wanyu/IteraTeR_v2 huggingface dataset:
Original:  We demonstrate that the max-flow min-cut theorem is equivalent to the Euler characteristic description of orientable divisors , and extend this characterization to the setting of partial orientations  . Efficient algorithms for computing break divisors and constructing partial orientations are presented.
Revised:  We prove that the max-flow min-cut theorem is equivalent to the Euler characteristic description of orientable divisors , and extend this characterization to the setting of partial orientations . Efficient algorithms for computing break divisors and constructing partial orientations are presented.
Edit intent: coherence
Original:  We demonstrate that the max-flow min-cut theorem is equivalent to the Euler characteristic description of orientable divisors , and extend this characterization to the setting of partial orientations  . Efficient algorithms for computing break divisors and constructing partial orientations are presented.
Revised:  We demonstrate that the max-flow min-cut theorem is equivalent to the Euler characteristic description of orientable divisors and extend this characterization to the setting of partial orientations . Efficient algorithms for computing break divisors and constructing partial orientations are presented.
Edit intent: fluency
Original: We develop a new framework for investigating linear equivalence of divisors on graphs using a generalization of Gioan's cycle-cocycle reversal system for partial orientations.
Revised: We develop a new framework for investigating linear equivalence of divisors on graphs using a generalization of Gioan's cycle--cocycle reversal system for partial orientations.
Edit intent: fluency
Original:  We then show that the Baker-Norine rank of a partially orientable divisor is one less than the minimum number of directed paths which need to be reversed in the generalized cycle-cocycle reversal system to produce an acyclic partial orientation.
Revised:  We then show that the Baker--Norine rank of a partially orientable divisor is one less than the minimum number of directed paths which need to be reversed in the generalized cycle-cocycle reversal system to produce an acyclic partial orientation.
Edit intent: fluency
Original:  We then show that the Baker-Norine rank of a partially orientable divisor is one less than the minimum number of directed paths which need to be reversed in the generalized cycle-cocycle reversal system to produce an acyclic partial orientation.
Revised:  We then show that the Baker-Norine rank of a partially orientable divisor is one less than the minimum number of directed paths which need to be reversed in the generalized cycle--cocycle reversal system to produce an acyclic partial orientation.
Edit intent: fluency
Original:  These results are applied in providing new proofs of the Riemann-Roch theorem for graphs as well as Luo's topological characterization of rank-determining sets.
Revised:  These results are applied in providing new proofs of the Riemann--Roch theorem for graphs as well as Luo's topological characterization of rank-determining sets.
Edit intent: fluency
Original:  Furthermore, we demonstrate that Pic^{g-1}(G) is canonically isomorphic as a Pic^{0}(G)-torsor to the equivalence classes of full orientations in the cycle-cocycle reversal system acted on my directed path reversals.
Revised:  Furthermore, we demonstrate that Pic^{g-1}(G) is canonically isomorphic as a Pic^{0}(G)-torsor to the equivalence classes of full orientations in the cycle--cocycle reversal system acted on my directed path reversals.
Edit intent: fluency
Original:  Furthermore, we demonstrate that Pic^{g-1}(G) is canonically isomorphic as a Pic^{0}(G)-torsor to the equivalence classes of full orientations in the cycle-cocycle reversal system acted on my directed path reversals.
Revised:  Furthermore, we demonstrate that Pic^{g-1}(G) is canonically isomorphic as a Pic^{0}(G)-torsor to the equivalence classes of full orientations in the cycle-cocycle reversal system acted on by directed path reversals.
Edit intent: fluency
Original:  We propose a new cooperative cognitive relaying protocol that allows the ST to relay a fraction of the undelivered primary packets.
Revised:  We propose a new cooperative cognitive relaying protocol that allows the SU to relay a fraction of the undelivered primary packets.
Edit intent: clarity
Original:  We consider an interference channel model (or a multipacket reception (MPR) channel model), where concurrent transmissions can survive  with certain probability characterized by the complement of channel outages.
