wanyu/IteraTeR_v2 huggingface dataset:
Original:  As a result, if a uniform time grid \{t_k=k\tau, k \in \mathbb{N\} } is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij at the very first time-step , through the initial values of both u and v at stencil nodes ;
Revised:  As a result, if a uniform time grid \} } is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij at the very first time-step , through the initial values of both u and v at stencil nodes ;
Edit intent: fluency
Original:  As a result, if a uniform time grid \{t_k=k\tau, k \in \mathbb{N\} } is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij at the very first time-step , through the initial values of both u and v at stencil nodes ;
Revised:  As a result, if a uniform time grid \{t_k=k\tau, k \in \mathbb{N\} } is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1) the %DIFDELCMD < {\bf %%% x _{ij at the very first time-step , through the initial values of both u and v at stencil nodes ;
Edit intent: clarity
Original:  As a result, if a uniform time grid \{t_k=k\tau, k \in \mathbb{N\} } is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij at the very first time-step , through the initial values of both u and v at stencil nodes ;
Revised:  As a result, if a uniform time grid \{t_k=k\tau, k \in \mathbb{N\} } is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1) the calculation of u(%DIFDELCMD < {\bf %%% x solution calculation at the very first time-step , through the initial values of both u and v at stencil nodes ;
Edit intent: clarity
Original:  As a result, if a uniform time grid \{t_k=k\tau, k \in \mathbb{N\} } is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij at the very first time-step , through the initial values of both u and v at stencil nodes ;
Revised:  As a result, if a uniform time grid \{t_k=k\tau, k \in \mathbb{N\} } is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij at the very first time-step through the initial values of both u and v at stencil nodes ;
Edit intent: fluency
Original:  As a result, if a uniform time grid \{t_k=k\tau, k \in \mathbb{N\} } is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij at the very first time-step , through the initial values of both u and v at stencil nodes ;
Revised:  As a result, if a uniform time grid \{t_k=k\tau, k \in \mathbb{N\} } is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij at the very first time-step , through the initial conditions ;
Edit intent: clarity
Original:  (a2) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij,t_{n+1}),n }%DIFDELCMD < \ge %%% 1, at the second and next time-steps , through a generated two-step  scheme.
Revised:  (a2) the %DIFDELCMD < {\bf %%% x _{ij,t_{n+1}),n }%DIFDELCMD < \ge %%% 1, at the second and next time-steps , through a generated two-step scheme.
Edit intent: clarity
Original:  (a2) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij,t_{n+1}),n }%DIFDELCMD < \ge %%% 1, at the second and next time-steps , through a generated two-step  scheme.
Revised:  (a2) the calculation of u(%DIFDELCMD < {\bf %%% x ,t_{n+1}),n }%DIFDELCMD < \ge %%% 1, at the second and next time-steps , through a generated two-step scheme.
Edit intent: fluency
Original:  (a2) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij,t_{n+1}),n }%DIFDELCMD < \ge %%% 1, at the second and next time-steps , through a generated two-step  scheme.
Revised:  (a2) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij,t_{n+1}),n }%DIFDELCMD < \ge %%% 1, at the second and next time-steps using a generated two-step scheme.
Edit intent: clarity
Original:  (a2) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij,t_{n+1}),n }%DIFDELCMD < \ge %%% 1, at the second and next time-steps , through a generated two-step  scheme.
Revised:  (a2) the calculation of u(%DIFDELCMD < {\bf %%% x _{ij,t_{n+1}),n }%DIFDELCMD < \ge %%% 1, at the second and next time-steps , through a generated two-step numerical scheme.
Edit intent: clarity
Original:  Three computational stencils (with 5, 9 and 13 space points) are built using the proposed approach.
Revised:  Three computational stencils (with five, nine and 13 space points) are built using the proposed approach.
