wanyu/IteraTeR_v2 huggingface dataset:
Original:  these relations should have small degree with respect to an input degree shift s \in \mathbb{Z . For an arbitrary input shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov  form with a cost of O ~(m^{ w-1 } \sigma)
Revised:  these relations should have small degree with respect to an input degree shift . For an arbitrary input shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov form with a cost of O ~(m^{ w-1 } \sigma)
Edit intent: clarity
Original:  these relations should have small degree with respect to an input degree shift s \in \mathbb{Z . For an arbitrary input shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov  form with a cost of O ~(m^{ w-1 } \sigma)
Revised:  these relations should have small degree with respect to an input degree shift s \in \mathbb{Z . For an arbitrary shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov form with a cost of O ~(m^{ w-1 } \sigma)
Edit intent: fluency
Original:  operations in the base field , where w is the exponent of matrix multiplication , and the notation O ~(\cdot) omits logarithmic terms . Earlier works, in the case of Hermite-Pad\'e approximation Zhou and Labahn, 2012 and in the general interpolation case Jeannerod et al., 2015 , compute non-normalized bases.
Revised:  operations in the base field , where w is the exponent of matrix multiplication and the notation O ~(\cdot) omits logarithmic terms . Earlier works, in the case of Hermite-Pad\'e approximation Zhou and Labahn, 2012 and in the general interpolation case Jeannerod et al., 2015 , compute non-normalized bases.
Edit intent: fluency
Original:  operations in the base field , where w is the exponent of matrix multiplication , and the notation O ~(\cdot) omits logarithmic terms . Earlier works, in the case of Hermite-Pad\'e approximation Zhou and Labahn, 2012 and in the general interpolation case Jeannerod et al., 2015 , compute non-normalized bases.
Revised:  operations in the base field , where w is the exponent of matrix multiplication , and the notation O ~(\cdot) indicates that logarithmic terms are omitted . Earlier works, in the case of Hermite-Pad\'e approximation Zhou and Labahn, 2012 and in the general interpolation case Jeannerod et al., 2015 , compute non-normalized bases.
Edit intent: clarity
Original:  Since for arbitrary shifts such bases may have size \Theta(m^2 \sigma), the cost bound O \tilde{~}(m^{ w-1 } \sigma) was feasible only with restrictive assumptions on s for ensuring small output sizes.
Revised:  Since for arbitrary shifts such bases may have size \Theta(m^2 \sigma), the cost bound O \tilde{~}(m^{ w-1 } \sigma) was feasible only with restrictive assumptions on the shift that ensure small output sizes.
Edit intent: clarity
Original:  We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediary bases.
Revised:  We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediate bases.
Edit intent: fluency
Original:  We find that none of the considered methods can consistently outperform the others when it comes to detecting and locating the change points .
Revised:  We find that none of the considered methods can consistently outperform the others when it comes to detecting and locating the expected change points in empirical temporal networks .
Edit intent: clarity
Original: We give a Las Vegas algorithm which computes the shifted Popov form of a nonsingular polynomial matrix A \in \mathbb{K^{ m \times m } in expected O}%DIFDELCMD < ~%%% (m^w\mathcal{O}\mathcal{O}\mathcal{O}\le \lceil \sigma(A)
Revised: We give a Las Vegas algorithm which computes the shifted Popov form of an m \times m nonsingular polynomial matrix A \in \mathbb{K^{ m \times m } in expected O}%DIFDELCMD < ~%%% (m^w\mathcal{O}\mathcal{O}\mathcal{O}\le \lceil \sigma(A)
Edit intent: fluency
Original: We give a Las Vegas algorithm which computes the shifted Popov form of a nonsingular polynomial matrix A \in \mathbb{K^{ m \times m } in expected O}%DIFDELCMD < ~%%% (m^w\mathcal{O}\mathcal{O}\mathcal{O}\le \lceil \sigma(A)
Revised: We give a Las Vegas algorithm which computes the shifted Popov form of a nonsingular polynomial matrix A \in \mathbb{K^{ m \times m } in expected O}%DIFDELCMD < ~%%% (m^w\mathcal{O}\mathcal{O}\mathcal{O}\le \lceil \sigma
Edit intent: coherence
Original:  /m \rceil ) \subseteq O ( m^w deg(A) ) operationsin K, where deg(A) is the degree of A, \sigma(A) is some quantity such that \sigma(A) / m is bounded from above by both the average row degree and the average column degree of A, w is the exponent of matrixmultiplication, and O (\cdot )indicates that logarithmic factors are omitted.