Revised:  We consider an interference channel model (or a multipacket reception (MPR) channel model), where concurrent transmissions can survive from interference with certain probability characterized by the complement of channel outages.
Edit intent: clarity
Original:  Our numerical results show the benefits of the cooperation, the receivers' MPR capabilities, and the secondary energy queue arrival rate on the system performance from  network layer standpoint.
Revised:  Our numerical results show the benefits of cooperation, receivers' MPR capabilities, and the secondary energy queue arrival rate on the system performance from network layer standpoint.
Edit intent: fluency
Original:  Our numerical results show the benefits of the cooperation, the receivers' MPR capabilities, and the secondary energy queue arrival rate on the system performance from  network layer standpoint.
Revised:  Our numerical results show the benefits of the cooperation, the receivers' MPR capabilities, and secondary energy queue arrival rate on the system performance from network layer standpoint.
Edit intent: fluency
Original:  Our numerical results show the benefits of the cooperation, the receivers' MPR capabilities, and the secondary energy queue arrival rate on the system performance from  network layer standpoint.
Revised:  Our numerical results show the benefits of the cooperation, the receivers' MPR capabilities, and the secondary energy queue arrival rate on the system performance from a network layer standpoint.
Edit intent: fluency
Original: In this paper, we investigate the maximum secondary throughput of a saturated rechargeable secondary user (SU) sharing the spectrum with a primary user (PU).
Revised: In this paper, we investigate the maximum throughput of a saturated rechargeable secondary user (SU) sharing the spectrum with a primary user (PU).
Edit intent: fluency
Original:  The proposed protocol exploits the primary queue burstiness and receivers' MPR capabilities .
Revised:  The proposed protocol exploits the primary queue burstiness and receivers' MPR capability .
Edit intent: fluency
Original:  Our numerical results show the benefits of cooperation, receivers' MPR capabilities , and secondary energy queue arrival rate on the system performance from a network layer standpoint.
Revised:  Our numerical results show the benefits of cooperation, receivers' MPR capability , and secondary energy queue arrival rate on the system performance from a network layer standpoint.
Edit intent: fluency
Original:  We use this result of  Chang to prove a curious corollary under standard models of recombination: there exist, with high probability, individuals within a constant multiple of \log_2(N) generations into the past who are simultaneously (i) genealogical ancestors of {\em each} of the individuals at the present, and (ii) genetic ancestors to {\em none} of the individuals at the present.
Revised:  We use this result of J. Chang to prove a curious corollary under standard models of recombination: there exist, with high probability, individuals within a constant multiple of \log_2(N) generations into the past who are simultaneously (i) genealogical ancestors of {\em each} of the individuals at the present, and (ii) genetic ancestors to {\em none} of the individuals at the present.
Edit intent: clarity
Original:  However, to the author' s best knowledge this is the first theorem showing a collapse of the parity index that exploits the fact that a given automaton is unambiguous.
Revised:  However, to the authors' best knowledge this is the first theorem showing a collapse of the parity index that exploits the fact that a given automaton is unambiguous.
Edit intent: fluency
Original: Recently, large whole-genome sequencing projects have provided access to much of the rare variation in human populations .
Revised: Large whole-genome sequencing projects have provided access to much of the rare variation in human populations .
Edit intent: clarity
Original:  In this paper , we show how the age of rare variants can be estimated from patterns of haplotype sharing and how this information can detect and quantify historical relationships between populations.
Revised:  Here , we show how the age of rare variants can be estimated from patterns of haplotype sharing and how this information can detect and quantify historical relationships between populations.
Edit intent: coherence
Original:  In this paper , we show how the age of rare variants can be estimated from patterns of haplotype sharing and how this information can detect and quantify historical relationships between populations.
Revised:  In this paper , we show how the age of rare variants can be estimated from patterns of haplotype sharing and how these ages can be related to historical relationships between populations.
Edit intent: clarity
Original:  We investigate the distribution of the age of  f2 variants  in a worldwide sample sequenced by the 1,000 Genomes Project, revealing enormous variation across populations.
