wanyu/IteraTeR_v2 huggingface dataset:
Original:  The kernel distance is monotonic with the kernel density estimate of a measure (sublevel sets of the kernel distance are superlevel sets of the kernel density estimate), thus allowing us to examine the kernel density estimate in this manner.
Revised:  The kernel distance is monotonic with the kernel density estimate (sublevel sets of the kernel distance are superlevel sets of the kernel density estimate), thus allowing us to examine the kernel density estimate in this manner.
Edit intent: clarity
Original:  Moreover, we show it has several computational and stability advantages over the distance to a measure .
Revised:  We show it has several computational and stability advantages over the distance to a measure .
Edit intent: coherence
Original: A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected intersection with all connected components of S. Hence, this  object, introduced by Canny, is used to answer connectivity queries (with applications, for instance, to motion planning) but has also become central , since it is used in many high-level algorithmsof effective real algebraic geometry .
Revised: A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected intersection with all connected components of S. Hence, this kind of object, introduced by Canny, is used to answer connectivity queries (with applications, for instance, to motion planning) but has also become central , since it is used in many high-level algorithmsof effective real algebraic geometry .
Edit intent: clarity
Original: A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected intersection with all connected components of S. Hence, this  object, introduced by Canny, is used to answer connectivity queries (with applications, for instance, to motion planning) but has also become central , since it is used in many high-level algorithmsof effective real algebraic geometry .
Revised: A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected intersection with all connected components of S. Hence, this object, introduced by Canny, can be used to answer connectivity queries (with applications, for instance, to motion planning) but has also become central , since it is used in many high-level algorithmsof effective real algebraic geometry .
Edit intent: coherence
Original: A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected intersection with all connected components of S. Hence, this  object, introduced by Canny, is used to answer connectivity queries (with applications, for instance, to motion planning) but has also become central , since it is used in many high-level algorithmsof effective real algebraic geometry .
Revised: A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected intersection with all connected components of S. Hence, this object, introduced by Canny, is used to answer connectivity queries (with applications, for instance, to motion planning) but has also become of central importance in effective real algebraic geometry , since it is used in many high-level algorithmsof effective real algebraic geometry .
Edit intent: clarity
Original: A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected intersection with all connected components of S. Hence, this  object, introduced by Canny, is used to answer connectivity queries (with applications, for instance, to motion planning) but has also become central , since it is used in many high-level algorithmsof effective real algebraic geometry .
Revised: A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected intersection with all connected components of S. Hence, this object, introduced by Canny, is used to answer connectivity queries (with applications, for instance, to motion planning) but has also become central , since it is used in many higher-level algorithms .
Edit intent: coherence
Original:  For a long time, the best known complexity result for computing roadmaps, given by Basu, Pollack and Roy, was s^{d+1} D^{O(n^2)}  where the input is given by s polynomials of degree D in n variables, with d \le n the dimension of an associated geometric object.
Revised:  For a long time, the best known complexity result for computing roadmaps, due to Basu, Pollack and Roy, was s^{d+1} D^{O(n^2)} where the input is given by s polynomials of degree D in n variables, with d \le n the dimension of an associated geometric object.
Edit intent: clarity
Original:  For a long time, the best known complexity result for computing roadmaps, given by Basu, Pollack and Roy, was s^{d+1} D^{O(n^2)}  where the input is given by s polynomials of degree D in n variables, with d \le n the dimension of an associated geometric object.
Revised:  For a long time, the best known complexity result for computing roadmaps, given by Basu, Pollack and Roy, was s^{d+1} D^{O(n^2)} , where the input is given by s polynomials of degree D in n variables, with d \le n the dimension of an associated geometric object.
Edit intent: fluency
Original:  This enabled more freedom for the design of algorithms computing roadmaps and led  to a first probabilistic roadmap algorithm for smooth and bounded real hypersurfaces running in time (nD)^{O(n^{1.5})}.
Revised:  This gave us more freedom for the design of algorithms computing roadmaps and led to a first probabilistic roadmap algorithm for smooth and bounded real hypersurfaces running in time (nD)^{O(n^{1.5})}.
Edit intent: clarity
Original:  This enabled more freedom for the design of algorithms computing roadmaps and led  to a first probabilistic roadmap algorithm for smooth and bounded real hypersurfaces running in time (nD)^{O(n^{1.5})}.
Revised:  This enabled more freedom for the design of algorithms computing roadmaps and led us to a first probabilistic roadmap algorithm for smooth and bounded real hypersurfaces running in time (nD)^{O(n^{1.5})}.