Edit intent: fluency
Original:   It is demonstrated by simulation that utilizing both procedures (a1) and (a2) are important for the accuracy of stencil computations, and that using the conventional first time-step calculation procedure instead of (a1) can lead to a significant loss of accuracy even for later time-steps .
Revised:  It is demonstrated by simulation that using the new first time-step calculation procedure instead of (a1) can lead to a significant loss of accuracy even for later time-steps .
Edit intent: clarity
Original:   It is demonstrated by simulation that utilizing both procedures (a1) and (a2) are important for the accuracy of stencil computations, and that using the conventional first time-step calculation procedure instead of (a1) can lead to a significant loss of accuracy even for later time-steps .
Revised:  It is demonstrated by simulation that utilizing both procedures (a1) and (a2) are important for the accuracy of stencil computations, and that using the conventional first time-step calculation procedure (a1) can lead to a significant loss of accuracy even for later time-steps .
Edit intent: coherence
Original:   It is demonstrated by simulation that utilizing both procedures (a1) and (a2) are important for the accuracy of stencil computations, and that using the conventional first time-step calculation procedure instead of (a1) can lead to a significant loss of accuracy even for later time-steps .
Revised:  It is demonstrated by simulation that utilizing both procedures (a1) and (a2) are important for the accuracy of stencil computations, and that using the conventional first time-step calculation procedure instead of (a1) instead of the conventional one can provide a significant improvement of accuracy even for later time-steps .
Edit intent: clarity
Original:   It is demonstrated by simulation that utilizing both procedures (a1) and (a2) are important for the accuracy of stencil computations, and that using the conventional first time-step calculation procedure instead of (a1) can lead to a significant loss of accuracy even for later time-steps .
Revised:  It is demonstrated by simulation that utilizing both procedures (a1) and (a2) are important for the accuracy of stencil computations, and that using the conventional first time-step calculation procedure instead of (a1) can lead to a significant loss of accuracy even for later time steps .
Edit intent: fluency
Original: A novel approach to building explicit  stencil computation schemes for the transient 2D scalar wave equation is proposed and implemented.
Revised: An approach to building explicit stencil computation schemes for the transient 2D scalar wave equation is proposed and implemented.
Edit intent: clarity
Original:  For the purpose of constructing a two-step time-marching algorithm, an additional integral representation formula is derived that relates the exact solution values at three time points .
Revised:  For the purpose of constructing a two-step time-marching algorithm, a modified integral representation formula is derived that relates the exact solution values at three time points .
Edit intent: clarity
Original:  For the purpose of constructing a two-step time-marching algorithm, an additional integral representation formula is derived that relates the exact solution values at three time points .
Revised:  For the purpose of constructing a two-step time-marching algorithm, an additional integral representation formula involving three time levels is also employed .
Edit intent: clarity
Original:  It is shown that integrals in the two representation formulas are easily calculated if the initial conditions and the solution at a current time level as functions of spatial coordinates are approximated by stencil interpolation polynomials in the neighborhood of any point in a 2D Cartesian grid.
Revised:  It is shown that integrals in the two representation formulas are exactly calculated if the initial conditions and the solution at a current time level as functions of spatial coordinates are approximated by stencil interpolation polynomials in the neighborhood of any point in a 2D Cartesian grid.
Edit intent: clarity
Original:  It is shown that integrals in the two representation formulas are easily calculated if the initial conditions and the solution at a current time level as functions of spatial coordinates are approximated by stencil interpolation polynomials in the neighborhood of any point in a 2D Cartesian grid.
Revised:  It is shown that integrals in the two representation formulas are easily calculated if the initial conditions and the sought solution at each time level as functions of spatial coordinates are approximated by stencil interpolation polynomials in the neighborhood of any point in a 2D Cartesian grid.
