wanyu/IteraTeR_v2 huggingface dataset:
Original:  They showed that if |f(A \times B)| %DIFDELCMD < \le %%% cn, for some constant c that), for every pair of finite sets A,B\mathbb R, with |A|=|B|=n (where the constant of proportionality } depends on {\rm deg} f , and for n%DIFDELCMD < \ge %%% n_0(c), for sufficiently large threshold n_0(c) that depends on c, then f must be of one of the special forms f(u,v)=h(\varphi(u)+\psi(v)), or f(u,v)=h(\varphi(u)\cdot\psi(v)), for some univariate polynomials \varphi,\psi,h over {\mathbb R}.
Revised:  They showed that if |f(A , B)| %DIFDELCMD < \le %%% cn, for some constant c that), for every pair of finite sets A,B\mathbb R, with |A|=|B|=n (where the constant of proportionality } depends on {\rm deg} f , and for n%DIFDELCMD < \ge %%% n_0(c), for sufficiently large threshold n_0(c) that depends on c, then f must be of one of the special forms f(u,v)=h(\varphi(u)+\psi(v)), or f(u,v)=h(\varphi(u)\cdot\psi(v)), for some univariate polynomials \varphi,\psi,h over {\mathbb R}.
Edit intent: clarity
Original:  They showed that if |f(A \times B)| %DIFDELCMD < \le %%% cn, for some constant c that), for every pair of finite sets A,B\mathbb R, with |A|=|B|=n (where the constant of proportionality } depends on {\rm deg} f , and for n%DIFDELCMD < \ge %%% n_0(c), for sufficiently large threshold n_0(c) that depends on c, then f must be of one of the special forms f(u,v)=h(\varphi(u)+\psi(v)), or f(u,v)=h(\varphi(u)\cdot\psi(v)), for some univariate polynomials \varphi,\psi,h over {\mathbb R}.
Revised:  They showed that if |f(A \times B)| %DIFDELCMD < \le %%% cn, for some constant c that), for every pair of finite sets A,B\mathbb R, with |A|=|B|=n (where the constant of proportionality } depends on {\rm deg} f %DIFDELCMD < \ge %%% n_0(c), for sufficiently large threshold n_0(c) that depends on c, then f must be of one of the special forms f(u,v)=h(\varphi(u)+\psi(v)), or f(u,v)=h(\varphi(u)\cdot\psi(v)), for some univariate polynomials \varphi,\psi,h over {\mathbb R}.
Edit intent: coherence
Original:  A refined analysis and generalizations of this result are given by Elekes and Szab\'o ESz%DIFDELCMD < ]%%% . In a variant of this setup, we have, in addition to A, B and f, another set C of n real numbers, and we denote by M the number of triples (a,b,c) \in A\times B\times C such that c=f(a, b).
Revised:  ESz%DIFDELCMD < ]%%% . In a variant of this setup, we have, in addition to A, B and f, another set C of n real numbers, and we denote by M the number of triples (a,b,c) \in A\times B\times C such that c=f(a, b).
Edit intent: coherence
Original:  A refined analysis and generalizations of this result are given by Elekes and Szab\'o ESz%DIFDELCMD < ]%%% . In a variant of this setup, we have, in addition to A, B and f, another set C of n real numbers, and we denote by M the number of triples (a,b,c) \in A\times B\times C such that c=f(a, b).
Revised:  A refined analysis and generalizations of this result are given by Elekes and Szab\'o %DIFDELCMD < ]%%% . In a variant of this setup, we have, in addition to A, B and f, another set C of n real numbers, and we denote by M the number of triples (a,b,c) \in A\times B\times C such that c=f(a, b).
Edit intent: clarity
Original:  We show that either M= O(n^{11/6})  , where the constant of proportionality depends on {\rm deg} f, or else f has one of the above special forms.
Revised:  We show that either M= O(n^{11/6}) , where the constant of proportionality depends on {\rm deg} f, unless f has one of the above special forms.
Edit intent: coherence
Original:  We show that either M= O(n^{11/6})  , where the constant of proportionality depends on {\rm deg} f, or else f has one of the above special forms.
Revised:  We show that either M= O(n^{11/6}) , where the constant of proportionality depends on {\rm deg} f, or else f has one of the aforementioned special forms.
Edit intent: fluency
Original:  This improves the results in ER00,ESz%DIFDELCMD < ]%%% , by making the bound explicit, with an exponent that is independent of the degree of f.
Revised:  This ER00,ESz%DIFDELCMD < ]%%% , by making the bound explicit, with an exponent that is independent of the degree of f.
