wanyu/IteraTeR_full_sent huggingface dataset:
Original: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Revised: We study the effects of rounding on the moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Edit intent: meaning-changed
Original: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Revised: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable . Specifically, given a random variable X and its rounded counterpart (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Edit intent: clarity
Original: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Revised: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), we study | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Edit intent: clarity
Original: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Revised: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Edit intent: fluency
Original: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Revised: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd \mathbf{E | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Edit intent: meaning-changed
Original: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Revised: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | %DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Edit intent: coherence
Original: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Revised: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% (x)-x| \leq \delta which depend linearly on \epsilon and \delta.
Edit intent: fluency
Original: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta. We then show that if the perturbation corresponds to\operatorname{rd} }\to rounding to the nearest point in some fixed discrete set , the bounds on the difference of moments can be improved to quadratic in many cases.
Revised: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when | %DIFDELCMD < {\rm %%% rd (x)-x | \leq \epsilon |x| or |%DIFDELCMD < {\rm %%% rd for non-negative integer k. We consider the case that the rounding function\operatorname{rd} }\to rounding to the nearest point in some fixed discrete set , the bounds on the difference of moments can be improved to quadratic in many cases.
Edit intent: meaning-changed
Original:  We then show that if the perturbation corresponds to\operatorname{rd} }\to rounding to the nearest point in some fixed discrete set , the bounds on the difference of moments can be improved to quadratic in many cases.
Revised:  We then show that if the perturbation corresponds to\operatorname{rd} : \mathbb{R}\to rounding to the nearest point in some fixed discrete set , the bounds on the difference of moments can be improved to quadratic in many cases.
Edit intent: meaning-changed
Original:  We then show that if the perturbation corresponds to\operatorname{rd} }\to rounding to the nearest point in some fixed discrete set , the bounds on the difference of moments can be improved to quadratic in many cases.
Revised:  We then show that if the perturbation corresponds to\operatorname{rd} }\to\mathbb{F rounding to the nearest point in some fixed discrete set , the bounds on the difference of moments can be improved to quadratic in many cases.
Edit intent: meaning-changed
Original:  We then show that if the perturbation corresponds to\operatorname{rd} }\to rounding to the nearest point in some fixed discrete set , the bounds on the difference of moments can be improved to quadratic in many cases. When the points in this fixed set are uniformly spaced, our analysis can be viewed as a generalization of Sheppard's corrections. We discuss how our bounds can be used to balance measurement error with sample error in a rigorous way, as well as how they can be used to generalize classical numerical analysis results. The frameworks developed in our analysis can be applied to a wider range of applications than those studied in this paper and may be of general interest or (ii) rounding randomly to either the nearest larger or smaller point in this same set with probabilities proportional to the distances to these points.
Revised:  We then show that if the perturbation corresponds to\operatorname{rd} }\to rounding to the nearest point in some discrete set \mathbb{F or (ii) rounding randomly to either the nearest larger or smaller point in this same set with probabilities proportional to the distances to these points.
Edit intent: meaning-changed
Original:  In both cases, we show, under reasonable assumptions on the density function of X, how to compute a constant C such that |\mathbb{E}[X^k] - \mathbb{E}[\operatorname{rd}(X)^{k}]| < C \epsilon^2, provided }\operatorname{rd}[ |X^k-\operatorname{rd}(X)^{k}| ] are also given} .
Revised:  In both cases, we show, under reasonable assumptions on the density function of X, how to compute a constant C such that |\mathbb{E}[X^k] - \mathbb{E}[\operatorname{rd}(X)^{k}]| < C \epsilon^2, provided }\operatorname{rd}(x) - x| \leq \epsilon E(x). Asymptotic and non-asymptotic bounds for the absolute moments \mathbb{E[ |X^k-\operatorname{rd}(X)^{k}| ] are also given} .
Edit intent: meaning-changed
Original:  We adopt a one-parameter model of \phi _i' infections per cluster, dividing any daily count n_i into n_i/\phi _i' ' clusters', for 'country' i.
Revised:  We adopt a one-parameter model of \phi '_i infections per cluster, dividing any daily count n_i into n_i/\phi _i' ' clusters', for 'country' i.
Edit intent: fluency
Original:  We adopt a one-parameter model of \phi _i' infections per cluster, dividing any daily count n_i into n_i/\phi _i' ' clusters', for 'country' i.