Edit intent: clarity
Original:  As a result, if a uniform time grid is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1 ) the solution calculation at the very first time-step through the initial conditions;
Revised:  As a result, if a uniform time grid is chosen, the proposed time-marching algorithm consists of two numerical procedures: 1 ) the solution calculation at the very first time-step through the initial conditions;
Edit intent: fluency
Original:  As a result, if a uniform time grid is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1 ) the solution calculation at the very first time-step through the initial conditions;
Revised:  As a result, if a uniform time grid is chosen, the proposed time-marching algorithm consists of two numerical procedures: (a1 ) the solution calculation at the first time-step through the initial conditions;
Edit intent: clarity
Original:  (a2 ) the solution calculation at the second and next time-steps using a generated two-step numerical scheme.
Revised:  2 ) the solution calculation at the second and next time-steps using a generated two-step numerical scheme.
Edit intent: coherence
Original:  The obtained numerical schemes are compared with  corresponding finite-difference methods available in the literature.
Revised:  The obtained numerical schemes are compared with the corresponding finite-difference methods available in the literature.
Edit intent: fluency
Original:  The obtained numerical schemes are compared with  corresponding finite-difference methods available in the literature.
Revised:  The obtained numerical schemes are compared with corresponding finite-difference schemes available in the literature.
Edit intent: clarity
Original:  It is demonstrated by simulation that using the new first time-step calculation procedure (a1) instead of the conventional one can provide a significant improvement of accuracy even for later time steps.
Revised:  It is demonstrated that using the new first time-step calculation procedure (a1) instead of the conventional one can provide a significant improvement of accuracy even for later time steps.
Edit intent: clarity
Original: An approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation without using finite-difference approximations is proposed and implemented.
Revised: A new approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation without using finite-difference approximations is proposed and implemented.
Edit intent: clarity
Original: An approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation without using finite-difference approximations is proposed and implemented.
Revised: An approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation is implemented.
Edit intent: clarity
Original:  It is based on using the integral representation formula ( Poisson's formula ) that provides the exact solution of the initial-value problem for the transient 2D scalar wave equation at any time point through the initial conditions.
Revised:  It is based on using Poisson's formula ) that provides the exact solution of the initial-value problem for the transient 2D scalar wave equation at any time point through the initial conditions.
Edit intent: clarity
Original:  As a result, if a uniform time grid is chosen, the proposed time-marching algorithm consists of two numerical procedures: 1) the solution calculation at the first time-step through the initial conditions ;
Revised:  As a result, if a uniform time grid is chosen, the proposed time-marching algorithm consists of two numerical procedures: 1) the solution calculation at the first time-step procedure incorporating the initial conditions ;
Edit intent: clarity
Original:  2) the solution calculation at the second and next time-steps using a generated two-step numerical scheme .
Revised:  2) the solution calculation at the second and next time-steps .
Edit intent: clarity
Original:  Three particular explicit stencil schemes (with five, nine and 13 space points) are built using the proposed approach.
Revised:  Three particular explicit stencil schemes (with five, nine and 13 space points) are constructed using this approach.
Edit intent: clarity
Original:  As our main result, we identify the precise borderline of tractability.
Revised:  As our main result, we identify the borderline of tractability.
Edit intent: clarity
Original:  By using Gr\"{o}bner bases techniques, we generalize Schaefer's dichotomy theorem [STOC, 1978] which classifies all Constraint Satisfaction Problems over the Boolean domain to be either in P or NP-hard.
Revised:  By using Gr\"{o}bner bases techniques, we extend Schaefer's dichotomy theorem [STOC, 1978] which classifies all Constraint Satisfaction Problems over the Boolean domain to be either in P or NP-hard.
Edit intent: clarity
Original:  Moreover, our result implies necessary and sufficient conditions (assuming P%DIFDELCMD < \not %%% = NP) for the efficient computation of Theta Body SDP relaxations, identifying therefore the precise borderline of tractability for constraint language problems.