Revised:  We investigate the distribution of the age of f2 variants ) in a worldwide sample sequenced by the 1,000 Genomes Project, revealing enormous variation across populations.
Edit intent: fluency
Original:  We investigate the distribution of the age of  f2 variants  in a worldwide sample sequenced by the 1,000 Genomes Project, revealing enormous variation across populations.
Revised:  We investigate the distribution of the age of f2 variants in a worldwide sample sequenced by the 1000 Genomes Project, revealing enormous variation across populations.
Edit intent: fluency
Original:  The median age of  f2 variants shared within continents is 50 to 160 generations for Europe and Asia, and 170 to 320 generations for Africa.
Revised:  The median age of haplotypes carrying f2 variants shared within continents is 50 to 160 generations for Europe and Asia, and 170 to 320 generations for Africa.
Edit intent: clarity
Original:  The median age of  f2 variants shared within continents is 50 to 160 generations for Europe and Asia, and 170 to 320 generations for Africa.
Revised:  The median age of f2 variants is 50 to 160 generations for Europe and Asia, and 170 to 320 generations for Africa.
Edit intent: clarity
Original:  The median age of  f2 variants shared within continents is 50 to 160 generations for Europe and Asia, and 170 to 320 generations for Africa.
Revised:  The median age of f2 variants shared within continents is 50 to 160 generations across populations within Europe or Asia, and 170 to 320 generations for Africa.
Edit intent: clarity
Original:  Variants shared between continents are much older with median ages  ranging from 320 to 670 generations between Europe and Asia, and 1,000 to 2,400 generations between African and Non-African populations .
Revised:  Variants shared between continents are much older with median ages ranging from 320 to 670 generations .
Edit intent: clarity
Original: In this work, we consider Corporate Governance  ties among companies from a multiple network perspective.
Revised: In this work, we consider Corporate Governance (CG) ties among companies from a multiple network perspective.
Edit intent: clarity
Original:  Such a structurenaturally arises from the close interrelation between the Shareholding Network  and the Board of Directors network .
Revised:  Such a structure naturally arises from the close interrelation between the Shareholding Network and the Board of Directors network .
Edit intent: fluency
Original:  Inorder to capture the simultaneous effects on both networks on Corporate Governance , we propose to model the Corporate Governance multiple network structure via tensor analysis.
Revised:  Inorder to capture the simultaneous effects of both networks on Corporate Governance , we propose to model the Corporate Governance multiple network structure via tensor analysis.
Edit intent: fluency
Original:  Inorder to capture the simultaneous effects on both networks on Corporate Governance , we propose to model the Corporate Governance multiple network structure via tensor analysis.
Revised:  Inorder to capture the simultaneous effects on both networks on CG , we propose to model the Corporate Governance multiple network structure via tensor analysis.
Edit intent: clarity
Original:  Inorder to capture the simultaneous effects on both networks on Corporate Governance , we propose to model the Corporate Governance multiple network structure via tensor analysis.
Revised:  Inorder to capture the simultaneous effects on both networks on Corporate Governance , we propose to model the CG multiple network structure via tensor analysis.
Edit intent: clarity
Original:  After providing some empirical results from the Italian financial market in the univariate case, we will show that a tensor-based multiple network approach can reveal important information.
Revised:  By providing some empirical results from the Italian financial market in the univariate case, we will show that a tensor-based multiple network approach can reveal important information.
Edit intent: fluency
Original:  After providing some empirical results from the Italian financial market in the univariate case, we will show that a tensor-based multiple network approach can reveal important information.
Revised:  After providing some empirical results from the Italian financial market in the univariate case, we then show that a tensor-based multiple network approach can reveal important information.
Edit intent: clarity
Original:  After providing some empirical results from the Italian financial market in the univariate case, we will show that a tensor-based multiple network approach can reveal important information.
Revised:  After providing some empirical results from the Italian financial market in the univariate case, we will show that a tensor--based multiple network approach can reveal important information.