Edit intent: fluency
Original:  In this paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets such that the output size and the running time are polynomial in n ^{ d \log( d )} D^{n\log(d), where d }%DIFDELCMD < \le %%% n is the dimension of the algebraic sets defined by the input equations .
Revised:  In this paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets such that the output size and the running time are polynomial in n ^{ n \log( d )} D^{n\log(d), where d }%DIFDELCMD < \le %%% n is the dimension of the algebraic sets defined by the input equations .
Edit intent: fluency
Original:  In this paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets such that the output size and the running time are polynomial in n ^{ d \log( d )} D^{n\log(d), where d }%DIFDELCMD < \le %%% n is the dimension of the algebraic sets defined by the input equations .
Revised:  In this paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets such that the output size and the running time are polynomial in n ^{ d \log( n )} D^{n\log(d), where d }%DIFDELCMD < \le %%% n is the dimension of the algebraic sets defined by the input equations .
Edit intent: fluency
Original:  In this paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets such that the output size and the running time are polynomial in n ^{ d \log( d )} D^{n\log(d), where d }%DIFDELCMD < \le %%% n is the dimension of the algebraic sets defined by the input equations .
Revised:  In this paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets such that the output size and the running time are polynomial in n ^{ d \log( d )} , where d }%DIFDELCMD < \le %%% n is the dimension of the algebraic sets defined by the input equations .
Edit intent: coherence
Original: A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected intersection with all connected components of S. Hence, this kind of object, introduced by Canny, can be used to answer connectivity queries (with applications, for instance, to motion planning) but has also become of central importance in effective real algebraic geometry, since it is used in many higher-level algorithms.
Revised: A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected intersection with all connected components of S. Hence, this kind of object, introduced by Canny, can be used to answer connectivity queries (with applications, for instance, to motion planning) but has also become of central importance in effective real algebraic geometry, since it is used in higher-level algorithms.
Edit intent: clarity
Original:  For a long time, the best known complexity result for computing roadmaps, due to Basu, Pollack and Roy, was s^{d+1 D^{O(n^2)}, where the input is given by s polynomials of degree D in n variables, with d }%DIFDELCMD < \le %%% n the dimension of an associated geometric object.
Revised:  D^{O(n^2)}, where the input is given by s polynomials of degree D in n variables, with d }%DIFDELCMD < \le %%% n the dimension of an associated geometric object.
Edit intent: clarity
Original:  In this }  paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets such that the output size and the running time are polynomial in (nD)^{n\log( n )}  . More precisely, the running time of the algorithm is essentially subquadratic in the output size.
Revised:  In this } paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets . Its output size and the running time are polynomial in (nD)^{n\log( n )} . More precisely, the running time of the algorithm is essentially subquadratic in the output size.
Edit intent: clarity
Original:  In this }  paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets such that the output size and the running time are polynomial in (nD)^{n\log( n )}  . More precisely, the running time of the algorithm is essentially subquadratic in the output size.
Revised:  In this } paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets such that the output size and running time are polynomial in (nD)^{n\log( n )} . More precisely, the running time of the algorithm is essentially subquadratic in the output size.
Edit intent: fluency
Original:  Even under these extra assumptions, it is the first roadmap algorithm with output size and running time polynomial in (nD)^{n\log( n )}.
Revised:  Even under our assumptions, it is the first roadmap algorithm with output size and running time polynomial in (nD)^{n\log( n )}.
Edit intent: clarity
Original:  In this paper, we first analyzes the effect of scarcity of spectrum on the market structure theoretically, where asymmetric-valued spectrum blocks are auctioned off to mobile network operators (MNOs) via a first-price sealed-bid auction .
Revised:  In this paper, we first analyzes the effect of scarcity of spectrum on the market structure theoretically, where asymmetric-valued spectrum blocks are auctioned off to them via a first-price sealed-bid auction .
Edit intent: clarity
Original: We study a stochastic matrix model to understand the mechanics of risk-spreading ( i.e., bet-hedging) by dispersion.
Revised: We study a stochastic matrix model to understand the mechanics of risk-spreading ( or bet-hedging) by dispersion.
Edit intent: clarity
Original:  Moreover, we found that the arithmetic  average of the long-term growth rate converges to the geometric one, because the sample size is finite.
Revised:  Moreover, we found that the arithmetic ensemble average of the long-term growth rate converges to the geometric one, because the sample size is finite.
Edit intent: clarity
Original:  In this study, focusing on phenotype prediction, I show that by properly selecting a small number of loci , a significant amount of missing heritability can be revealed .
Revised:  In this study, focusing on phenotype prediction, the method is able to reveal a significant amount of missing heritability by properly selecting a small number of loci , a significant amount of missing heritability can be revealed .
