wanyu/IteraTeR_v2 huggingface dataset:
Original:  As an immediate consequence, this gives optimal amortised decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, such as planar graphs , minor free graphs , and more .
Revised:  As an immediate consequence, this gives optimal amortised decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs , and more .
Edit intent: clarity
Original:  As an immediate consequence, this gives optimal amortised decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, such as planar graphs , minor free graphs , and more .
Revised:  As an immediate consequence, this gives optimal amortised decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, such as planar graphs , minor free graphs .
Edit intent: coherence
Original: We compute minimal solutions of the rational interpolation problem in terms of different notions of degrees associated to these functions.
Revised: We explore connections between the approach of solving the rational interpolation problem in terms of different notions of degrees associated to these functions.
Edit intent: clarity
Original:  In all the cases, the rational interpolating functions with smallest degree can be computed via the Extended Euclidean Algorithm and syzygies of polynomials .
Revised:  In all the cases, the rational interpolating functions with smallest degree can be computed via the Extended Euclidean Algorithm .
Edit intent: clarity
Original:  As a by-product, we describe the minimal degree in a mu-basis of a polynomial planar parametrization in terms of a "critical" degree arising in the EEA.
Revised:  As a by-product, we describe the minimal degree in a \mu-basis of a polynomial planar parametrization in terms of a "critical" degree arising in the EEA.
Edit intent: fluency
Original: In this paper, we propose new and accurate finite difference methods to discretize the d-dimensional ( d >=\ge 1) tempered  fractional Laplacian and apply them to study the tempered effects on the solution of the fractional problems .
Revised: In this paper, we propose an accurate finite difference methods to discretize the d-dimensional ( d >=\ge 1) tempered fractional Laplacian and apply them to study the tempered effects on the solution of the fractional problems .
Edit intent: clarity
Original: In this paper, we propose new and accurate finite difference methods to discretize the d-dimensional ( d >=\ge 1) tempered  fractional Laplacian and apply them to study the tempered effects on the solution of the fractional problems .
Revised: In this paper, we propose new and accurate finite difference method to discretize the d-dimensional ( d >=\ge 1) tempered fractional Laplacian and apply them to study the tempered effects on the solution of the fractional problems .
Edit intent: fluency
Original: In this paper, we propose new and accurate finite difference methods to discretize the d-dimensional ( d >=\ge 1) tempered  fractional Laplacian and apply them to study the tempered effects on the solution of the fractional problems .
Revised: In this paper, we propose new and accurate finite difference methods to discretize the d-dimensional ( d >=\ge 1) tempered fractional Laplacian and apply it to study the tempered effects on the solution of the fractional problems .
Edit intent: clarity
Original:  Our finite difference methodshave an accuracy of O(h^\epsilon), for u \in C^{0, \alpha+\epsilon} ( R^d \Omega ) if \alpha < 1 (or u \in C^{1, \alpha-1+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\Omega ) if \alpha >= \ge 1) with \epsilon > 0, suggesting the minimum consistency conditions.
Revised:  Our finite difference methodshave an accuracy of O(h^\epsilon), for u \in C^{0, \alpha+\epsilon} ( \Omega ) if \alpha < 1 (or u \in C^{1, \alpha-1+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\Omega ) if \alpha >= \ge 1) with \epsilon > 0, suggesting the minimum consistency conditions.
Edit intent: fluency
Original:  Our finite difference methodshave an accuracy of O(h^\epsilon), for u \in C^{0, \alpha+\epsilon} ( R^d \Omega ) if \alpha < 1 (or u \in C^{1, \alpha-1+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\Omega ) if \alpha >= \ge 1) with \epsilon > 0, suggesting the minimum consistency conditions.
Revised:  Our finite difference methodshave an accuracy of O(h^\epsilon), for u \in C^{0, \alpha+\epsilon} ( R^d \Omega ) if \alpha < 1 (or u \in C^{1, \alpha-1+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (\Omega ) if \alpha >= \ge 1) with \epsilon > 0, suggesting the minimum consistency conditions.
Edit intent: fluency
Original:  Our finite difference methodshave an accuracy of O(h^\epsilon), for u \in C^{0, \alpha+\epsilon} ( R^d \Omega ) if \alpha < 1 (or u \in C^{1, \alpha-1+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\Omega ) if \alpha >= \ge 1) with \epsilon > 0, suggesting the minimum consistency conditions.
Revised:  Our finite difference methodshave an accuracy of O(h^\epsilon), for u \in C^{0, \alpha+\epsilon} ( R^d \Omega ) if \alpha < 1 (or u \in C^{1, \alpha-1+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\Omega ) if \alpha \ge 1) with \epsilon > 0, suggesting the minimum consistency conditions.
