wanyu/IteraTeR_v2 huggingface dataset:
Original:  The smallness of the parameters in (H5)-(H6) enable an a priori and a posteriori error control along with a best approximation result for semilinear elliptic problems with trilinear nonlinearity.
Revised:  The smallness of the parameters in (H5)-(H6) enable an a priori and a posteriori error control of conforming and nonconforming discretisations of a class of semilinear elliptic problems with trilinear nonlinearity.
Edit intent: clarity
Original: The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of 2D Navier-Stokes problem and in the von Karmanrm\'{a}n } equations.
Revised: The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of 2D Navier-Stokes problem and in the von K\'{arm\'{a}n } equations.
Edit intent: fluency
Original:  This paper establishes a best-approximation a  priori error analysis and an a  posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity.
Revised:  This paper establishes a best-approximation a ~ priori error analysis and an a posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity.
Edit intent: fluency
Original:  This paper establishes a best-approximation a  priori error analysis and an a  posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity.
Revised:  This paper establishes a best-approximation a priori error analysis and an a ~ posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity.
Edit intent: fluency
Original:  The paper identifies six parameters and sufficient conditions for the local a  priori and a  posteriori error control of conforming and nonconforming discretisations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications.
Revised:  The paper identifies six parameters and sufficient conditions for the local a ~ priori and a posteriori error control of conforming and nonconforming discretisations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications.
Edit intent: fluency
Original:  The paper identifies six parameters and sufficient conditions for the local a  priori and a  posteriori error control of conforming and nonconforming discretisations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications.
Revised:  The paper identifies six parameters and sufficient conditions for the local a priori and a ~ posteriori error control of conforming and nonconforming discretisations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications.
Edit intent: fluency
Original:  This leads to new best-approximation error estimates and to a  posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM.
Revised:  This leads to new best-approximation error estimates and to a ~ posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM.
Edit intent: fluency
Original:  This definition finds motivation in edge casing, as a k-gap-planar graph can be drawn crossing-free after introducing at most k local gaps per edge.
Revised:  This definition is motivated by applications in edge casing, as a k-gap-planar graph can be drawn crossing-free after introducing at most k local gaps per edge.
Edit intent: clarity
Original:  We obtain results on the maximum density , drawability of complete graphs, complexity of the recognition problem, and relationships with other families of beyond-planar graphs.
Revised:  We present results on the maximum density , drawability of complete graphs, complexity of the recognition problem, and relationships with other families of beyond-planar graphs.
Edit intent: clarity
Original:  We obtain results on the maximum density , drawability of complete graphs, complexity of the recognition problem, and relationships with other families of beyond-planar graphs.
Revised:  We obtain results on the maximum density of k-gap-planar graphs, their relationship to other classes of beyond-planar graphs, characterization of k-gap-planar complete graphs, and the computational complexity of recognizing k-gap-planar graphs.
Edit intent: clarity
Original:  We propose an analytic method to solve a translation model based on the inhomogeneous totally asymmetric simple exclusion process, which allow us to unveil simple design principles of nucleotide sequences determining protein production rates.
Revised:  We propose an analytic method to solve a translation model based on the inhomogeneous totally asymmetric simple exclusion process, which allows us to unveil simple design principles of nucleotide sequences determining protein production rates.
Edit intent: fluency
Original:  Tremendous effort has been invested to understand their structure and functions, among which centrality is  one of the most effective approaches .
Revised:  Considerable effort has been invested to understand their structure and functions, among which centrality is one of the most effective approaches .
Edit intent: clarity
Original:  Tremendous effort has been invested to understand their structure and functions, among which centrality is  one of the most effective approaches .
Revised:  Tremendous effort has been invested to understand their structure and functions, among which centrality is identified as one of the most effective approaches .
Edit intent: clarity
Original:  Tremendous effort has been invested to understand their structure and functions, among which centrality is  one of the most effective approaches .
Revised:  Tremendous effort has been invested to understand their structure and functions, among which centrality is one of the most critical metrics .