Revised:  We adopt a one-parameter model of \phi _i' infections per cluster, dividing any daily count n_i into n_i/\phi '_i ' clusters', for 'country' i.
Edit intent: fluency
Original:  We assume that n_i/\phi _i' on a given day j is drawn from a Poisson distribution whose mean is robustly estimated from the four neighbouring days, and calculate the inferred Poisson probability P _{ij of the observation.
Revised:  We assume that n_i/\phi '_i on a given day j is drawn from a Poisson distribution whose mean is robustly estimated from the four neighbouring days, and calculate the inferred Poisson probability P _{ij of the observation.
Edit intent: fluency
Original:  We assume that n_i/\phi _i' on a given day j is drawn from a Poisson distribution whose mean is robustly estimated from the four neighbouring days, and calculate the inferred Poisson probability P _{ij of the observation.
Revised:  We assume that n_i/\phi _i' on a given day j is drawn from a Poisson distribution whose mean is robustly estimated from the four neighbouring days, and calculate the inferred Poisson probability P '_{ij of the observation.
Edit intent: fluency
Original:  All are consistent with the \phi_i model . Clustering increases with total infection count for the full sequences: \phi_i \sim\sqrt{N_i 28-, 14- and 7-day least noisy sequences for several countries are best modelled as sub-Poissonian, suggesting a distinct epidemiological family.
Revised:  Most are found to be consistent with the \phi_i model . Clustering increases with total infection count for the full sequences: \phi_i \sim\sqrt{N_i 28-, 14- and 7-day least noisy sequences for several countries are best modelled as sub-Poissonian, suggesting a distinct epidemiological family.
Edit intent: clarity
Original:  All are consistent with the \phi_i model . Clustering increases with total infection count for the full sequences: \phi_i \sim\sqrt{N_i 28-, 14- and 7-day least noisy sequences for several countries are best modelled as sub-Poissonian, suggesting a distinct epidemiological family.
Revised:  All are consistent with the \phi_i model , the 28-, 14- and 7-day least noisy sequences for several countries are best modelled as sub-Poissonian, suggesting a distinct epidemiological family.
Edit intent: clarity
Original:  The 28-day sequences of DZ , BY, TR, AE have strongly sub-Poissonian preferred models , with \phi_i^{28} < 0.5;
Revised:  The 28-day least noisy sequence of DZ (Algeria) has a preferred model that is strongly sub-Poissonian preferred models , with \phi_i^{28} < 0.5;
Edit intent: meaning-changed
Original:  The 28-day sequences of DZ , BY, TR, AE have strongly sub-Poissonian preferred models , with \phi_i^{28} < 0.5; and FI, SA , RU, AL, IR have \phi_i^{28} < 3.0.
Revised:  The 28-day sequences of DZ , BY, TR, AE have strongly sub-Poissonian preferred models , with \phi_i^{28} < 0.1. TJ, TR , RU, AL, IR have \phi_i^{28} < 3.0.
Edit intent: meaning-changed
Original:  and FI, SA , RU, AL, IR have \phi_i^{28} < 3.0. Independent verification may be warranted for those countries with unusually low clustering < 0.05) correlation was found between the lack of media freedom in a country, as represented by a high Reporters sans frontieres Press Freedom Index (PFI^{2020}), and the lack of statistical noise in the country's daily counts.
Revised:  and FI, SA , RU, AL, IR have \phi_i^{28} < 0.5. A statistically significant (P^{\tau < 0.05) correlation was found between the lack of media freedom in a country, as represented by a high Reporters sans frontieres Press Freedom Index (PFI^{2020}), and the lack of statistical noise in the country's daily counts.
Edit intent: meaning-changed
Original:  For the gap edit distance problem, we give a greedy algorithm that distinguishes in time O(n{k}+k^2)  between length-n input strings with edit distance at most k and those with edit distance more than 4k^2 .
Revised:  For the gap edit distance problem, we give an O(n{k}+k^2) between length-n input strings with edit distance at most k and those with edit distance more than 4k^2 .
Edit intent: clarity
Original:  For the gap edit distance problem, we give a greedy algorithm that distinguishes in time O(n{k}+k^2)  between length-n input strings with edit distance at most k and those with edit distance more than 4k^2 .
Revised:  For the gap edit distance problem, we give a greedy algorithm that distinguishes in time O(n{k}+k^2) -time greedy algorithm that distinguishes between length-n input strings with edit distance at most k and those with edit distance more than 4k^2 .