Edit intent: clarity
Original:  The threshold bound O(n^{11/6 . Several applications of our results to problems of these kinds will be presented.
Revised:  The threshold bound O(n^{11/6 . Several applications of our results to problems of these kinds are presented.
Edit intent: fluency
Original:  For example, we show that the number of distinct distances between n points lying on a constant-degree parametric algebraic curve  , in any dimension, is \Omega(n^{4/3}), extending the result in PdZ%DIFDELCMD < ] %DIFDELCMD < %%% and improving the bound in Cha .
Revised:  For example, we show that the number of distinct distances between n points lying on a constant-degree parametric algebraic curve , in any dimension, is \Omega(n^{4/3}), extending the result PdZ%DIFDELCMD < ] %DIFDELCMD < %%% and improving the bound in Cha .
Edit intent: fluency
Original:  For example, we show that the number of distinct distances between n points lying on a constant-degree parametric algebraic curve  , in any dimension, is \Omega(n^{4/3}), extending the result in PdZ%DIFDELCMD < ] %DIFDELCMD < %%% and improving the bound in Cha .
Revised:  For example, we show that the number of distinct distances between n points lying on a constant-degree parametric algebraic curve , in any dimension, is \Omega(n^{4/3}), extending the result in PdZ%DIFDELCMD < ] %DIFDELCMD < %%% and improving the bound Cha .
Edit intent: fluency
Original:  This has an impact on the movement of the whole crowd especially at bottlenecks where slower movement and smaller steps can be observed . State-of-the-art pedestrian motion models automatically reduce speed in dense crowds simply because there is no space where the virtual pedestrians could advance .
Revised:  This has an impact on the movement of the whole crowd especially at bottlenecks where slower movement and smaller steps can be observed . State-of-the-art pedestrian motion models automatically reduce speed in dense crowds simply because there is no space where the pedestrians could go .
Edit intent: clarity
Original:  We reformulate the problem of correct stride adaptation as an optimization problem on a disk around the pedestrian.
Revised:  We reformulate the problem as an optimisation problem on a disk around the pedestrian.
Edit intent: clarity
Original:  He or she seeks the position that is most attractive in a sense of balanced goals between the search for targets, the need of space of individual pedestrians and the need to keep a distance from obstacles.
Revised:  Each pedestrian seeks the position that is most attractive in a sense of balanced goals between the search for targets, the need of space of individual pedestrians and the need to keep a distance from obstacles.
Edit intent: clarity
Original:  He or she seeks the position that is most attractive in a sense of balanced goals between the search for targets, the need of space of individual pedestrians and the need to keep a distance from obstacles.
Revised:  He or she seeks the position that is most attractive in a sense of balanced goals between the search for targets, the need for individual space and the need to keep a distance from obstacles.
Edit intent: clarity
Original:  The  result is a fully automatic adjustment that simplifies calibration , and gives visually natural results and an excellent fit to measured experimental data.
Revised:  The result is a fully automatic adjustment that simplifies calibration , and gives visually natural results with an excellent fit to measured experimental data.
Edit intent: coherence
Original:  In the two appendices , we give proofs of our main results and guidance for the implementation of the proposed model and the Fourier inversion results by means of one- and two-dimensional FFT methods.
Revised:  In the two appendices , we give guidance for the implementation of the proposed model and the Fourier inversion results by means of one- and two-dimensional FFT methods.
Edit intent: coherence
Original:  As a consequence, our annealing BP finds (asymptotically) a minimum fractional vertex cover on any graph.
Revised:  LABP finds (asymptotically) a minimum fractional vertex cover on any graph.
Edit intent: clarity
Original:  Using loop calculus, we also give an exact (also intractable for general graphs) expression of the partition function for matchings in term of our BP messages which can be used to improve mean-field approximations.
Revised:  Using loop calculus, we also give an exact (also intractable for general graphs) expression of the partition function for matching in term of our BP messages which can be used to improve mean-field approximations.
Edit intent: fluency
Original:  We moreover illustrate in the application to an enzymatic network, the relation between the exact implicit dose-response curve we obtain symbolically and the standard hysteretic diagram provided by a numerical solver.
Revised:  We moreover illustrate in the application to an enzymatic network, the relation between the exact implicit dose-response curve we obtain symbolically and the standard hysteresis diagram provided by a numerical solver.
Edit intent: fluency
Original:  In particular, we take a conjecture raised by Medina in his dissertation that says if \Phi%DIFDELCMD < \land%%% \varphi defines an%DIFDELCMD < \NP%%% -complete problems via fops and \varphi is a first-order formula , then it must be the case that %DIFDELCMD < \mod%%% (\Phi) is also%DIFDELCMD < \NP%%% -complete .