Revised:  Moreover, our result implies necessary and sufficient conditions %DIFDELCMD < \not %%% = NP) for the efficient computation of Theta Body SDP relaxations, identifying therefore the precise borderline of tractability for constraint language problems.
Edit intent: clarity
Original:  Moreover, our result implies necessary and sufficient conditions (assuming P%DIFDELCMD < \not %%% = NP) for the efficient computation of Theta Body SDP relaxations, identifying therefore the precise borderline of tractability for constraint language problems.
Revised:  Moreover, our result implies necessary and sufficient conditions (assuming P%DIFDELCMD < \not %%% for the efficient computation of Theta Body SDP relaxations, identifying therefore the precise borderline of tractability for constraint language problems.
Edit intent: clarity
Original:  Moreover, our result implies necessary and sufficient conditions (assuming P%DIFDELCMD < \not %%% = NP) for the efficient computation of Theta Body SDP relaxations, identifying therefore the precise borderline of tractability for constraint language problems.
Revised:  Moreover, our result implies necessary and sufficient conditions (assuming P%DIFDELCMD < \not %%% = NP) for the efficient computation of Theta Body SDP relaxations, identifying therefore the borderline of tractability for constraint language problems.
Edit intent: clarity
Original:  Two example cases of a free falling wedge and cylinder are simulated to validate both FD implementations, and numerical results are compared against benchmark cases in literature.
Revised:  Example cases of a free falling wedge and cylinder are simulated to validate both FD implementations, and numerical results are compared against benchmark cases in literature.
Edit intent: clarity
Original:  Two example cases of a free falling wedge and cylinder are simulated to validate both FD implementations, and numerical results are compared against benchmark cases in literature.
Revised:  Two example cases of a free falling wedge and cylinder are simulated , and the numerical results are compared against benchmark cases in literature.
Edit intent: clarity
Original:  We show how the Gaussian belief propagation solver  can be modified to handle non-symmetric positive definite matricesand introduce a new algorithm for matrix inversion that corresponds to the generalized belief propagation derived from the cluster variation method  or Kikuchi approximation  .
Revised:  We show how the Gaussian belief propagation solver can be modified to handle nonsymmetric matrices. Furthermore, we introduce a new algorithm for matrix inversion that corresponds to the generalized belief propagation derived from the cluster variation method or Kikuchi approximation .
Edit intent: coherence
Original:  We show how the Gaussian belief propagation solver  can be modified to handle non-symmetric positive definite matricesand introduce a new algorithm for matrix inversion that corresponds to the generalized belief propagation derived from the cluster variation method  or Kikuchi approximation  .
Revised:  We show how the Gaussian belief propagation solver can be modified to handle non-symmetric positive definite matricesand introduce a new algorithm for matrix inversion that corresponds to the generalized belief propagation derived from the cluster variation method ( or Kikuchi approximation .
Edit intent: fluency
Original:  We show how the Gaussian belief propagation solver  can be modified to handle non-symmetric positive definite matricesand introduce a new algorithm for matrix inversion that corresponds to the generalized belief propagation derived from the cluster variation method  or Kikuchi approximation  .
Revised:  We show how the Gaussian belief propagation solver can be modified to handle non-symmetric positive definite matricesand introduce a new algorithm for matrix inversion that corresponds to the generalized belief propagation derived from the cluster variation method or Kikuchi approximation ) .
Edit intent: fluency
Original:  We relate these algorithms to LU and block LU decompositions and provide some guarantees based on theorems from the theory of belief propagation.
Revised:  We relate these algorithms to LU and block LU decompositions and provide certain guarantees based on theorems from the theory of belief propagation.
Edit intent: clarity
Original:  All proposed algorithms are compared with classical solvers (e.g., Gauss-Seidel , BiCGSTAB) for an examples of linear elliptic equations.
Revised:  All proposed algorithms are compared with classical solvers (e.g., Gauss-Seidel and BiCGSTAB) with application to linear elliptic equations.
