wanyu/IteraTeR_v2 huggingface dataset:
Original: Previously we have introduced the cosine product-to-sum identity  \prodm = 1^M {\cos \left( {t{{{2^m}}}} \right)} = 1{{{2^{M - 1}}}}\summ = 1^{{2^{M - 1}}} {\cos \left( {{2m - 1}{{{2^M}}}t} \right)} and applied it for sampling  as an incomplete cosine expansion of the sinc function in order to obtain a rational approximation of the Voigt/complex error function that with only 16 summation terms can provide accuracy {\sim 10^{ - 14}}.
Revised: Previously we have introduced the cosine product-to-sum identity 17 \prodm = 1^M {\cos \left( {t{{{2^m}}}} \right)} = 1{{{2^{M - 1}}}}\summ = 1^{{2^{M - 1}}} {\cos \left( {{2m - 1}{{{2^M}}}t} \right)} and applied it for sampling as an incomplete cosine expansion of the sinc function in order to obtain a rational approximation of the Voigt/complex error function that with only 16 summation terms can provide accuracy {\sim 10^{ - 14}}.
Edit intent: coherence
Original:  A Matlab code validating these results is presented.
Revised:  A MATLAB code validating these results is presented.
Edit intent: fluency
Original:  Different from the stochastic Allen-Cahn equation, the presence of the unbounded elliptic operator in the front of the cubic nonlinearity in the underlying model makes the error analysis much more challenging and demanding.
Revised:  Different from the stochastic Allen-Cahn equation, the presence of the unbounded elliptic operator in front of the cubic nonlinearity in the underlying model makes the error analysis much more challenging and demanding.
Edit intent: fluency
Original:  Anal. 56(2), 708-731, 2018], strong convergence of the semi-discrete and fully discrete finite element methods are , respectively, proved for the Cahn-Hilliard-Cook (CHC) equation, but without convergence rates revealed.
Revised:  Anal. 56(2), 708-731, 2018], strong convergence of the semi-discrete and fully discrete finite element methods is , respectively, proved for the Cahn-Hilliard-Cook (CHC) equation, but without convergence rates revealed.
Edit intent: fluency
Original:  More accurately, strong convergence rates of the full discretization are analyzed, realized by using Galerkin finite element methods based on piecewise continuous polynomials of degree at most r-1, r \geq 2 for the spatial discretization and the backward Euler method for the temporal discretization.
Revised:  More accurately, strong convergence rates of a full discretization are analyzed, realized by using Galerkin finite element methods based on piecewise continuous polynomials of degree at most r-1, r \geq 2 for the spatial discretization and the backward Euler method for the temporal discretization.
Edit intent: fluency
Original:  More accurately, strong convergence rates of the full discretization are analyzed, realized by using Galerkin finite element methods based on piecewise continuous polynomials of degree at most r-1, r \geq 2 for the spatial discretization and the backward Euler method for the temporal discretization.
Revised:  More accurately, strong convergence rates of the full discretization are obtained, based on Galerkin finite element methods based on piecewise continuous polynomials of degree at most r-1, r \geq 2 for the spatial discretization and the backward Euler method for the temporal discretization.
Edit intent: clarity
Original:  More accurately, strong convergence rates of the full discretization are analyzed, realized by using Galerkin finite element methods based on piecewise continuous polynomials of degree at most r-1, r \geq 2 for the spatial discretization and the backward Euler method for the temporal discretization.
Revised:  More accurately, strong convergence rates of the full discretization are analyzed, realized by using Galerkin finite element methods for the spatial discretization and the backward Euler method for the temporal discretization.
Edit intent: coherence
Original: The Euler equations for an incompressible fluid on a (rotating) sphere are a fundamental model of oceanic and atmospheric dynamics.
Revised: The incompressible Euler equations on a (rotating) sphere are a fundamental model of oceanic and atmospheric dynamics.
