wanyu/IteraTeR_v2 huggingface dataset:
Original: In a typical regular expression crossword puzzle (or regexcrossword, for short) , you are given two nonempty lists R_1,\ldots,R_m and C_1,\ldots,C_n of regular expressions over some alphabet, and your goal is to fill in an m\times n grid with letters from that alphabet so that the string formed by the ith row is in L(R_i), and the string formed by the jth column is in L(C_j), for all 1\le i\le m and 1\le j\le n. Such a grid is a solution to the puzzle.
Revised: In a typical regular expression (regex) crossword puzzle , you are given two nonempty lists R_1,\ldots,R_m and C_1,\ldots,C_n of regular expressions over some alphabet, and your goal is to fill in an m\times n grid with letters from that alphabet so that the string formed by the ith row is in L(R_i), and the string formed by the jth column is in L(C_j), for all 1\le i\le m and 1\le j\le n. Such a grid is a solution to the puzzle.
Edit intent: clarity
Original: We generalize \beta-skeletons , well-known neighborhood graphs for point sets, to sets of line segments  and present algorithms for computing such skeletons , both circle and lens-based, for the entire range of \beta values.
Revised:  \beta-skeletons , well-known neighborhood graphs for point sets, to sets of line segments and present algorithms for computing such skeletons , both circle and lens-based, for the entire range of \beta values.
Edit intent: coherence
Original: We generalize \beta-skeletons , well-known neighborhood graphs for point sets, to sets of line segments  and present algorithms for computing such skeletons , both circle and lens-based, for the entire range of \beta values.
Revised: We generalize \beta-skeletons are well-known neighborhood graphs for point sets, to sets of line segments and present algorithms for computing such skeletons , both circle and lens-based, for the entire range of \beta values.
Edit intent: fluency
Original: We generalize \beta-skeletons , well-known neighborhood graphs for point sets, to sets of line segments  and present algorithms for computing such skeletons , both circle and lens-based, for the entire range of \beta values.
Revised: We generalize \beta-skeletons , well-known neighborhood graphs for a set of points. We extend this notion to sets of line segments and present algorithms for computing such skeletons , both circle and lens-based, for the entire range of \beta values.
Edit intent: clarity
Original: We generalize \beta-skeletons , well-known neighborhood graphs for point sets, to sets of line segments  and present algorithms for computing such skeletons , both circle and lens-based, for the entire range of \beta values.
Revised: We generalize \beta-skeletons , well-known neighborhood graphs for point sets, to sets of line segments and present algorithms computing such skeletons , both circle and lens-based, for the entire range of \beta values.
Edit intent: fluency
Original: We generalize \beta-skeletons , well-known neighborhood graphs for point sets, to sets of line segments  and present algorithms for computing such skeletons , both circle and lens-based, for the entire range of \beta values.
Revised: We generalize \beta-skeletons , well-known neighborhood graphs for point sets, to sets of line segments and present algorithms for computing such skeletons for the entire range of \beta values.
Edit intent: coherence
Original:  In particular, for \beta%DIFDELCMD < \ge %%%  1  the \beta-skeleton for a set S on n segments in the Euclidean plane can be constructed in O(n^2 \alpha (n) \log n) time and the construction relies on the Delaunay triangulation for S  . When 0 %DIFDELCMD < \le %%% \beta%DIFDELCMD < \le %%%  1, the \beta-skeleton can be constructed in a O(n ^4) time.
Revised:  %DIFDELCMD < \ge %%% 1 the \beta-skeleton for a set S on n segments in the Euclidean plane can be constructed in O(n^2 \alpha (n) \log n) time and the construction relies on the Delaunay triangulation for S . When 0 %DIFDELCMD < \le %%% \beta%DIFDELCMD < \le %%% 1, the \beta-skeleton can be constructed in a O(n ^4) time.
Edit intent: coherence
Original:  In particular, for \beta%DIFDELCMD < \ge %%%  1  the \beta-skeleton for a set S on n segments in the Euclidean plane can be constructed in O(n^2 \alpha (n) \log n) time and the construction relies on the Delaunay triangulation for S  . When 0 %DIFDELCMD < \le %%% \beta%DIFDELCMD < \le %%%  1, the \beta-skeleton can be constructed in a O(n ^4) time.
Revised:  In particular, for \beta%DIFDELCMD < \ge %%% 1 computes the \beta-skeleton for a set S on n segments in the Euclidean plane can be constructed in O(n^2 \alpha (n) \log n) time and the construction relies on the Delaunay triangulation for S . When 0 %DIFDELCMD < \le %%% \beta%DIFDELCMD < \le %%% 1, the \beta-skeleton can be constructed in a O(n ^4) time.
Edit intent: clarity
Original:  In particular, for \beta%DIFDELCMD < \ge %%%  1  the \beta-skeleton for a set S on n segments in the Euclidean plane can be constructed in O(n^2 \alpha (n) \log n) time and the construction relies on the Delaunay triangulation for S  . When 0 %DIFDELCMD < \le %%% \beta%DIFDELCMD < \le %%%  1, the \beta-skeleton can be constructed in a O(n ^4) time.