Edit intent: clarity
Original:  While various metrics of centrality have been proposed for single-layer networks, a general framework of studying centrality in multiplayer networks is lacking.
Revised:  While various forms of centrality have been proposed for single-layer networks, a general framework of studying centrality in multiplayer networks is lacking.
Edit intent: clarity
Original:  While various metrics of centrality have been proposed for single-layer networks, a general framework of studying centrality in multiplayer networks is lacking.
Revised:  While various metrics of centrality have been proposed for single-layer networks, a general framework for studying centrality in multiplayer networks is lacking.
Edit intent: fluency
Original:  While various metrics of centrality have been proposed for single-layer networks, a general framework of studying centrality in multiplayer networks is lacking.
Revised:  While various metrics of centrality have been proposed for single-layer networks, a general framework of studying centrality in multilayer networks is lacking.
Edit intent: clarity
Original:  Here we introduce a mathematical framework to study eigenvector multicentrality, which enables us to quantify the relationship between interlayer influences and eigenvector multicentrality, providing an analytical tool to describe how eigenvector multicentralities of nodes propagate among different layers.
Revised:  Here we introduce a mathematical framework to study eigenvector multicentrality, which enables us to quantify the relationship between interlayer influence and eigenvector multicentrality, providing an analytical tool to describe how eigenvector multicentralities of nodes propagate among different layers.
Edit intent: fluency
Original:  We show how to model the multilayer influences by choosing appropriate influence weight functions and design algorithms to calculate eigenvector multicentrality in various typical scenarios.
Revised:  We show how to model the multilayer influence by choosing appropriate influence weight functions and design algorithms to calculate eigenvector multicentrality in various typical scenarios.
Edit intent: fluency
Original:  We apply this framework to analyze several empirical multilayer networks, finding that it can quantify the influences among layers and describe the structure-function relationship of multilayer networks very well.
Revised:  We apply this framework to analyze several empirical multilayer networks, finding that it can quantify the influence among layers and describe the structure-function relationship of multilayer networks very well.
Edit intent: fluency
Original:  While various forms of centrality have been proposed for single-layer networks, a general framework for studying centrality in multilayer networks is lacking.
Revised:  While various centrality measures have been proposed for single-layer networks, a general framework for studying centrality in multilayer networks is lacking.
Edit intent: clarity
Original:  Here we introduce a mathematical framework to study eigenvector multicentrality, which enables us to quantify the relationship between interlayer influence and eigenvector multicentrality, providing an analytical tool to describe how eigenvector multicentralities of nodes propagate among different layers.
Revised:  Here we introduce a mathematical framework to study eigenvector multicentrality, which enables the quantification of the impact of interlayer influence on multicentrality, providing an analytical tool to describe how eigenvector multicentralities of nodes propagate among different layers.
Edit intent: clarity
Original:  Here we introduce a mathematical framework to study eigenvector multicentrality, which enables us to quantify the relationship between interlayer influence and eigenvector multicentrality, providing an analytical tool to describe how eigenvector multicentralities of nodes propagate among different layers.
Revised:  Here we introduce a mathematical framework to study eigenvector multicentrality, which enables us to quantify the relationship between interlayer influence and eigenvector multicentrality, providing a systematic way to describe how eigenvector multicentralities of nodes propagate among different layers.
Edit intent: clarity
Original:  Here we introduce a mathematical framework to study eigenvector multicentrality, which enables us to quantify the relationship between interlayer influence and eigenvector multicentrality, providing an analytical tool to describe how eigenvector multicentralities of nodes propagate among different layers.
Revised:  Here we introduce a mathematical framework to study eigenvector multicentrality, which enables us to quantify the relationship between interlayer influence and eigenvector multicentrality, providing an analytical tool to describe how multicentrality propagates across different layers.
Edit intent: clarity
Original:  Further, the framework is flexible for integrating prior knowledge of the interplay among layers so as to attain a tailored eigenvector multicentrality for varying scenarios.