Edit intent: clarity
Original: The Euler equations for an incompressible fluid on a (rotating) sphere are a fundamental model of oceanic and atmospheric dynamics.
Revised: The Euler equations for an incompressible fluid on a (rotating) sphere are fundamental in models of oceanic and atmospheric dynamics.
Edit intent: fluency
Original:  the statistical mechanics theory predicts that a steady vorticity configuration should eventually appear , but detailed numerical results in the literature contradict this theory, yielding instead an unsteadiness.
Revised:  statistical mechanics predict a steady vorticity configuration should eventually appear , but detailed numerical results in the literature contradict this theory, yielding instead an unsteadiness.
Edit intent: coherence
Original:  the statistical mechanics theory predicts that a steady vorticity configuration should eventually appear , but detailed numerical results in the literature contradict this theory, yielding instead an unsteadiness.
Revised:  the statistical mechanics theory predicts that a steady vorticity configuration , but detailed numerical results in the literature contradict this theory, yielding instead an unsteadiness.
Edit intent: clarity
Original:  Based on the geometric quantization spatial discretization of the viscosity equation combined with symplectic integration of Lie--Poisson systems, we develop here an efficient numerical method for long-time simulations that preserve the complete underlying geometric features of the exact flow, in particular exact conservation of Casimirs.
Revised:  Based on the geometric quantization spatial discretization of the viscosity equation combined with symplectic integration of Lie--Poisson systems, we develop here an efficient numerical method for long-time simulations that preserve the geometric features of the exact flow, in particular exact conservation of Casimirs.
Edit intent: clarity
Original:  Based on the geometric quantization spatial discretization of the viscosity equation combined with symplectic integration of Lie--Poisson systems, we develop here an efficient numerical method for long-time simulations that preserve the complete underlying geometric features of the exact flow, in particular exact conservation of Casimirs.
Revised:  Based on the geometric quantization spatial discretization of the viscosity equation combined with symplectic integration of Lie--Poisson systems, we develop here an efficient numerical method for long-time simulations that preserve the complete underlying geometric features of the exact flow, in particular conservation of Casimirs.
Edit intent: fluency
Original: The incompressible Euler equations on a (rotating) sphere are fundamental in models of oceanic and atmospheric dynamics.
Revised: The incompressible Euler equations on a sphere are fundamental in models of oceanic and atmospheric dynamics.
Edit intent: clarity
Original:  The long-time behavior of solutions is largely unknown;
Revised:  The long-time behaviour of solutions is largely unknown;
Edit intent: fluency
Original:  statistical mechanics predict a steady vorticity configuration, but detailed numerical results in the literature contradict this theory, yielding instead persistent unsteadiness.
Revised:  statistical mechanics predicts a steady vorticity configuration, but detailed numerical results in the literature contradict this theory, yielding instead persistent unsteadiness.
Edit intent: fluency
Original:  Hyperviscosity, however, destroys the underlying geometry of the phase flow (such as conservation of Casimir functions), and therefore might affect the qualitative long-time behavior .
Revised:  Hyperviscosity, however, destroys the underlying geometry of the phase flow (such as conservation of Casimir functions), and therefore might affect the qualitative long-time behaviour .
Edit intent: fluency
Original:  The method is based on spatial discretization obtained by geometric quantization, combined with symplectic Lie--Poisson time integration.
Edit intent: coherence
Original: The incompressible  Euler equations on a sphere are fundamental in models of oceanic and atmospheric dynamics .
Revised: The incompressible Euler equations on a sphere constitute a fundamental model in hydrodynamics .
Edit intent: clarity
Original:  These numerical results contradict both the statistical mechanics theory and  previous numerical results suggesting that the generic behaviour should be 4 coherent vortex structures .
Revised:  These numerical results contradict the statistical mechanics theory and previous numerical results suggesting that the generic behaviour should be 4 coherent vortex structures .