Edit intent: fluency
Original:  We try to find the correspondence between the histograms of gene lengths and the distributions of hyperbolic type for some bacteria, as %DIFDELCMD < {\em %DIFDELCMD < %%% Borelia burgdorferi ,%DIFDELCMD < {\em %%% Escherichia coli and%DIFDELCMD < {\em %%% Saccharomyces cerevisiae .
Revised:  We present the number of repetitions of genes in tne genomes for some bacteria, as %DIFDELCMD < {\em %DIFDELCMD < %%% Borelia burgdorferi ,%DIFDELCMD < {\em %%% Escherichia coli and%DIFDELCMD < {\em %%% Saccharomyces cerevisiae .
Edit intent: clarity
Original:  We try to find the correspondence between the histograms of gene lengths and the distributions of hyperbolic type for some bacteria, as %DIFDELCMD < {\em %DIFDELCMD < %%% Borelia burgdorferi ,%DIFDELCMD < {\em %%% Escherichia coli and%DIFDELCMD < {\em %%% Saccharomyces cerevisiae .
Revised:  We try to find the correspondence between the histograms of gene lengths and the distributions of hyperbolic type for some bacteria, as %DIFDELCMD < {\em %DIFDELCMD < %%% Borelia burgdorferi %DIFDELCMD < {\em %%% Escherichia coli and%DIFDELCMD < {\em %%% Saccharomyces cerevisiae .
Edit intent: fluency
Original:  We try to find the correspondence between the histograms of gene lengths and the distributions of hyperbolic type for some bacteria, as %DIFDELCMD < {\em %DIFDELCMD < %%% Borelia burgdorferi ,%DIFDELCMD < {\em %%% Escherichia coli and%DIFDELCMD < {\em %%% Saccharomyces cerevisiae .
Revised:  We try to find the correspondence between the histograms of gene lengths and the distributions of hyperbolic type for some bacteria, as %DIFDELCMD < {\em %DIFDELCMD < %%% Borelia burgdorferi ,%DIFDELCMD < {\em %%% and%DIFDELCMD < {\em %%% Saccharomyces cerevisiae .
Edit intent: coherence
Original:  We try to find the correspondence between the histograms of gene lengths and the distributions of hyperbolic type for some bacteria, as %DIFDELCMD < {\em %DIFDELCMD < %%% Borelia burgdorferi ,%DIFDELCMD < {\em %%% Escherichia coli and%DIFDELCMD < {\em %%% Saccharomyces cerevisiae .
Revised:  We try to find the correspondence between the histograms of gene lengths and the distributions of hyperbolic type for some bacteria, as %DIFDELCMD < {\em %DIFDELCMD < %%% Borelia burgdorferi ,%DIFDELCMD < {\em %%% Escherichia coli %DIFDELCMD < {\em %%% Saccharomyces cerevisiae .
Edit intent: coherence
Original:  Our certificates can be verified in Monte-Carlo bit complexity (n^2 \log ||A|| \lognormA )^{1+o(1)}, where \log ||A|| \lognormA is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc.
Revised:  Our certificates can be verified in Monte-Carlo bit complexity (n^2 \lognormA )^{1+o(1)}, where \log ||A|| \lognormA is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc.
Edit intent: fluency
Original:  Our certificates can be verified in Monte-Carlo bit complexity (n^2 \log ||A|| \lognormA )^{1+o(1)}, where \log ||A|| \lognormA is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc.
Revised:  Our certificates can be verified in Monte-Carlo bit complexity (n^2 \log ||A|| \lognormA )^{1+o(1)}, where \lognormA is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc.
Edit intent: fluency
Original:   ISSAC 2011] subject to computational hardness assumptions.
Revised:  \ ISSAC 2011] subject to computational hardness assumptions.
Edit intent: fluency
Original:  The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size N, meaning N times a factor N^{o(1)}, i.e., a factor N^{\eta(N)} with \lim _{N\to \infty \to  \eta(N) = 0. We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an n\times n dense integer matrix A.
Revised:  The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size N, meaning N times a factor N^{o(1)}, i.e., a factor N^{\eta(N)} with \lim \_ \to \eta(N) = 0. We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an n\times n dense integer matrix A.
