primarily on missing nodes, and suggests that network statistics derived from subsampled data are not suitable estimators for the same network statistics describing the overall network topology.
Edit intent: coherence
Original:  Our aim here is to generate scaling methods to predict true network statistics  from only partial knowledge of nodes, links, or weights.
Revised:  We generate scaling methods to predict true network statistics from only partial knowledge of nodes, links, or weights.
Edit intent: coherence
Original:  We analyse the feasibility of a particular loading mechanism, active symplasmic loading, also called the polymer trap mechanism, where sucrose is transformed into heavier sugars, such as raffinose and stachyose, in the intermediary-type companion cells bordering the sieve elements in the minor veins of the phloem.
Revised:  We analyze the feasibility of a particular loading mechanism, active symplasmic loading, also called the polymer trap mechanism, where sucrose is transformed into heavier sugars, such as raffinose and stachyose, in the intermediary-type companion cells bordering the sieve elements in the minor veins of the phloem.
Edit intent: fluency
Original:  The value of \mu \in%DIFDELCMD < \left(%%%  1,3 %DIFDELCMD < \right) %%% that maximises the search efficiency can shift substantially depending on two modelling parameters in the foraging process: the number of steps and the intensity of returning, which can represent a number of general internal or external hidden constraints in foraging.
Revised:  The value of \mu %DIFDELCMD < \left(%%% 1,3 %DIFDELCMD < \right) %%% that maximises the search efficiency can shift substantially depending on two modelling parameters in the foraging process: the number of steps and the intensity of returning, which can represent a number of general internal or external hidden constraints in foraging.
Edit intent: fluency
Original:  The value of \mu \in%DIFDELCMD < \left(%%%  1,3 %DIFDELCMD < \right) %%% that maximises the search efficiency can shift substantially depending on two modelling parameters in the foraging process: the number of steps and the intensity of returning, which can represent a number of general internal or external hidden constraints in foraging.
Revised:  The value of \mu \in%DIFDELCMD < \left(%%% 1,3 %DIFDELCMD < \right) %%% ), depending on the search efficiency can shift substantially depending on two modelling parameters in the foraging process: the number of steps and the intensity of returning, which can represent a number of general internal or external hidden constraints in foraging.
Edit intent: clarity
Original:  Our model provides a new understanding of the origin of L\'{e}vy-flight foraging  as well as the substantial difference of the \mu-values observed in empirical data, and therefore opens a new avenue for analysing trajectory and movement patterns of animals .
Revised:  Our model provides a new perspective on L\'{e}vy-flight foraging as well as the substantial difference of the \mu-values observed in empirical data, and therefore opens a new avenue for analysing trajectory and movement patterns of animals .
Edit intent: clarity
Original:  Our model provides a new understanding of the origin of L\'{e}vy-flight foraging  as well as the substantial difference of the \mu-values observed in empirical data, and therefore opens a new avenue for analysing trajectory and movement patterns of animals .
Revised:  Our model provides a new understanding of the origin of L\'{e}vy-flight foraging as well as offers a realistic framework for analysing animal movement patterns from empirical data .
Edit intent: clarity
Original: We study novel arithmetic algorithms on a canonical number representation based on the Catalan family of combinatorial objects  .
Revised: We describe arithmetic algorithms on a canonical number representation based on the Catalan family of combinatorial objects .
Edit intent: clarity
Original:  Our algorithms work on a generic {\em  representation that we illustrate on instances  like ordered binary and multiway trees , balanced parentheses languages as well as the usual bitstring-based natural numbers seen through the same generic interface as members of the Catalan family.
Revised:  Our algorithms work on a {\em representation that we illustrate on instances like ordered binary and multiway trees , balanced parentheses languages as well as the usual bitstring-based natural numbers seen through the same generic interface as members of the Catalan family.
Edit intent: fluency
Original:  Our algorithms work on a generic {\em  representation that we illustrate on instances  like ordered binary and multiway trees , balanced parentheses languages as well as the usual bitstring-based natural numbers seen through the same generic interface as members of the Catalan family.
Revised:  Our algorithms work on a generic {\em representation that we illustrate on instances members of the Catalan family, like ordered binary and multiway trees , balanced parentheses languages as well as the usual bitstring-based natural numbers seen through the same generic interface as members of the Catalan family.
Edit intent: clarity
Original:  For numbers corresponding to Catalan objects of low representation complexity, our algorithms provide super-exponential gains while their average and worst case complexity is within constant factors of their traditional counterparts  .
Revised:  For numbers corresponding to Catalan objects of low representation complexity, our algorithms provide super-exponential gains while their average and worst case complexity is within constant factors of their traditional counterparts , our algorithms provide super-exponential gains for numbers corresponding to Catalan objects of low representation size .
Edit intent: clarity