Edit intent: clarity
Original:  These numerical results contradict both the statistical mechanics theory and  previous numerical results suggesting that the generic behaviour should be 4 coherent vortex structures .
Revised:  These numerical results contradict both the statistical mechanics theory and previous numerical results , suggesting 4 coherent vortex structures .
Edit intent: clarity
Original:  Furthermore, through integrability theory for point vortex dynamics on the sphere , we present a theoretical model which describe the mechanism by which the three observed regimes appear.
Revised:  Furthermore, through integrability theory for point vortex dynamics on the sphere we present a theoretical model which describe the mechanism by which the three observed regimes appear.
Edit intent: fluency
Original:  Furthermore, through integrability theory for point vortex dynamics on the sphere , we present a theoretical model which describe the mechanism by which the three observed regimes appear.
Revised:  Furthermore, through integrability theory for point vortex dynamics on the sphere , we present a theoretical model which describes the mechanism by which the three observed regimes appear.
Edit intent: fluency
Original:  Instead, during the design phase of network functions, programmers should be given a easier way to specify checks up front, all without having to account for and keep track of these consequences at each and every step during the development cycle.
Revised:  Instead, during the design phase of network functions, programmers should be given an easier way to specify checks up front, all without having to account for and keep track of these consequences at each and every step during the development cycle.
Edit intent: fluency
Original:  We present a short  self-contained  purely combinatorial proof which also proves EPPA for the class of integer valued antipodal metric spaces of diameter 3, answering a question of Aranda et al.
Revised:  We present a short , self-contained purely combinatorial proof which also proves EPPA for the class of integer valued antipodal metric spaces of diameter 3, answering a question of Aranda et al.
Edit intent: fluency
Original:  We present a short  self-contained  purely combinatorial proof which also proves EPPA for the class of integer valued antipodal metric spaces of diameter 3, answering a question of Aranda et al.
Revised:  We present a short self-contained , purely combinatorial proof which also proves EPPA for the class of integer valued antipodal metric spaces of diameter 3, answering a question of Aranda et al.
Edit intent: fluency
Original:  The class of two-graphs is an important new example which behaves differently from all the other known classes with EPPA: Two-graphs do not have the amalgamation property with automorphisms (APA), their Ramsey expansion has to add a graph, it is not known if they have ample generics and coherent EPPA and even EPPA itself cannot be proved using the Herwig--Lascar theorem.
Revised:  The class of two-graphs is an important new example which behaves differently from all the other known classes with EPPA: Two-graphs do not have the amalgamation property with automorphisms (APA), their Ramsey expansion has to add a graph, it is not known if they have coherent EPPA and even EPPA itself cannot be proved using the Herwig--Lascar theorem.
Edit intent: clarity
Original:  In the past years, various works have used diffusion MRI for automatic classification of Alzheimers disease .
Revised:  In past years, various works have used diffusion MRI for automatic classification of Alzheimers disease .
Edit intent: fluency
Original:  In the past years, various works have used diffusion MRI for automatic classification of Alzheimers disease .
Revised:  In the past years, various works have used diffusion MRI for automatic classification of AD .
Edit intent: clarity
Original:  First , voxel-wise features generally gave better performances than regional features.
Revised:  First , voxel-wise features generally gives better performance than regional features.
Edit intent: fluency
Original:  In particular, the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient.
Revised:  In particular, the resulting error estimator constitutes an upper bound for the error and is shown to be local efficient.
Edit intent: clarity
Original:  Our architecture learns to dynamically construct plans using a learned state-transition model by selecting and traversing between simulated states and actions to maximize valuable information before acting.
Revised:  Our architecture learns to dynamically construct plans using a learned state-transition model by selecting and traversing between simulated states and actions to maximize information before acting.
Edit intent: clarity
Original:  The co-action between the variational inequalities and the Holder stability estimates is also discussed for both the linear and non-linear problems.
Revised:  The co-action between the variational inequalities and the Holder stability estimates is also discussed for both the linear and nonlinear problems.