Edit intent: meaning-changed
Original:  For the gap edit distance problem, we give a greedy algorithm that distinguishes in time O(n{k}+k^2)  between length-n input strings with edit distance at most k and those with edit distance more than 4k^2 .
Revised:  For the gap edit distance problem, we give a greedy algorithm that distinguishes in time O(n{k}+k^2) between length-n input strings with edit distance at most k and those with edit distance exceeding (3k+5)k .
Edit intent: meaning-changed
Original:  This is an improvement and a simplification upon the main result of Goldenberg, Krauthgamer, Saha, FOCS 2019], where the k vs \Theta(k^2) gap edit distance problem is solved in O(n{k}+k^3) time.
Revised:  This is an improvement and a simplification upon the result of Goldenberg, Krauthgamer, Saha, FOCS 2019], where the k vs \Theta(k^2) gap edit distance problem is solved in O(n{k}+k^3) time.
Edit intent: clarity
Original:  This is an improvement and a simplification upon the main result of Goldenberg, Krauthgamer, Saha, FOCS 2019], where the k vs \Theta(k^2) gap edit distance problem is solved in O(n{k}+k^3) time.
Revised:  This is an improvement and a simplification upon the main result of Goldenberg, Krauthgamer, and Saha FOCS 2019], where the k vs \Theta(k^2) gap edit distance problem is solved in O(n{k}+k^3) time.
Edit intent: fluency
Original:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}}  +k^2+ \frac{k{\alpha}}  \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}}  +k^3).
Revised:  We further generalize our result to solve the k vs k' gap edit distance problem in time O( \frac{n{\alpha}} +k^2+ \frac{k{\alpha}} \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}} +k^3).
Edit intent: clarity
Original:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}}  +k^2+ \frac{k{\alpha}}  \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}}  +k^3).
Revised:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( {\alpha}} +k^2+ \frac{k{\alpha}} \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}} +k^3).
Edit intent: fluency
Original:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}}  +k^2+ \frac{k{\alpha}}  \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}}  +k^3).
Revised:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}} \frac{nk +k^2+ \frac{k{\alpha}} \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}} +k^3).
Edit intent: meaning-changed
Original:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}}  +k^2+ \frac{k{\alpha}}  \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}}  +k^3).
Revised:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}} +k^2+ {\alpha}} \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}} +k^3).
Edit intent: fluency
Original:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}}  +k^2+ \frac{k{\alpha}}  \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}}  +k^3).
Revised:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}} +k^2+ \frac{k{\alpha}} \frac{k^2 \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}} +k^3).
Edit intent: meaning-changed
Original:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}}  +k^2+ \frac{k{\alpha}}  \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}}  +k^3).
Revised:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}} +k^2+ \frac{k{\alpha}} \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( {\alpha}} +k^3).
Edit intent: fluency
Original:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}}  +k^2+ \frac{k{\alpha}}  \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}}  +k^3).
Revised:  We further generalize our result to solve the k vs \alpha k gap edit distance problem in time O( \frac{n{\alpha}} +k^2+ \frac{k{\alpha}} \sqrt{nk}), strictly improving upon the previously known bound \tilde{O}( \frac{n{\alpha}} \frac{nk +k^3).
Edit intent: meaning-changed
Original:  Finally, we show that if the input strings do not have long highly periodic substrings, then the gap edit distance problem can be solved in sublinear time within any factor \alpha>1.\tilde{O}{\epsilon^2 k}+k^2\ell).
Revised:  Finally, we show that if the input strings do not have long highly periodic substrings, then already the k vs (1+\epsilon)k gap edit distance problem can be solved in sublinear time within any factor \alpha>1.\tilde{O}{\epsilon^2 k}+k^2\ell).
Edit intent: meaning-changed
Original:  Finally, we show that if the input strings do not have long highly periodic substrings, then the gap edit distance problem can be solved in sublinear time within any factor \alpha>1.\tilde{O}{\epsilon^2 k}+k^2\ell).
Revised:  Finally, we show that if the input strings do not have long highly periodic substrings, then the gap edit distance problem can be solved in sublinear time . Specifically, if the strings contain no substring of length \ell with period at most 2k, then the running time we achieve is\tilde{O}{\epsilon^2 k}+k^2\ell).