Revised:  /m \rceil ( m^w deg(A) ) operationsin K, where deg(A) is the degree of A, \sigma(A) is some quantity such that \sigma(A) / m is bounded from above by both the average row degree and the average column degree of A, w is the exponent of matrixmultiplication, and O (\cdot )indicates that logarithmic factors are omitted.
Edit intent: coherence
Original:  /m \rceil ) \subseteq O ( m^w deg(A) ) operationsin K, where deg(A) is the degree of A, \sigma(A) is some quantity such that \sigma(A) / m is bounded from above by both the average row degree and the average column degree of A, w is the exponent of matrixmultiplication, and O (\cdot )indicates that logarithmic factors are omitted.
Revised:  /m \rceil ) \subseteq O ( m^w deg(A) ) operationsin K, where deg(A) is the degree of A, \sigma(A) is some quantity such that \sigma(A) / m bounded from above by both the average row degree and the average column degree of A, w is the exponent of matrixmultiplication, and O (\cdot )indicates that logarithmic factors are omitted.
Edit intent: fluency
Original:  /m \rceil ) \subseteq O ( m^w deg(A) ) operationsin K, where deg(A) is the degree of A, \sigma(A) is some quantity such that \sigma(A) / m is bounded from above by both the average row degree and the average column degree of A, w is the exponent of matrixmultiplication, and O (\cdot )indicates that logarithmic factors are omitted.
Revised:  /m \rceil ) \subseteq O ( m^w deg(A) ) operationsin K, where deg(A) is the degree of A, \sigma(A) is some quantity such that \sigma(A) / m is bounded from above by both the average row degree and the average column degree of (\cdot )indicates that logarithmic factors are omitted.
Edit intent: clarity
Original:  This improves\mathcal{O}\lceil \rceil upon the cost bound of the fastest known algorithms for row reduction and Hermite form computation, which are deterministic.
Revised:  This improves\mathcal{O}\lceil \rceil), improving upon the cost bound of the fastest known algorithms for row reduction and Hermite form computation, which are deterministic.
Edit intent: fluency
Original:  This improves\mathcal{O}\lceil \rceil upon the cost bound of the fastest known algorithms for row reduction and Hermite form computation, which are deterministic.
Revised:  This improves\mathcal{O}\lceil \rceil upon the cost of the fastest known algorithms for row reduction and Hermite form computation, which are deterministic.
Edit intent: clarity
Original:  This improves\mathcal{O}\lceil \rceil upon the cost bound of the fastest known algorithms for row reduction and Hermite form computation, which are deterministic.
Revised:  This improves\mathcal{O}\lceil \rceil upon the cost bound of the fastest previously known algorithm for row reduction and Hermite form computation, which are deterministic.
Edit intent: clarity
Original:  It remains to find the basis in shifted Popov form of this solution set: we give a deterministic algorithm for this problem in O%DIFDELCMD < \tilde{~}%%% (m\widetilde{\mathcal{O}} ^{ w-1 } \sigma)  operations, where m is the number of unknowns and \sigma is the sum of the degrees of the moduli.
Revised:  It remains to find the basis in shifted Popov form of this set. We give a deterministic algorithm for this problem in O%DIFDELCMD < \tilde{~}%%% (m\widetilde{\mathcal{O}} ^{ w-1 } \sigma) operations, where m is the number of unknowns and \sigma is the sum of the degrees of the moduli.
Edit intent: clarity
Original:  It remains to find the basis in shifted Popov form of this solution set: we give a deterministic algorithm for this problem in O%DIFDELCMD < \tilde{~}%%% (m\widetilde{\mathcal{O}} ^{ w-1 } \sigma)  operations, where m is the number of unknowns and \sigma is the sum of the degrees of the moduli.