Edit intent: clarity
Original:  In this study, focusing on phenotype prediction, I show that by properly selecting a small number of loci , a significant amount of missing heritability can be revealed .
Revised:  In this study, focusing on phenotype prediction, I show that by properly selecting a small number of loci with implicit predictive ability .
Edit intent: clarity
Original:  In this article , we have employed an iterative interpolation scheme for the construction of scaling functions in a two- dimensional mesh that is a finite collection of rectangles .
Revised:  In this article , we have employed an iterative interpolation scheme for the construction of scaling functions in a two- dimensional mesh that is a finite collection of rectangular elements .
Edit intent: clarity
Original:  We show that the discretization error is controlled by the order of the scaling function.
Revised:  We show that the discretization error is controlled by the order of the basis function.
Edit intent: clarity
Original:  The potential of this novel technique has been verified with some representative examples of the Poisson equation.
Revised:  The potential of this novel technique has been demonstrated with some representative examples of the Poisson equation.
Edit intent: clarity
Original:  We have extended the technique for solving nonlinear advection-diffusion equations, and simulated a shear driven flow in a square cavity at CFL = 2.5 ( Courant Friedrichs Lewy) and Re = 1  000 (Reynolds number) .
Revised:  We have extended the technique for solving nonlinear advection-diffusion equations, and simulated a shear driven flow in a square cavity at CFL = 2.5 ( Courant Friedrichs Lewy) and Re = 1 \, 000 (Reynolds number) .
Edit intent: fluency
Original:  The Indian buffet process (IBP) is such an approach that can be used to define a prior distribution on infinite binary features, where the exchangeability among subjects is assumed.
Revised:  The Indian buffet process (IBP) is such an example that can be used to define a prior distribution on infinite binary features, where the exchangeability among subjects is assumed.
Edit intent: clarity
Original:  Phylogenetic Indian buffet process (pIBP), a derivative of IBP, enables the modeling of non-exchangeability among subjects through a stochastic process on a rooted tree, which is similar to that used in phylogenetics, to describe relationships among the subjects.
Revised:  The phylogenetic Indian buffet process (pIBP), a derivative of IBP, enables the modeling of non-exchangeability among subjects through a stochastic process on a rooted tree, which is similar to that used in phylogenetics, to describe relationships among the subjects.
Edit intent: fluency
Original:  In this paper, we study both theoretical properties and practical usefulness of IBP and pIBP for binary factor models.
Revised:  In this paper, we study the theoretical properties of IBP and pIBP for binary factor models.
Edit intent: clarity
Original:  For theoretical analysis, we established the posterior convergence rates for both IBP and pIBP and substantiated the theoretical results through simulation studies.
Revised:  For theoretical analysis, we established the posterior convergence rates for both IBP and pIBP and substantiate the theoretical results through simulation studies.
Edit intent: fluency
Original:  We  quantify this selective advantage theoretically and present analytical formulas for the average growth of the fastest species and its fixation probability .
Revised:  We theoretically quantify this selective advantage theoretically and present analytical formulas for the average growth of the fastest species and its fixation probability .
Edit intent: fluency
Original:  We  quantify this selective advantage theoretically and present analytical formulas for the average growth of the fastest species and its fixation probability .
Revised:  We quantify this selective advantage and present analytical formulas for the average growth of the fastest species and its fixation probability .
Edit intent: clarity
Original:  If S is a set of n points in %DIFDELCMD < \real%%%  ^d, we say that S is (k,G)-clusterable if it can be partitioned into k clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object G. In this paper, as an application of Helly's theorem, by taking a constant size sample from S, we present a testing algorithm for (k,G)-clustering, i.e. , to distinguish between two cases:
Revised:  If S is a set of n points in %DIFDELCMD < \real%%% R ^d, we say that S is (k,G)-clusterable if it can be partitioned into k clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object G. In this paper, as an application of Helly's theorem, by taking a constant size sample from S, we present a testing algorithm for (k,G)-clustering, i.e. , to distinguish between two cases:
Edit intent: clarity
Original:  If S is a set of n points in %DIFDELCMD < \real%%%  ^d, we say that S is (k,G)-clusterable if it can be partitioned into k clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object G. In this paper, as an application of Helly's theorem, by taking a constant size sample from S, we present a testing algorithm for (k,G)-clustering, i.e. , to distinguish between two cases:
Revised:  If S is a set of n points in %DIFDELCMD < \real%%% ^d, we say that S is (k,G)-clusterable if it can be partitioned into k clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object G. In this paper, as an application of Helly's theorem, by taking a constant size sample from S, we present a testing algorithm for (k,G)-clustering, i.e. , to distinguish between two cases:
Edit intent: fluency
Original:  when S is (k,G)-clusterable  and when it is \epsilon-far from being (k,G)-clusterable.