Revised:  This framework can leverage prior knowledge about the interplay among layers so as to attain a tailored eigenvector multicentrality for varying scenarios.
Edit intent: clarity
Original:  Further, the framework is flexible for integrating prior knowledge of the interplay among layers so as to attain a tailored eigenvector multicentrality for varying scenarios.
Revised:  Further, the framework is flexible for integrating prior knowledge of the interplay among layers to better characterize multicentrality for varying scenarios.
Edit intent: clarity
Original:  We show how to model the multilayer influence by choosing appropriate influence weight functions and design algorithms to calculate eigenvector multicentrality in various typical scenarios.
Revised:  We show how to model multilayer influence by choosing appropriate influence weight functions and design algorithms to calculate eigenvector multicentrality in various typical scenarios.
Edit intent: fluency
Original:  We show how to model the multilayer influence by choosing appropriate influence weight functions and design algorithms to calculate eigenvector multicentrality in various typical scenarios.
Revised:  We show how to model the multilayer influence by choosing appropriate functions of interlayer influence and design algorithms to calculate eigenvector multicentrality in various typical scenarios.
Edit intent: clarity
Original:  We apply this framework to analyze several empirical multilayer networks, finding that it can quantify the influence among layers and describe the structure-function relationship of multilayer networks very well .
Revised:  We apply this framework to analyze several empirical multilayer networks, and the results corroborate that it can quantify the influence among layers and describe the structure-function relationship of multilayer networks very well .
Edit intent: coherence
Original:  We apply this framework to analyze several empirical multilayer networks, finding that it can quantify the influence among layers and describe the structure-function relationship of multilayer networks very well .
Revised:  We apply this framework to analyze several empirical multilayer networks, finding that it can quantify the influence among layers and multicentrality of nodes effectively .
Edit intent: clarity
Original:  In this article, we first discuss the principles for supporting URLLC in the perspective to the traditional assumptions and models applied in communication/information theory.
Revised:  In this article, we first discuss the principles for supporting URLLC from the perspective of the traditional assumptions and models applied in communication/information theory.
Edit intent: fluency
Original:  We then discuss how this principles are applied in various elements of the system design, such as use of various diversity sources, design of packets and access protocols.
Revised:  We then discuss how these principles are applied in various elements of the system design, such as use of various diversity sources, design of packets and access protocols.
Edit intent: fluency
Original:  Important messages are that there is a need to optimize the transmission of signaling information, as well as  for a lean use of various sources of diversity.
Revised:  The important messages are that there is a need to optimize the transmission of signaling information, as well as for a lean use of various sources of diversity.
Edit intent: coherence
Original:  Important messages are that there is a need to optimize the transmission of signaling information, as well as  for a lean use of various sources of diversity.
Revised:  Important messages are that there is a need to optimize the transmission of signaling information, as well as a need for a lean use of various sources of diversity.
Edit intent: fluency
Original:  The properties of GRNs were varied from ordered, through critical, to chaotic by  node in-degree K.
Revised:  The properties of GRNs from ordered, through critical, to chaotic by node in-degree K.
Edit intent: clarity
Original: A simple DNA-based data storage scheme is demonstrated in which information is written using " address-encoded " oligonucleotides to encode two bits .
Revised: A simple DNA-based data storage scheme is demonstrated in which information is written using " addressing " oligonucleotides to encode two bits .
Edit intent: clarity
Original: A simple DNA-based data storage scheme is demonstrated in which information is written using " address-encoded " oligonucleotides to encode two bits .
Revised: A simple DNA-based data storage scheme is demonstrated in which information is written using " address-encoded " oligonucleotides .
Edit intent: clarity
Original:  Despite its importance, all proofs of the correctness of Strassen's famous 1969 algorithm to multiply two 2x2 matrices with only seven multiplications involve some more or less tedious calculations such as explicitly multiplying specific 2x2 matrices, expanding expressions to cancel terms with opposing signs, or expanding tensors over the standard basis.