Revised:  It remains to find the basis in shifted Popov form of this solution set: we give a deterministic algorithm for this %DIFDELCMD < \tilde{~}%%% (m\widetilde{\mathcal{O}} ^{ w-1 } \sigma) operations, where m is the number of unknowns and \sigma is the sum of the degrees of the moduli.
Edit intent: clarity
Original:  It remains to find the basis in shifted Popov form of this solution set: we give a deterministic algorithm for this problem in O%DIFDELCMD < \tilde{~}%%% (m\widetilde{\mathcal{O}} ^{ w-1 } \sigma)  operations, where m is the number of unknowns and \sigma is the sum of the degrees of the moduli.
Revised:  It remains to find the basis in shifted Popov form of this solution set: we give a deterministic algorithm for this problem in O%DIFDELCMD < \tilde{~}%%% (m\widetilde{\mathcal{O}}(m ^{ w-1 } \sigma) operations, where m is the number of unknowns and \sigma is the sum of the degrees of the moduli.
Edit intent: fluency
Original:  It remains to find the basis in shifted Popov form of this solution set: we give a deterministic algorithm for this problem in O%DIFDELCMD < \tilde{~}%%% (m\widetilde{\mathcal{O}} ^{ w-1 } \sigma)  operations, where m is the number of unknowns and \sigma is the sum of the degrees of the moduli.
Revised:  It remains to find the basis in shifted Popov form of this solution set: we give a deterministic algorithm for this problem in O%DIFDELCMD < \tilde{~}%%% (m\widetilde{\mathcal{O}} ^{ \omega-1 } \sigma) operations, where m is the number of unknowns and \sigma is the sum of the degrees of the moduli.
Edit intent: clarity
Original:  It remains to find the basis in shifted Popov form of this solution set: we give a deterministic algorithm for this problem in O%DIFDELCMD < \tilde{~}%%% (m\widetilde{\mathcal{O}} ^{ w-1 } \sigma)  operations, where m is the number of unknowns and \sigma is the sum of the degrees of the moduli.
Revised:  It remains to find the basis in shifted Popov form of this solution set: we give a deterministic algorithm for this problem in O%DIFDELCMD < \tilde{~}%%% (m\widetilde{\mathcal{O}} ^{ w-1 } \sigma) field operations, where m is the number of unknowns and \sigma is the sum of the degrees of the moduli.
Edit intent: clarity
Original:  Finally, we present applications for high-frequency arbitrage based on LOB resiliency analysis and find that both long and short arbitrage strategies we design can achieve significantly positive returns.
Edit intent: coherence
Original: Certificates to a linear algebra computation are additional data structures for each output, which can be used by a-possibly randomized- verification algorithm that proves the correctness of each output.
Revised: Computational problem certificates are additional data structures for each output, which can be used by a-possibly randomized- verification algorithm that proves the correctness of each output.
Edit intent: clarity
Original: Certificates to a linear algebra computation are additional data structures for each output, which can be used by a-possibly randomized- verification algorithm that proves the correctness of each output.
Revised: Certificates to a linear algebra computation are additional data structures for each output, which can be used by a-possibly randomized-verification algorithm that proves the correctness of each output.
Edit intent: fluency
Original:  In this paper, we give an algorithm that compute a certificate for the minimal polynomial of sparse or structured n x n matrices over an abstract field, of sufficiently large cardinality, whose Monte Carlo verification complexity requires a single matrix-vector multiplication and a linear number of extra field operations.
Revised:  In this paper, we give an algorithm that computes a certificate for the minimal polynomial of sparse or structured n x n matrices over an abstract field, of sufficiently large cardinality, whose Monte Carlo verification complexity requires a single matrix-vector multiplication and a linear number of extra field operations.
Edit intent: fluency
Original:  In this paper, we give an algorithm that compute a certificate for the minimal polynomial of sparse or structured n x n matrices over an abstract field, of sufficiently large cardinality, whose Monte Carlo verification complexity requires a single matrix-vector multiplication and a linear number of extra field operations.
Revised:  In this paper, we give an algorithm that compute a certificate for the minimal polynomial of sparse or structured nxn matrices over an abstract field, of sufficiently large cardinality, whose Monte Carlo verification complexity requires a single matrix-vector multiplication and a linear number of extra field operations.