Revised:  In particular, we take a conjecture raised by Medina in his dissertation that says if %DIFDELCMD < \land%%% \varphi defines an%DIFDELCMD < \NP%%% -complete problems via fops and \varphi is a first-order formula , then it must be the case that %DIFDELCMD < \mod%%% (\Phi) is also%DIFDELCMD < \NP%%% -complete .
Edit intent: clarity
Original:  In particular, we take a conjecture raised by Medina in his dissertation that says if \Phi%DIFDELCMD < \land%%% \varphi defines an%DIFDELCMD < \NP%%% -complete problems via fops and \varphi is a first-order formula , then it must be the case that %DIFDELCMD < \mod%%% (\Phi) is also%DIFDELCMD < \NP%%% -complete .
Revised:  In particular, we take a conjecture raised by Medina in his dissertation that says if \Phi%DIFDELCMD < \land%%% %DIFDELCMD < \NP%%% -complete problems via fops and \varphi is a first-order formula , then it must be the case that %DIFDELCMD < \mod%%% (\Phi) is also%DIFDELCMD < \NP%%% -complete .
Edit intent: clarity
Original:  In particular, we take a conjecture raised by Medina in his dissertation that says if \Phi%DIFDELCMD < \land%%% \varphi defines an%DIFDELCMD < \NP%%% -complete problems via fops and \varphi is a first-order formula , then it must be the case that %DIFDELCMD < \mod%%% (\Phi) is also%DIFDELCMD < \NP%%% -complete .
Revised:  In particular, we take a conjecture raised by Medina in his dissertation that says if \Phi%DIFDELCMD < \land%%% \varphi defines an%DIFDELCMD < \NP%%% -complete problems via fops , then it must be the case that %DIFDELCMD < \mod%%% (\Phi) is also%DIFDELCMD < \NP%%% -complete .
Edit intent: coherence
Original:  However, we can solve in the affirmative a weaker claim that says that all "consistent" first-order sentences of form \forall%DIFDELCMD < \bar %DIFDELCMD < %%% x\theta(%DIFDELCMD < \bar %%% x), where \theta is a quantifier-free formula with free variables in%DIFDELCMD < \bar %%% x, can be safely eliminated without the fear of losing completeness.
Revised:  However, we can solve in the affirmative a weaker claim that says that all "consistent" first-order sentences %DIFDELCMD < \bar %DIFDELCMD < %%% x\theta(%DIFDELCMD < \bar %%% x), where \theta is a quantifier-free formula with free variables in%DIFDELCMD < \bar %%% x, can be safely eliminated without the fear of losing completeness.
Edit intent: clarity
Original:  However, we can solve in the affirmative a weaker claim that says that all "consistent" first-order sentences of form \forall%DIFDELCMD < \bar %DIFDELCMD < %%% x\theta(%DIFDELCMD < \bar %%% x), where \theta is a quantifier-free formula with free variables in%DIFDELCMD < \bar %%% x, can be safely eliminated without the fear of losing completeness.
Revised:  However, we can solve in the affirmative a weaker claim that says that all "consistent" first-order sentences of form \forall%DIFDELCMD < \bar %DIFDELCMD < %%% %DIFDELCMD < \bar %%% x), where \theta is a quantifier-free formula with free variables in%DIFDELCMD < \bar %%% x, can be safely eliminated without the fear of losing completeness.
Edit intent: coherence
Original:  However, we can solve in the affirmative a weaker claim that says that all "consistent" first-order sentences of form \forall%DIFDELCMD < \bar %DIFDELCMD < %%% x\theta(%DIFDELCMD < \bar %%% x), where \theta is a quantifier-free formula with free variables in%DIFDELCMD < \bar %%% x, can be safely eliminated without the fear of losing completeness.
Revised:  However, we can solve in the affirmative a weaker claim that says that all "consistent" first-order sentences of form \forall%DIFDELCMD < \bar %DIFDELCMD < %%% x\theta(%DIFDELCMD < \bar %%% x), where \theta is a quantifier-free formula with free variables in%DIFDELCMD < \bar %%% can be safely eliminated without the fear of losing completeness.
Edit intent: fluency
Original:  Our methods are quite general and can be applied to complexity classes other than %DIFDELCMD < \NP %%%  (in this paper: to %DIFDELCMD < \NLSPACE%%% ,%DIFDELCMD < \PTIME%%% , and%DIFDELCMD < \coNP%%%  ), provided the class has a complete problem satisfying a certain combinatorial property.