Edit intent: fluency
Original: In this article, we look for a new kind of smoothness concept, i.e. Holder smoothness condition for finding the convergence rates of linear and nonlinear inverse problems in Hilbert spaces.
Revised: In this article, we look for a new kind of smoothness concept, i.e. the Holder stability estimates for determination of convergence rates of linear and nonlinear inverse problems in Hilbert spaces.
Edit intent: clarity
Original:  For linear problems, the convergence rates are obtained without the use of the  spectral theory and results obtained are similar to 3%DIFDELCMD < ]%%% . The co-action between the variational inequalities and the Holder stability estimates is also discussed for both the linear and nonlinear problems.
Revised:  For linear problems, the convergence rates without the use of the spectral theory and results obtained are similar to 3%DIFDELCMD < ]%%% . The co-action between the variational inequalities and the Holder stability estimates is also discussed for both the linear and nonlinear problems.
Edit intent: clarity
Original:  For linear problems, the convergence rates are obtained without the use of the  spectral theory and results obtained are similar to 3%DIFDELCMD < ]%%% . The co-action between the variational inequalities and the Holder stability estimates is also discussed for both the linear and nonlinear problems.
Revised:  For linear problems, the convergence rates are obtained without the use of the spectral theory and 3%DIFDELCMD < ]%%% . The co-action between the variational inequalities and the Holder stability estimates is also discussed for both the linear and nonlinear problems.
Edit intent: coherence
Original:  For linear problems, the convergence rates are obtained without the use of the  spectral theory and results obtained are similar to 3%DIFDELCMD < ]%%% . The co-action between the variational inequalities and the Holder stability estimates is also discussed for both the linear and nonlinear problems.
Revised:  For linear problems, the convergence rates are obtained without the use of the spectral theory and results obtained are similar to %DIFDELCMD < ]%%% . The co-action between the variational inequalities and the Holder stability estimates is also discussed for both the linear and nonlinear problems.
Edit intent: clarity
Original:  For linear problems, the convergence rates are obtained without the use of the  spectral theory and results obtained are similar to 3%DIFDELCMD < ]%%% . The co-action between the variational inequalities and the Holder stability estimates is also discussed for both the linear and nonlinear problems.
Revised:  For linear problems, the convergence rates are obtained without the use of the spectral theory and results obtained are similar to 3%DIFDELCMD < ]%%% . The co-action between the variational inequalities and the Holder stability estimates is also discussed for both the linear and nonlinear problems.
Edit intent: fluency
Original: In this article, we look for a new kind of smoothness concept, i.e. the Holder stability estimates for  determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces.
Revised: In this paper, we apply a new kind of smoothness concept, i.e. the Holder stability estimates for determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces.
Edit intent: clarity
Original: In this article, we look for a new kind of smoothness concept, i.e. the Holder stability estimates for  determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces.
Revised: In this article, we look for a new kind of smoothness concept, i.e. H\"older stability estimates for determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces.
Edit intent: fluency
Original: In this article, we look for a new kind of smoothness concept, i.e. the Holder stability estimates for  determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces.
Revised: In this article, we look for a new kind of smoothness concept, i.e. the Holder stability estimates for the determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces.
Edit intent: fluency
Original:  For linear inverse problems, we obtain the convergence rates without the use of the classical concept of spectral theory and for non-linear inverse problems, we obtain the results by incorporating both Holder stability estimates as well as the inhomogeneous variational inequality.
Revised:  For linear inverse problems, we obtain the convergence rates without incorporating the classical concept of spectral theory and for non-linear inverse problems, we obtain the results by incorporating both Holder stability estimates as well as the inhomogeneous variational inequality.
Edit intent: coherence
Original:  We also consider Lavrentiev's regularization method for non-linear inverse problems in Hilbert spaces and find the convergence rates of its regularized solutions by incorporating Holder stability estimates as well as inhomogeneous variational inequality.