Revised:  Despite its importance, all proofs of the correctness of Strassen's famous 1969 algorithm to multiply two 2x2 matrices with only seven multiplications involve some basis-dependent calculations such as explicitly multiplying specific 2x2 matrices, expanding expressions to cancel terms with opposing signs, or expanding tensors over the standard basis.
Edit intent: clarity
Original:  This is why the proof is nontrivial to memorize and why many presentations of the proof avoid showing all the details and leave a significant amount of verifications to the reader.
Revised:  This makes the proof nontrivial to memorize and why many presentations of the proof avoid showing all the details and leave a significant amount of verifications to the reader.
Edit intent: coherence
Original:  This is why the proof is nontrivial to memorize and why many presentations of the proof avoid showing all the details and leave a significant amount of verifications to the reader.
Revised:  This is why the proof is nontrivial to memorize and many presentations of the proof avoid showing all the details and leave a significant amount of verifications to the reader.
Edit intent: fluency
Original:  As shown by this paper, RIOT achieves comparable results to the other approaches but does so in a fraction of the time .
Revised:  As shown by this paper, RIOT achieves comparable results to other methods .
Edit intent: clarity
Original:  In fixed template CSPs , denoted CSP({\bf \Gamma}), the right side structure {\bf \Gamma} is fixed and the left side structure { %DIFDELCMD < \bf %%%  R} is unconstrained.
Revised:  In a fixed template CSP , denoted CSP({\bf \Gamma}), the right side structure {\bf \Gamma} is fixed and the left side structure { %DIFDELCMD < \bf %%% R} is unconstrained.
Edit intent: clarity
Original:  In the last 2 decades it was impressively revealed that those templates%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from%DIFDELCMD < {\bf %%% R to%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Revised:  %DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from%DIFDELCMD < {\bf %%% R to%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Edit intent: coherence
Original:  In the last 2 decades it was impressively revealed that those templates%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from%DIFDELCMD < {\bf %%% R to%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Revised:  In the last 2 decades it was impressively revealed that those templates%DIFDELCMD < {\bf %DIFDELCMD < %%% are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from%DIFDELCMD < {\bf %%% R to%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Edit intent: fluency
Original:  In the last 2 decades it was impressively revealed that those templates%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from%DIFDELCMD < {\bf %%% R to%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Revised:  In the last 2 decades it was impressively revealed that those templates%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma %DIFDELCMD < {\bf %%% R to%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Edit intent: coherence
Original:  In the last 2 decades it was impressively revealed that those templates%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from%DIFDELCMD < {\bf %%% R to%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Revised:  In the last 2 decades it was impressively revealed that those templates%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from%DIFDELCMD < {\bf %%% to%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Edit intent: clarity
Original:  In the last 2 decades it was impressively revealed that those templates%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from%DIFDELCMD < {\bf %%% R to%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Revised:  In the last 2 decades it was impressively revealed that those templates%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from%DIFDELCMD < {\bf %%% R %DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Edit intent: coherence
Original:  In the last 2 decades it was impressively revealed that those templates%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from%DIFDELCMD < {\bf %%% R to%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Revised:  In the last 2 decades it was impressively revealed that those templates%DIFDELCMD < {\bf %DIFDELCMD < %%% \Gamma are tractable that are preserved under certain polymorphisms p_1, ..., p_k. Moreover, the set of all homomorphisms from%DIFDELCMD < {\bf %%% R to%DIFDELCMD < {\bf %DIFDELCMD < %%% are preserved under the same p_1, ..., p_k. I.e. p_1, ..., p_k not only guide us in the decision problem, but also define algebraic properties of the solutions set.
Edit intent: fluency
Original:  We study this formulation and prove that the latter problem itself is an instance of a certain fixed-template CSP, over another template {\bf \Gamma}^{\mathcal B}.
Revised:  We study this problem and show that it can be reformulated as an instance of a certain fixed-template CSP, over another template {\bf \Gamma}^{\mathcal B}.
Edit intent: clarity