Revised:  Our methods are quite general and can be applied to complexity classes other than %DIFDELCMD < \NP %%% (in this paper: to %DIFDELCMD < \NLSPACE%%% %DIFDELCMD < \PTIME%%% , and%DIFDELCMD < \coNP%%% ), provided the class has a complete problem satisfying a certain combinatorial property.
Edit intent: fluency
Original:  Our methods are quite general and can be applied to complexity classes other than %DIFDELCMD < \NP %%%  (in this paper: to %DIFDELCMD < \NLSPACE%%% ,%DIFDELCMD < \PTIME%%% , and%DIFDELCMD < \coNP%%%  ), provided the class has a complete problem satisfying a certain combinatorial property.
Revised:  Our methods are quite general and can be applied to complexity classes other than %DIFDELCMD < \NP %%% (in this paper: to %DIFDELCMD < \NLSPACE%%% ,%DIFDELCMD < \PTIME%%% %DIFDELCMD < \coNP%%% ), provided the class has a complete problem satisfying a certain combinatorial property.
Edit intent: coherence
Original:  Finally, we discuss how this approach differs from Approximate Bayesian Computation (ABC), another method commonly used to generate simulation-based likelihood approximations.
Revised:  Finally, we discuss how this approach differs from approximate Bayesian computation (ABC), another method commonly used to generate simulation-based likelihood approximations.
Edit intent: fluency
Original:  There are several popular implementations of the methodincluding Eigenvector Method,  Least Squares Method,  Chi Squares Method and others.
Revised:  There are several popular implementations of this method, including the Eigenvector Method, Least Squares Method, Chi Squares Method and others.
Edit intent: fluency
Original:  There are several popular implementations of the methodincluding Eigenvector Method,  Least Squares Method,  Chi Squares Method and others.
Revised:  There are several popular implementations of the methodincluding Eigenvector Method, the Least Squares Method, Chi Squares Method and others.
Edit intent: clarity
Original:  There are several popular implementations of the methodincluding Eigenvector Method,  Least Squares Method,  Chi Squares Method and others.
Revised:  There are several popular implementations of the methodincluding Eigenvector Method, Least Squares Method, the Chi Squares Method and others.
Edit intent: clarity
Original:  Each of the above methods comes with one or more inconsistency indices helping to decide whether the consistency of input guarantees obtaining the reliable output,  taking the correct decision.
Revised:  Each of the above methods comes with one or more inconsistency indices that help to decide whether the consistency of input guarantees obtaining the reliable output, taking the correct decision.
Edit intent: clarity
Original:  Each of the above methods comes with one or more inconsistency indices helping to decide whether the consistency of input guarantees obtaining the reliable output,  taking the correct decision.
Revised:  Each of the above methods comes with one or more inconsistency indices helping to decide whether the consistency of input guarantees obtaining a reliable output, taking the correct decision.
Edit intent: fluency
Original:  Each of the above methods comes with one or more inconsistency indices helping to decide whether the consistency of input guarantees obtaining the reliable output,  taking the correct decision.
Revised:  Each of the above methods comes with one or more inconsistency indices helping to decide whether the consistency of input guarantees obtaining the reliable output, thus taking the correct decision.
Edit intent: coherence
Original:  The article explore the relationship between the inconsistency of input and the discrepancy of output.
Revised:  This article explores the relationship between the inconsistency of input and the discrepancy of output.
Edit intent: fluency
Original:  The article explore the relationship between the inconsistency of input and the discrepancy of output.
Revised:  The article explore the relationship between inconsistency of input and the discrepancy of output.
Edit intent: fluency
Original:  The article explore the relationship between the inconsistency of input and the discrepancy of output.
Revised:  The article explore the relationship between the inconsistency of input and discrepancy of output.
Edit intent: fluency
Original:  A discrepancy defined as the discrepancy index describes how far the obtained results correspond to the single expert's assessments .
Revised:  A discrepancy , defined by a discrepancy index, describes how far the obtained results correspond to the single expert's assessments .
Edit intent: clarity
Original:  A discrepancy defined as the discrepancy index describes how far the obtained results correspond to the single expert's assessments .
Revised:  A discrepancy defined as the discrepancy index describes how far the obtained results correspond to the single expert's assesments .
Edit intent: fluency
Original:  On the basis of the inconsistency and discrepancy indices  two properties of the weigh deriving procedure are formulated.