Edit intent: clarity
Original:  The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size N, meaning N times a factor N^{o(1)}, i.e., a factor N^{\eta(N)} with \lim _{N\to \infty \to  \eta(N) = 0. We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an n\times n dense integer matrix A.
Revised:  The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size N, meaning N times a factor N^{o(1)}, i.e., a factor N^{\eta(N)} with \lim _{N\to \infty N\to \eta(N) = 0. We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an n\times n dense integer matrix A.
Edit intent: fluency
Original:  The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size N, meaning N times a factor N^{o(1)}, i.e., a factor N^{\eta(N)} with \lim _{N\to \infty \to  \eta(N) = 0. We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an n\times n dense integer matrix A.
Revised:  The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size N, meaning N times a factor N^{o(1)}, i.e., a factor N^{\eta(N)} with \lim _{N\to \infty \to \infty \eta(N) = 0. We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an n\times n dense integer matrix A.
Edit intent: fluency
Original:  }\det\left( A_{\backslash\left\{ j\right\} }-\lambda _{k}I\right) \] }\left( \right) \left( \right) ^{\prime}}\left( \right) \left( \lambda\right) }{d\lambda} and A_{\backslash\left\{ j\right\} } is obtained from A by removal of row j and column j, suggests us to consider the square eigenvector component as a graph centrality metric for node j that } reflects the impact of the removal of node m from the graph at a different eigenfrequencyof that graph matrix .
Revised:  }\det\left( A_{\backslash\left\{ j\right\} }-\lambda _{k}I\right) \] }\left( \lambda\right) \left( \right) ^{\prime}}\left( \right) \left( \lambda\right) }{d\lambda} and A_{\backslash\left\{ j\right\} } is obtained from A by removal of row j and column j, suggests us to consider the square eigenvector component as a graph centrality metric for node j that } reflects the impact of the removal of node m from the graph at a different eigenfrequencyof that graph matrix .
Edit intent: fluency
Original:  }\det\left( A_{\backslash\left\{ j\right\} }-\lambda _{k}I\right) \] }\left( \right) \left( \right) ^{\prime}}\left( \right) \left( \lambda\right) }{d\lambda} and A_{\backslash\left\{ j\right\} } is obtained from A by removal of row j and column j, suggests us to consider the square eigenvector component as a graph centrality metric for node j that } reflects the impact of the removal of node m from the graph at a different eigenfrequencyof that graph matrix .
Revised:  }\det\left( A_{\backslash\left\{ j\right\} }-\lambda _{k}I\right) \] }\left( \right) =\det\left( \right) ^{\prime}}\left( \right) \left( \lambda\right) }{d\lambda} and A_{\backslash\left\{ j\right\} } is obtained from A by removal of row j and column j, suggests us to consider the square eigenvector component as a graph centrality metric for node j that } reflects the impact of the removal of node m from the graph at a different eigenfrequencyof that graph matrix .
Edit intent: coherence
Original:  }\det\left( A_{\backslash\left\{ j\right\} }-\lambda _{k}I\right) \] }\left( \right) \left( \right) ^{\prime}}\left( \right) \left( \lambda\right) }{d\lambda} and A_{\backslash\left\{ j\right\} } is obtained from A by removal of row j and column j, suggests us to consider the square eigenvector component as a graph centrality metric for node j that } reflects the impact of the removal of node m from the graph at a different eigenfrequencyof that graph matrix .
Revised:  }\det\left( A_{\backslash\left\{ j\right\} }-\lambda _{k}I\right) \] }\left( \right) \left( A-\lambda I\right) ^{\prime}}\left( \right) \left( \lambda\right) }{d\lambda} and A_{\backslash\left\{ j\right\} } is obtained from A by removal of row j and column j, suggests us to consider the square eigenvector component as a graph centrality metric for node j that } reflects the impact of the removal of node m from the graph at a different eigenfrequencyof that graph matrix .
Edit intent: fluency