Edit intent: clarity
Original:  Also, we show how Gaussian belief propagation can be used as multigrid smoother, resulting in a far more robust solver than the one based on the Gauss-Seidel iterative method.
Revised:  We also show how the Gaussian belief propagation can be used as multigrid smoother, resulting in a far more robust solver than the one based on the Gauss-Seidel iterative method.
Edit intent: coherence
Original:  Also, we show how Gaussian belief propagation can be used as multigrid smoother, resulting in a far more robust solver than the one based on the Gauss-Seidel iterative method.
Revised:  Also, we show how Gaussian belief propagation can be used as multigrid smoother, resulting in a substantially more robust solver than the one based on the Gauss-Seidel iterative method.
Edit intent: clarity
Original:  All proposed algorithms are compared with classical solvers (e.g., Gauss-Seidel and BiCGSTAB) with application to linear elliptic equations.
Revised:  All proposed algorithms are compared with classical solvers (e.g., Gauss-Seidel , BiCGSTAB) with application to linear elliptic equations.
Edit intent: fluency
Original:  Its construction is a bottleneck in many scenarios  and settling the complexity of this task is one of the most important unsolved problems in sequence analysis that has remained open for 25 years.
Revised:  Its construction is a bottleneck in many scenarios , and settling the complexity of this task is one of the most important unsolved problems in sequence analysis that has remained open for 25 years.
Edit intent: fluency
Original:  Given a binary string of length n  occupying O(n/\log n) machine words of space , the BWT construction algorithm due to Hon et al. ( FOCS 2003 ) runs in O(n) time and O(n/\log n) space.
Revised:  Given a binary string of length n , occupying O(n/\log n) machine words of space , the BWT construction algorithm due to Hon et al. ( FOCS 2003 ) runs in O(n) time and O(n/\log n) space.
Edit intent: fluency
Original:  Given a binary string of length n  occupying O(n/\log n) machine words of space , the BWT construction algorithm due to Hon et al. ( FOCS 2003 ) runs in O(n) time and O(n/\log n) space.
Revised:  Given a binary string of length n occupying O(n/\log n) machine words , the BWT construction algorithm due to Hon et al. ( FOCS 2003 ) runs in O(n) time and O(n/\log n) space.
Edit intent: clarity
Original: User-generated content on siteslike Wikipedia is produced by volunteers who not only create content, but invest time and effort in quality control by reviewing others' contributions . Because of a perception  that privacy enhancing tools are a source of vandalism, spam, and abuse , many user-generated sites like Wikipedia block contributions from anonymity-seeking editors who use proxies like Tor .
Revised: User-generated content on siteslike Wikipedia is produced by volunteers who not only create content, but invest time and effort in quality control by reviewing others' contributions . Because of a perception among the Wikipedia community that privacy enhancing tools are a source of vandalism, spam, and abuse , many user-generated sites like Wikipedia block contributions from anonymity-seeking editors who use proxies like Tor .
Edit intent: clarity
Original: User-generated content on siteslike Wikipedia is produced by volunteers who not only create content, but invest time and effort in quality control by reviewing others' contributions . Because of a perception  that privacy enhancing tools are a source of vandalism, spam, and abuse , many user-generated sites like Wikipedia block contributions from anonymity-seeking editors who use proxies like Tor .
Revised: User-generated content on siteslike Wikipedia is produced by volunteers who not only create content, but invest time and effort in quality control by reviewing others' contributions . Because of a perception that privacy enhancing tools are a source of vandalism, spam, and abuse .
Edit intent: coherence
Original:  While blocking anonymity seeking editors is perceived to be effective in stemming abuse, collateral damage in the form of unrealized valuable contributions from anonymity seeking editors is typically invisible.
Revised:  While blocking editors that use these tools is done in an attempt to stem abuse, collateral damage in the form of unrealized valuable contributions from anonymity seeking editors is typically invisible.
Edit intent: clarity