Revised:  On the basis of the inconsistency and discrepancy indices , two properties of the weigh deriving procedure are formulated.
Edit intent: fluency
Original:  On the basis of the inconsistency and discrepancy indices  two properties of the weigh deriving procedure are formulated.
Revised:  On the basis of the inconsistency and discrepancy indices two properties of the weight deriving procedure are formulated.
Edit intent: fluency
Original:  These properties are proven for the pair: Eigenvector Method and the Koczkodaj's Inconsistency Index.
Revised:  These properties are proven for Eigenvector Method and the Koczkodaj's Inconsistency Index.
Edit intent: clarity
Original:  These properties are proven for the pair: Eigenvector Method and the Koczkodaj's Inconsistency Index.
Revised:  These properties are proven for the pair: Eigenvector Method and Koczkodaj's Inconsistency Index.
Edit intent: fluency
Original:  A few estimates using the Koczkodaj's Inconsistency Index for the principal eigenvalue, the Saaty's inconsistency index and the Condition of Order Preservation are also provided.
Revised:  Several estimates using Koczkodaj's Inconsistency Index for the principal eigenvalue, the Saaty's inconsistency index and the Condition of Order Preservation are also provided.
Edit intent: fluency
Original:  A few estimates using the Koczkodaj's Inconsistency Index for the principal eigenvalue, the Saaty's inconsistency index and the Condition of Order Preservation are also provided.
Revised:  A few estimates using the Koczkodaj's Inconsistency Index for a principal eigenvalue, the Saaty's inconsistency index and the Condition of Order Preservation are also provided.
Edit intent: fluency
Original:  A few estimates using the Koczkodaj's Inconsistency Index for the principal eigenvalue, the Saaty's inconsistency index and the Condition of Order Preservation are also provided.
Revised:  A few estimates using the Koczkodaj's Inconsistency Index for the principal eigenvalue, Saaty's inconsistency index and the Condition of Order Preservation are also provided.
Edit intent: fluency
Original:  Each of the above methods comes with one or more inconsistency indices that help to decide whether the consistency of input guarantees obtaining a reliable output, thus taking the correct decision.
Revised:  Each of the above methods comes with one or more inconsistency indices that help to decide whether the consistency of input guarantees obtaining a reliable output, thus taking the optimal decision.
Edit intent: clarity
Original:  A discrepancy, defined by a discrepancy index, describes how far the obtained results correspond to the single expert's assesments .
Revised:  A global ranking discrepancy describes to what extent the obtained results correspond to the single expert's assesments .
Edit intent: clarity
Original:  A discrepancy, defined by a discrepancy index, describes how far the obtained results correspond to the single expert's assesments .
Revised:  A discrepancy, defined by a discrepancy index, describes how far the obtained results correspond to the single expert's assessments .
Edit intent: fluency
Original: We extend the existing enumeration of neck tie knots to include tie knots with a textured front, tied with the narrow end of a tie.
Revised: We extend the existing enumeration of neck tie-knots to include tie-knots with a textured front, tied with the narrow end of a tie.
Edit intent: fluency
Original:  These tie knots have gained popularity in recent years, based on reconstructions of a costume detail from The Matrix Reloaded, and are explicitly ruled out in the enumeration by Fink and Mao (2000).
Revised:  These tie-knots have gained popularity in recent years, based on reconstructions of a costume detail from The Matrix Reloaded, and are explicitly ruled out in the enumeration by Fink and Mao (2000).
Edit intent: clarity
Original:  We show that the relaxed tie knot description language that comprehensively describes these extended tie knot classes is either context sensitive or context free.
Revised:  We show that the relaxed tie-knot description language that comprehensively describes these extended tie knot classes is either context sensitive or context free.
Edit intent: fluency
Original:  We show that the relaxed tie knot description language that comprehensively describes these extended tie knot classes is either context sensitive or context free.
Revised:  We show that the relaxed tie knot description language that comprehensively describes these extended tie-knot classes is context free.
Edit intent: clarity
Original:  It has a  sub-language that covers all the knots that  inspired the work , and that is regular.
Revised:  It has a regular sub-language that covers all the knots that inspired the work , and that is regular.
Edit intent: coherence
Original: We show that the Survey Propagation guided decimation algorithm fails to find satisfying assignments on random instances of the "Not-All-Equal-K-SAT" problem , well below the satisfiability threshold }\over   .
Revised: We show that the Survey Propagation-guided decimation algorithm fails to find satisfying assignments on random instances of the "Not-All-Equal-K-SAT" problem , well below the satisfiability threshold }\over .
Edit intent: fluency