Edit intent: fluency
Original:  We develop two algorithms using different approaches and with different advantages .
Revised:  We develop two algorithms using different approaches .
Edit intent: clarity
Original: We present a further development of the HYPERDIRE project : a set of Mathematica-based program packages for manipulations with Horn-type hypergeometric functions on the basis of differential equations.
Revised: We present a further extension of the HYPERDIRE project : a set of Mathematica-based program packages for manipulations with Horn-type hypergeometric functions on the basis of differential equations.
Edit intent: clarity
Original: We present a further development of the HYPERDIRE project : a set of Mathematica-based program packages for manipulations with Horn-type hypergeometric functions on the basis of differential equations.
Revised: We present a further development of the HYPERDIRE project , which is devoted to the creation of a set of Mathematica-based program packages for manipulations with Horn-type hypergeometric functions on the basis of differential equations.
Edit intent: coherence
Original:  Specifically, we present the implementation of  differential reduction for the Lauricella function F_C of three variables.
Revised:  Specifically, we present the implementation of the differential reduction for the Lauricella function F_C of three variables.
Edit intent: fluency
Original: Deep Convolutional Neural Network (CNN) enforces supervised information only at the output layer, and hidden layers are trained by back propagating the prediction error from the output layer without explicit supervision.
Revised: Deep Convolutional Neural Networks (CNN) enforces supervised information only at the output layer, and hidden layers are trained by back propagating the prediction error from the output layer without explicit supervision.
Edit intent: fluency
Original:  We propose a supervised feature learning approach, Label Consistency Neural Network, which enforces direct supervision in late hidden layers.
Revised:  We propose a supervised feature learning approach, Label Consistent Neural Network, which enforces direct supervision in late hidden layers.
Edit intent: fluency
Original:  This label consistency constraint not only alleviates the common problem of gradient vanishing and tends to faster convergence , but also makes the features derived from late hidden layers discriminative enough for classification even using a simple classifier such as k-NN classifier, since input signals from the same class will have very similar representations.
Revised:  This label consistency constraint alleviates the common problem of gradient vanishing and tends to faster convergence , but also makes the features derived from late hidden layers discriminative enough for classification even using a simple classifier such as k-NN classifier, since input signals from the same class will have very similar representations.
Edit intent: coherence
Original:  This label consistency constraint not only alleviates the common problem of gradient vanishing and tends to faster convergence , but also makes the features derived from late hidden layers discriminative enough for classification even using a simple classifier such as k-NN classifier, since input signals from the same class will have very similar representations.
Revised:  This label consistency constraint not only alleviates the common problem of gradient vanishing and tends to faster convergence ; it also makes the features derived from late hidden layers discriminative enough for classification even using a simple classifier such as k-NN classifier, since input signals from the same class will have very similar representations.
Edit intent: fluency
Original:  This label consistency constraint not only alleviates the common problem of gradient vanishing and tends to faster convergence , but also makes the features derived from late hidden layers discriminative enough for classification even using a simple classifier such as k-NN classifier, since input signals from the same class will have very similar representations.
Revised:  This label consistency constraint not only alleviates the common problem of gradient vanishing and tends to faster convergence , but also makes the features derived from late hidden layers discriminative enough for classification even using a simple k-NN classifier, since input signals from the same class will have very similar representations.
Edit intent: clarity
Original:  Experimental results demonstrate that our approach can achieve state-of-the-art performances on several public benchmarks for action and object category recognition.
Revised:  Experimental results demonstrate that our approach achieves state-of-the-art performances on several public benchmarks for action and object category recognition.
Edit intent: clarity
Original:  These topological markers  provide insights into mechanisms of functional activity underlying disease phenotype and could further be used for designing algorithms for clinical diagnosis of schizophrenia as well as its early detection .
Revised:  These topological markers may provide insights into mechanisms of functional activity underlying disease phenotype and could further be used for designing algorithms for clinical diagnosis of schizophrenia as well as its early detection .
Edit intent: fluency
Original: The observation that in large data the occurrence of first significant digits of numbers  is often governed by a logarithmically decreasing distribution , quite far from the ordinarily expected uniform distribution, is called Benford's law (BL) . It was first reported by S. Newcomb  and many decades later independently by F. Benford  .