Revised:  We also consider Lavrentiev's regularization method for non-linear inverse problems and determine its convergence rates by incorporating the H\"older stability estimates as well as inhomogeneous variational inequality.
Edit intent: clarity
Original:  The co-action between the variational inequalities and the Holder stability estimatesis also discussed for both linear and non-linear inverse problems .
Revised:  The co-action between the variational inequalities and the H\"older stability estimates .
Edit intent: clarity
Original: We study several problems concerning convex polygons whose vertices lie in a Cartesian product (for short, grid) of two sets of n real numbers  .
Revised: We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers .
Edit intent: clarity
Original: We study several problems concerning convex polygons whose vertices lie in a Cartesian product (for short, grid) of two sets of n real numbers  .
Revised: We study several problems concerning convex polygons whose vertices lie in a Cartesian product (for short, grid) of two sets of n real numbers ) .
Edit intent: fluency
Original:  First, we prove that every such grid contains a convex polygon with \Omega(log n) vertices and that this bound is tight up to a constant factor.
Revised:  First, we prove that every such grid contains \Omega(\log n) points in convex position and that this bound is tight up to a constant factor.
Edit intent: clarity
Original:  Second, we present polynomial-time algorithms for computing the largest convex chain in a grid that contains no two points of the same x-or y-coordinate.
Revised:  Second, we present polynomial-time algorithms for computing the largest convex chain in a grid that contains no two points with the same x- or y-coordinate.
Edit intent: clarity
Original:  We show how to efficiently approximate the maximum size of a supported convex polygon up to a factor of 2.
Revised:  We show that the maximum size of a supported convex polygon up to a factor of 2.
Edit intent: clarity
Original:  Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants.
Revised:  Finally, we present exponential bounds on the maximum number of point sets in convex position in such grids, and for some restricted variants.
Edit intent: clarity
Original:  We also introduce the second-order adaptive SST.
Revised:  We also introduce the 2nd-order adaptive SST.
Edit intent: fluency
Original:  The procedure relies on the formalism and on the mimetic properties of diagonal norm ,
Revised:  The procedure relies on the formalism and mimetic properties of diagonal norm ,
Edit intent: fluency
Original:  The procedure relies on the formalism and on the mimetic properties of diagonal norm ,
Revised:  The procedure relies on the formalism and on the mimetic properties of diagonal-norm ,
Edit intent: fluency
Original:  summation-by-parts and simultaneous-approximation-term operators, and it is as a generalization of previous works on discontinuous interface coupling  and solid wall boundary conditions  . Using the method of line approach , a semi-discrete entropy estimate for the entire domain is obtained when the proposed numerical imposition of boundary conditions are coupled with an entropy conservative or entropy stable discrete interior operator.
Revised:  summation-by-parts and simultaneous-approximation-term operators, and is a generalization of previous works on discontinuous interface coupling and solid wall boundary conditions . Using the method of line approach , a semi-discrete entropy estimate for the entire domain is obtained when the proposed numerical imposition of boundary conditions are coupled with an entropy conservative or entropy stable discrete interior operator.
Edit intent: fluency
Original:  summation-by-parts and simultaneous-approximation-term operators, and it is as a generalization of previous works on discontinuous interface coupling  and solid wall boundary conditions  . Using the method of line approach , a semi-discrete entropy estimate for the entire domain is obtained when the proposed numerical imposition of boundary conditions are coupled with an entropy conservative or entropy stable discrete interior operator.
Revised:  summation-by-parts and simultaneous-approximation-term operators, and it is as a generalization of previous works on discontinuous interface coupling and solid wall boundary conditions . Using the method of lines , a semi-discrete entropy estimate for the entire domain is obtained when the proposed numerical imposition of boundary conditions are coupled with an entropy conservative or entropy stable discrete interior operator.
Edit intent: clarity