Edit intent: meaning-changed
Original:  Finally, we show that if the input strings do not have long highly periodic substrings, then the gap edit distance problem can be solved in sublinear time within any factor \alpha>1.\tilde{O}{\epsilon^2 k}+k^2\ell).
Revised:  Finally, we show that if the input strings do not have long highly periodic substrings, then the gap edit distance problem can be solved in sublinear time within any factor \alpha>1.\tilde{O}(\frac{n{\epsilon^2 k}+k^2\ell).
Edit intent: meaning-changed
Original:  This generalizes the linear-time embedding of Chakraborty, Goldenberg, Koucky, STOC 2016], where the resultant Hamming distance is between k and k^2 . Our algorithm is based on a random walk over samples, which we believe will find other applications in sublinear-time algorithms.
Revised:  This generalizes the linear-time embedding of Chakraborty, Goldenberg, and Kouck\'y STOC 2016], where the resultant Hamming distance is between k and k^2 . Our algorithm is based on a random walk over samples, which we believe will find other applications in sublinear-time algorithms.
Edit intent: clarity
Original:  This generalizes the linear-time embedding of Chakraborty, Goldenberg, Koucky, STOC 2016], where the resultant Hamming distance is between k and k^2 . Our algorithm is based on a random walk over samples, which we believe will find other applications in sublinear-time algorithms.
Revised:  This generalizes the linear-time embedding of Chakraborty, Goldenberg, Koucky, STOC 2016], where the resultant Hamming distance is between \frac k2 and O(k^2) . Our algorithm is based on a random walk over samples, which we believe will find other applications in sublinear-time algorithms.
Edit intent: meaning-changed
Original: This paper aims at giving some equivalent conditions for that a collection of finitely many of Hermitian matrices can be simultaneously diagonalizable via congruence (SDC) by a nonsingular matrix.
Revised: This paper aims at solving the Hermitian SDC problem, i.e., that of a collection of finitely many of Hermitian matrices can be simultaneously diagonalizable via congruence (SDC) by a nonsingular matrix.
Edit intent: clarity
Original: This paper aims at giving some equivalent conditions for that a collection of finitely many of Hermitian matrices can be simultaneously diagonalizable via congruence (SDC) by a nonsingular matrix.
Revised: This paper aims at giving some equivalent conditions for thatsimultaneously diagonalizing via *-congruence a collection of finitely many of Hermitian matrices can be simultaneously diagonalizable via congruence (SDC) by a nonsingular matrix.
Edit intent: meaning-changed
Original: This paper aims at giving some equivalent conditions for that a collection of finitely many of Hermitian matrices can be simultaneously diagonalizable via congruence (SDC) by a nonsingular matrix.
Revised: This paper aims at giving some equivalent conditions for that a collection of finitely many (not need pairwise commute) Hermitian matrices. Theoretically, we provide some equivalent conditions for that such a matrix collection can be simultaneously diagonalizable via congruence (SDC) by a nonsingular matrix.
Edit intent: meaning-changed
Original:  It surprisingly turns out that one of such equivalent conditions applies the semidefinite programming (SDP) , which leads to a practical usefulness. As a consequence, this certainly solves such SDC-problem for collections of real symmetric matrices listed in J-B . Hiriart-Urruty, Potpourri of conjectures and open questions in nonlinear analysis and optimization,SIAM Review 49 (  2) , 2007 . Corresponding algorithms and illustrating examples by hand/coding \textsc{ are also presented.
Revised:  It surprisingly turns out that one of such equivalent conditions applies the semidefinite programming (SDP) J-B . Hiriart-Urruty, Potpourri of conjectures and open questions in nonlinear analysis and optimization,SIAM Review 49 ( 2) , 2007 . Corresponding algorithms and illustrating examples by hand/coding \textsc{ are also presented.
Edit intent: coherence
Original:  As a consequence, this certainly solves such SDC-problem for collections of real symmetric matrices listed in J-B . Hiriart-Urruty, Potpourri of conjectures and open questions in nonlinear analysis and optimization,SIAM Review 49 (  2) , 2007 . Corresponding algorithms and illustrating examples by hand/coding \textsc{ are also presented.
Revised:  As a consequence, this certainly solves such SDC-problem for collections of real symmetric matrices listed in . Hiriart-Urruty, Potpourri of conjectures and open questions in nonlinear analysis and optimization,SIAM Review 49 ( 2) , 2007 . Corresponding algorithms and illustrating examples by hand/coding \textsc{ are also presented.
Edit intent: coherence