Edit intent: fluency
Original:  This accuracy can be improved to O(h^2), for u \in C^{2, \alpha+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\bar{\Omega} ) if \alpha < 1 (or u \in C^{3, \alpha - 1 + \epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\bar{\Omega} ) if \alpha >= \ge 1).
Revised:  The accuracy can be improved to O(h^2), for u \in C^{2, \alpha+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\bar{\Omega} ) if \alpha < 1 (or u \in C^{3, \alpha - 1 + \epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\bar{\Omega} ) if \alpha >= \ge 1).
Edit intent: fluency
Original:  This accuracy can be improved to O(h^2), for u \in C^{2, \alpha+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\bar{\Omega} ) if \alpha < 1 (or u \in C^{3, \alpha - 1 + \epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\bar{\Omega} ) if \alpha >= \ge 1).
Revised:  This accuracy can be improved to O(h^2), for u \in C^{2, \alpha+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (\bar{\Omega} ) if \alpha < 1 (or u \in C^{3, \alpha - 1 + \epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\bar{\Omega} ) if \alpha >= \ge 1).
Edit intent: fluency
Original:  This accuracy can be improved to O(h^2), for u \in C^{2, \alpha+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\bar{\Omega} ) if \alpha < 1 (or u \in C^{3, \alpha - 1 + \epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\bar{\Omega} ) if \alpha >= \ge 1).
Revised:  This accuracy can be improved to O(h^2), for u \in C^{2, \alpha+\epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (R^d\bar{\Omega} ) if \alpha < 1 (or u \in C^{3, \alpha - 1 + \epsilon %DIFDELCMD < \MBLOCKRIGHTBRACE%%% (\bar{\Omega} ) if \alpha >= \ge 1).
Edit intent: fluency
Original:  Numerical results confirm our analytical conclusions and provide further insights of our methods in solving the tempered fractional Poisson problem.
Revised:  Numerical experiments confirm our analytical conclusions and provide further insights of our methods in solving the tempered fractional Poisson problem.
Edit intent: clarity
Original:  Numerical results confirm our analytical conclusions and provide further insights of our methods in solving the tempered fractional Poisson problem.
Revised:  Numerical results confirm our analytical results and provide insights in solving the tempered fractional Poisson problem.
Edit intent: clarity
Original:  It suggests that to achieve the second order of accuracy, our methods only require the solution u \in C^{1,1}( R^d \bar{\Omega} ) for any 0<\alpha<2.
Revised:  It suggests that to achieve the second order of accuracy, our method only requires the solution u \in C^{1,1}( R^d \bar{\Omega} ) for any 0<\alpha<2.
Edit intent: clarity
Original:  It suggests that to achieve the second order of accuracy, our methods only require the solution u \in C^{1,1}( R^d \bar{\Omega} ) for any 0<\alpha<2.
Revised:  It suggests that to achieve the second order of accuracy, our methods only require the solution u \in C^{1,1}( \bar{\Omega} ) for any 0<\alpha<2.
Edit intent: clarity
Original:  Moreover, if the solution  u \in C^{p, s}( R^d) for \bar{\Omega} 0< = p , s <= \le 1, our methods have the accuracy of O(h^{p+s}) in solving the tempered Poisson problem.
Revised:  Moreover, if the solution u \in C^{p, s}( \bar{\Omega} 0< = p , s <= \le 1, our methods have the accuracy of O(h^{p+s}) in solving the tempered Poisson problem.
Edit intent: clarity
Original:  Moreover, if the solution  u \in C^{p, s}( R^d) for \bar{\Omega} 0< = p , s <= \le 1, our methods have the accuracy of O(h^{p+s}) in solving the tempered Poisson problem.
Revised:  Moreover, if the solution u \in C^{p, s}( R^d) for \bar{\Omega} 0< s <= \le 1, our methods have the accuracy of O(h^{p+s}) in solving the tempered Poisson problem.
Edit intent: clarity
Original:  Moreover, if the solution  u \in C^{p, s}( R^d) for \bar{\Omega} 0< = p , s <= \le 1, our methods have the accuracy of O(h^{p+s}) in solving the tempered Poisson problem.
Revised:  Moreover, if the solution u \in C^{p, s}( R^d) for \bar{\Omega} 0< = p , s \le 1, our methods have the accuracy of O(h^{p+s}) in solving the tempered Poisson problem.