Revised:  when S is (k,G)-clusterable , and when it is \epsilon-far from being (k,G)-clusterable.
Edit intent: fluency
Original:  For k>1, we solve a weaker version of this problem.
Revised:  For k>1, we solve a weaker version of this problem.
Edit intent: fluency
Original:  Finally, as an application of  testing result, in clustering with outliers, we show that one can find the approximate clusters by querying a constant size sample, with high probability.
Revised:  Finally, as an application of our testing result, in clustering with outliers, we show that one can find the approximate clusters by querying a constant size sample, with high probability.
Edit intent: fluency
Original:  Finally, as an application of  testing result, in clustering with outliers, we show that one can find the approximate clusters by querying a constant size sample, with high probability.
Revised:  Finally, as an application of testing result, in clustering with outliers, we show that one can find the approximate clusters by querying a constant size sample, with high probability.
Edit intent: fluency
Original:  We use techniques of cluster algebra theory to design efficient subtraction-free algorithms for computing Schur functions and their skew, double, and supersymmetric analogues, thereby generalizing earlier results by P. Koev.
Revised:  We use cluster transformations to design efficient subtraction-free algorithms for computing Schur functions and their skew, double, and supersymmetric analogues, thereby generalizing earlier results by P. Koev.
Edit intent: clarity
Original:  We also develop such algorithms for  the generating functions of spanning trees .
Revised:  We also develop such algorithms for computing the generating functions of spanning trees .
Edit intent: clarity
Original: Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three arithmetic operations: addition, multiplication, and division.
Revised: Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division.
Edit intent: clarity
Original:  We also develop such algorithms for computing the generating functions of spanning trees, both directed and undirected.
Revised:  We develop such algorithms for computing the generating functions of spanning trees, both directed and undirected.
Edit intent: clarity
Original:  We also develop such algorithms for computing the generating functions of spanning trees, both directed and undirected.
Revised:  We also develop such algorithms for computing generating functions of spanning trees, both directed and undirected.
Edit intent: fluency
Original:  A comparison of this result to the lower bound due to M. Jerrum and M. Snir demonstrates that in subtraction-free computations, "division can be exponentially powerful." 
Revised:  A comparison to the lower bound due to M. Jerrum and M. Snir demonstrates that in subtraction-free computations, "division can be exponentially powerful." 
Edit intent: clarity
Original:  A comparison of this result to the lower bound due to M. Jerrum and M. Snir demonstrates that in subtraction-free computations, "division can be exponentially powerful." 
Revised:  A comparison of this result to the lower bound due to M. Jerrum and M. Snir shows that in subtraction-free computations, "division can be exponentially powerful." 
Edit intent: clarity
Original: This paper presents a tractable model for analyzing non-coherent joint-transmission base station (BS) cooperation, taking into account the irregular BS deployment typically encountered in practice.
Revised: This paper presents a tractable model for analyzing non-coherent joint transmission base station (BS) cooperation, taking into account the irregular BS deployment typically encountered in practice.
Edit intent: fluency
Original:  Besides cellular-network specific aspects such as BS density, channel fading, average path loss and interference, the model also captures relevant cooperation mechanisms including user-centric BS clustering and channel-dependent scheduling .
Revised:  Besides cellular-network specific aspects such as BS density, channel fading, average path loss and interference, the model also captures relevant cooperation mechanisms including user-centric BS clustering and channel-dependent cooperation activation .
Edit intent: clarity
Original:  Finally, it is shown that intra-cluster frequency reuse is favorable in moderately-loaded cells with generous triggering of a joint-transmission , while intra-cluster coordinated scheduling may be better in lightly-loaded cells with conservative triggering of a joint transmission .
Revised:  Finally, it is shown that intra-cluster frequency reuse is favorable in moderately loaded cells with generous triggering of a joint-transmission , while intra-cluster coordinated scheduling may be better in lightly-loaded cells with conservative triggering of a joint transmission .
Edit intent: fluency
Original:  Finally, it is shown that intra-cluster frequency reuse is favorable in moderately-loaded cells with generous triggering of a joint-transmission , while intra-cluster coordinated scheduling may be better in lightly-loaded cells with conservative triggering of a joint transmission .
Revised:  Finally, it is shown that intra-cluster frequency reuse is favorable in moderately-loaded cells with generous triggering of a joint-transmission , while intra-cluster coordinated scheduling may be better in lightly loaded cells with conservative triggering of a joint transmission .
Edit intent: fluency