Revised: The occurrence of first significant digits of numbers is often governed by a logarithmically decreasing distribution , quite far from the ordinarily expected uniform distribution, is called Benford's law (BL) . It was first reported by S. Newcomb and many decades later independently by F. Benford .
Edit intent: clarity
Original: The observation that in large data the occurrence of first significant digits of numbers  is often governed by a logarithmically decreasing distribution , quite far from the ordinarily expected uniform distribution, is called Benford's law (BL) . It was first reported by S. Newcomb  and many decades later independently by F. Benford  .
Revised: The observation that in large data the occurrence of first significant digits of numbers in large data is often governed by a logarithmically decreasing distribution , quite far from the ordinarily expected uniform distribution, is called Benford's law (BL) . It was first reported by S. Newcomb and many decades later independently by F. Benford .
Edit intent: fluency
Original: The observation that in large data the occurrence of first significant digits of numbers  is often governed by a logarithmically decreasing distribution , quite far from the ordinarily expected uniform distribution, is called Benford's law (BL) . It was first reported by S. Newcomb  and many decades later independently by F. Benford  .
Revised: The observation that in large data the occurrence of first significant digits of numbers is often governed by a logarithmically decreasing distribution called Benford's law (BL) . It was first reported by S. Newcomb and many decades later independently by F. Benford .
Edit intent: coherence
Original: The observation that in large data the occurrence of first significant digits of numbers  is often governed by a logarithmically decreasing distribution , quite far from the ordinarily expected uniform distribution, is called Benford's law (BL) . It was first reported by S. Newcomb  and many decades later independently by F. Benford  .
Revised: The observation that in large data the occurrence of first significant digits of numbers is often governed by a logarithmically decreasing distribution , quite far from the ordinarily expected uniform distribution, is called Benford's law (BL) , reported first by S. Newcomb and many decades later independently by F. Benford .
Edit intent: coherence
Original:  However,  huge swell in number of publications which the law has seen lately is an indication of its remarkable resurgence .
Revised:  However, an indication of its remarkable resurgence is the huge swell in number of publications which the law has seen lately is an indication of its remarkable resurgence .
Edit intent: fluency
Original:  We find that the monthly citations data of the articles citing Benford's paper remarkably follow the law but that of Newcomb's paper in some cases does/does not follow the law.
Revised:  We find that the citations data of the articles citing Benford's paper remarkably follow the law but that of Newcomb's paper in some cases does/does not follow the law.
Edit intent: clarity
Original: We give an upper bound on the size of a dominating set in a circulant graph on n notes and k distinct chord lengths.
Revised: We give an efficient construction of a reasonably small dominating set in a circulant graph on n notes and k distinct chord lengths.
Edit intent: clarity
Original:  This result is based on bounds on some double exponential sums  .
Revised:  This result is based on bounds on some double exponential sums . .
Edit intent: fluency
Original:  Here , we show that if G has girth at least 10, then G is b-continuous.
Revised:  In this article , we show that if G has girth at least 10, then G is b-continuous.
Edit intent: coherence
Original: In the convergence analysis of numerical methods for solving partial differential equations (such as finite element methods) one arrives at certain generalized eigenvalue problems, whose maximal eigenvalues need to be estimated as accurate as possible.
Revised: In the convergence analysis of numerical methods for solving partial differential equations (such as finite element methods) one arrives at certain generalized eigenvalue problems, whose maximal eigenvalues need to be estimated as accurately as possible.
Edit intent: fluency
Original:  In this paper , we initiate and develop the theory of semantic acyclicity under constraints.
Revised:  In this paper we initiate and develop the theory of semantic acyclicity under constraints.
Edit intent: fluency
Original:  We study the following natural problem: Given a CQ and a set of constraints, is the query semantically acyclic under the constraints, or, in other words, is the query equivalent to an acyclic one over all those databases that satisfy the set of constraints?
Revised:  More precisely, we study the following natural problem: Given a CQ and a set of constraints, is the query semantically acyclic under the constraints, or, in other words, is the query equivalent to an acyclic one over all those databases that satisfy the set of constraints?
Edit intent: coherence