Edit intent: clarity
Original:  Moreover, if the solution  u \in C^{p, s}( R^d) for \bar{\Omega} 0< = p , s <= \le 1, our methods have the accuracy of O(h^{p+s}) in solving the tempered Poisson problem.
Revised:  Moreover, if the solution u \in C^{p, s}( R^d) for \bar{\Omega} 0< = p , s <= \le 1, our method has the accuracy of O(h^{p+s}) in solving the tempered Poisson problem.
Edit intent: clarity
Original:  Finally, we apply them together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including nonlinear Schrodinger equation, Allen-Cahn equation , and Gray-Scott equations.
Revised:  Finally, we apply it together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including nonlinear Schrodinger equation, Allen-Cahn equation , and Gray-Scott equations.
Edit intent: coherence
Original:  Finally, we apply them together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including nonlinear Schrodinger equation, Allen-Cahn equation , and Gray-Scott equations.
Revised:  Finally, we apply them together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including nonlinear Schrodinger equation, Allen-Cahn equation and Gray-Scott equations.
Edit intent: fluency
Original:  Modern adversarial methods either directly alter pixel colors , or "paint" colors onto a 3D shapes.
Revised:  Current adversarial methods directly alter pixel colors , or "paint" colors onto a 3D shapes.
Edit intent: coherence
Original:  We show that the definition of fractality cannot be applied to networks with ' tree-like' structure and exponential growth rate of neighborhoods.
Revised:  We show that the definition of fractality cannot be applied to networks with ` tree-like' structure and exponential growth rate of neighborhoods.
Edit intent: fluency
Original: Let x=(x_1,x_2,\cdots,x_n)^T be a vector of non-zero algebraic numbers , the set R_x:=\{(k_1,k_2,\cdots,k_n)^T\inZ^n\;|\;x_1^{k_1x_2^{k_2}\cdots x_n^{k_n}=1\} is called }the multiplicative dependence lattice of x.
Revised: x_2^{k_2}\cdots x_n^{k_n}=1\} is called }the multiplicative dependence lattice of x.
Edit intent: clarity
Original:  In this paper we propose the conception of therank of a finite sequence of non-zero algebraic numbers, which turns out to be closely related to the rank of the lattice, and as well as to the complexity.
Revised:  In this paper we propose the conception of the of a finite sequence of non-zero algebraic numbers, which turns out to be closely related to the rank of the lattice, and as well as to the complexity.
Edit intent: fluency
Original: Estimating the impact of environmental processes on vertical reef development in geological timescales due to complex models and data with missing information is a very challenging task.
Revised: Estimating the impact of environmental processes on vertical reef development in geological time is a very challenging task.
Edit intent: clarity
Original:  PyReef-Core is a deterministic , carbonate stratigraphic forward model designed to simulate the key biological and physical processes that determine vertical  accretion and assemblage changes in  reef drill cores.
Revised:  PyReef-Core is a deterministic carbonate stratigraphic forward model designed to simulate the key biological and physical processes that determine vertical accretion and assemblage changes in reef drill cores.
Edit intent: fluency
Original:  PyReef-Core is a deterministic , carbonate stratigraphic forward model designed to simulate the key biological and physical processes that determine vertical  accretion and assemblage changes in  reef drill cores.
Revised:  PyReef-Core is a deterministic , carbonate stratigraphic forward model designed to simulate the key biological and environmental processes that determine vertical accretion and assemblage changes in reef drill cores.
Edit intent: clarity
Original:  PyReef-Core is a deterministic , carbonate stratigraphic forward model designed to simulate the key biological and physical processes that determine vertical  accretion and assemblage changes in  reef drill cores.
Revised:  PyReef-Core is a deterministic , carbonate stratigraphic forward model designed to simulate the key biological and physical processes that determine vertical reef accretion and assemblage changes in reef drill cores.
Edit intent: fluency
Original:  PyReef-Core is a deterministic , carbonate stratigraphic forward model designed to simulate the key biological and physical processes that determine vertical  accretion and assemblage changes in  reef drill cores.
Revised:  PyReef-Core is a deterministic , carbonate stratigraphic forward model designed to simulate the key biological and physical processes that determine vertical accretion and assemblage changes in fossil reef drill cores.
Edit intent: clarity
Original:  BayesReef provides insights into the complex posterior distributions of parameters in PyReef-Core and provides the groundwork for future research in this area.
Revised:  BayesReef provides insights into the complex posterior distributions of parameters in pyReef-Core, which provides the groundwork for future research in this area.
Edit intent: coherence
Original:  We derive a general kernelization method for this problem and thus obtain an upper bound for the complexity of its counting version of O( 2kR 2  ^{( 1-k ) R }) for number of variables R and clauses-to-variables ratio k . Combining this method with previous  results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16})  .
Revised:  We derive a general kernelization method for this problem and thus obtain an upper bound for the complexity of its counting version of O( 2 ^{( 1-k ) R }) for number of variables R and clauses-to-variables ratio k . Combining this method with previous results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}) .
Edit intent: clarity
Original:  We derive a general kernelization method for this problem and thus obtain an upper bound for the complexity of its counting version of O( 2kR 2  ^{( 1-k ) R }) for number of variables R and clauses-to-variables ratio k . Combining this method with previous  results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16})  .
Revised:  We derive a general kernelization method for this problem and thus obtain an upper bound for the complexity of its counting version of O( 2kR 2 ^{( 1-k ) r }) for number of variables R and clauses-to-variables ratio k . Combining this method with previous results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}) .
Edit intent: fluency
Original:  We derive a general kernelization method for this problem and thus obtain an upper bound for the complexity of its counting version of O( 2kR 2  ^{( 1-k ) R }) for number of variables R and clauses-to-variables ratio k . Combining this method with previous  results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16})  .
Revised:  We derive a general kernelization method for this problem and thus obtain an upper bound for the complexity of its counting version of O( 2kR 2 ^{( 1-k ) R }) for number of variables r and clauses-to-variables ratio k . Combining this method with previous results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}) .
Edit intent: fluency
Original:  We derive a general kernelization method for this problem and thus obtain an upper bound for the complexity of its counting version of O( 2kR 2  ^{( 1-k ) R }) for number of variables R and clauses-to-variables ratio k . Combining this method with previous  results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16})  .
Revised:  We derive a general kernelization method for this problem and thus obtain an upper bound for the complexity of its counting version of O( 2kR 2 ^{( 1-k ) R }) for number of variables R and clauses-to-variables ratio \kappa . Combining this method with previous results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}) .
Edit intent: clarity
Original:  We derive a general kernelization method for this problem and thus obtain an upper bound for the complexity of its counting version of O( 2kR 2  ^{( 1-k ) R }) for number of variables R and clauses-to-variables ratio k . Combining this method with previous  results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16})  .
Revised:  We derive a general kernelization method for this problem and thus obtain an upper bound for the complexity of its counting version of O( 2kR 2 ^{( 1-k ) R }) for number of variables R and clauses-to-variables ratio k . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}) .
Edit intent: clarity
Original: This paper illustrates the power of Gaussian Elimination by adapting it to the problem of Exact Satisfiability.
Revised: We use Gaussian Elimination by adapting it to the problem of Exact Satisfiability.
Edit intent: clarity
Original: This paper illustrates the power of Gaussian Elimination by adapting it to the problem of Exact Satisfiability.
Revised: This paper illustrates the power of Gaussian Elimination by adapting it to Exact Satisfiability.
Edit intent: clarity
Original:  For 1-in-3 SAT instances with non-negated literals we are able to obtain considerably smaller equivalent instances of 0/1 Integer Programming restricted to equality only .
Revised:  For 1-in-3 SAT instances with non-negated literals we obtain considerably smaller equivalent instances of 0/1 Integer Programming restricted to equality only .
Edit intent: clarity
Original:  For 1-in-3 SAT instances with non-negated literals we are able to obtain considerably smaller equivalent instances of 0/1 Integer Programming restricted to equality only .
Revised:  For 1-in-3 SAT instances with non-negated literals we are able to obtain considerably smaller equivalent instances of 0/1 Integer Programming restricted to equality .
Edit intent: clarity
Original:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-\kappa ) r }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Revised:  Thus we obtain an upper bound for the complexity of its counting version of O( 2K\R 2 \kappa r 2 ^{( 1-\kappa ) r }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Edit intent: clarity
Original:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-\kappa ) r }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Revised:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 ^{( 1-\kappa ) r }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Edit intent: coherence
Original:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-\kappa ) r }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Revised:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-K ) r }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Edit intent: clarity
Original:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-\kappa ) r }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Revised:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-\kappa ) R }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Edit intent: fluency
Original:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-\kappa ) r }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Revised:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-\kappa ) r }) for number of variables R and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Edit intent: fluency
Original:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-\kappa ) r }) for number of variables r and clauses-to-variables ratio \kappa . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Revised:  Thus we obtain an upper bound for the complexity of its counting version of O( \R 2 \kappa r 2 ^{( 1-\kappa ) r }) for number of variables r and clauses-to-variables ratio K . Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^{3|V|/8}) and O(4/3|V|2^{3|V|/16}).
Edit intent: clarity