Revised:  In particular, for \beta%DIFDELCMD < \ge %%% 1 the \beta-skeleton for a set S of n segments in the Euclidean plane can be constructed in O(n^2 \alpha (n) \log n) time and the construction relies on the Delaunay triangulation for S . When 0 %DIFDELCMD < \le %%% \beta%DIFDELCMD < \le %%% 1, the \beta-skeleton can be constructed in a O(n ^4) time.
Edit intent: clarity
Original:  In particular, for \beta%DIFDELCMD < \ge %%%  1  the \beta-skeleton for a set S on n segments in the Euclidean plane can be constructed in O(n^2 \alpha (n) \log n) time and the construction relies on the Delaunay triangulation for S  . When 0 %DIFDELCMD < \le %%% \beta%DIFDELCMD < \le %%%  1, the \beta-skeleton can be constructed in a O(n ^4) time.
Revised:  In particular, for \beta%DIFDELCMD < \ge %%% 1 the \beta-skeleton for a set S on n segments in the Euclidean plane in O(n^2 \alpha (n) \log n) time and the construction relies on the Delaunay triangulation for S . When 0 %DIFDELCMD < \le %%% \beta%DIFDELCMD < \le %%% 1, the \beta-skeleton can be constructed in a O(n ^4) time.
Edit intent: fluency
Original:  In particular, for \beta%DIFDELCMD < \ge %%%  1  the \beta-skeleton for a set S on n segments in the Euclidean plane can be constructed in O(n^2 \alpha (n) \log n) time and the construction relies on the Delaunay triangulation for S  . When 0 %DIFDELCMD < \le %%% \beta%DIFDELCMD < \le %%%  1, the \beta-skeleton can be constructed in a O(n ^4) time.
Revised:  In particular, for \beta%DIFDELCMD < \ge %%% 1 the \beta-skeleton for a set S on n segments in the Euclidean plane can be constructed in O(n^2 \alpha (n) \log n) time and the construction relies on the Delaunay triangulation for S . When 0 %DIFDELCMD < \le %%% %DIFDELCMD < \le %%% 1, the \beta-skeleton can be constructed in a O(n ^4) time.
Edit intent: fluency
Original:  In a special case of \beta = 1, which is a generalization of Gabriel Graph, the construction can be carried out in a O(n \log n) time .
Revised:  In the special case of \beta = 1, which is a generalization of Gabriel Graph, the construction can be carried out in a O(n \log n) time .
Edit intent: fluency
Original: We present a new, dynamical way to study powers ( or repetitions) in Sturmian words based on results from Diophantine approximation theory.
Revised: We present a new, dynamical way to study powers ( that is, repetitions) in Sturmian words based on results from Diophantine approximation theory.
Edit intent: clarity
Original:  As a result, we provide an alternative and shorter proof of a result by Damanik and Lenz characterizing powers in Sturmian words [Powers in Sturmian Sequences, European J. Combin.
Revised:  As a result, we provide an alternative and shorter proof of a result by Damanik and Lenz characterizing powers in Sturmian words [Powers in Sturmian sequences, Eur. J. Combin.
Edit intent: fluency
Original:  It is mathematically formulated on the basis of communicability distance functions that account for the "goodness" of communication between nodes in the network.
Revised:  It is mathematically formulated on the basis of communicability distance functions that account for the quality of communication between nodes in the network.
Edit intent: clarity
Original:  We also show that the communication-driven method outperforms the random mechanism in explaining the clustering coefficient, average path length, average communicability, and degree heterogeneity of a network .
Revised:  We also show that the communication-driven method outperforms the random mechanism in explaining the clustering coefficient, average path length, and average communicability .
Edit intent: clarity
Original:  The new method also displays some interesting features toward its use for optimizing communication and degree heterogeneity in networks.
Revised:  The new method also displays some interesting features with regards to optimizing communication in networks.
Edit intent: clarity
Original: Self-alignment concentrations, c(x), as functions of the length x of the identically matching maximal segments in the genomes of a variety of species, present typically power-law tails extending to the largest scales,  c(x) \propto x^{\alpha}, with similar or  different negative \alphas (<-2 as observed so far ).
Revised: The self-alignment concentrations, c(x), as functions of the length x of the identically matching maximal segments in the genomes of a variety of species, present typically power-law tails extending to the largest scales, c(x) \propto x^{\alpha}, with similar or different negative \alphas (<-2 as observed so far ).
Edit intent: fluency
Original: Self-alignment concentrations, c(x), as functions of the length x of the identically matching maximal segments in the genomes of a variety of species, present typically power-law tails extending to the largest scales,  c(x) \propto x^{\alpha}, with similar or  different negative \alphas (<-2 as observed so far ).
Revised: Self-alignment concentrations, c(x), as functions of the length , x, of the identically matching maximal segments in the genomes of a variety of species, present typically power-law tails extending to the largest scales, c(x) \propto x^{\alpha}, with similar or different negative \alphas (<-2 as observed so far ).
Edit intent: fluency
Original: Self-alignment concentrations, c(x), as functions of the length x of the identically matching maximal segments in the genomes of a variety of species, present typically power-law tails extending to the largest scales,  c(x) \propto x^{\alpha}, with similar or  different negative \alphas (<-2 as observed so far ).
Revised: Self-alignment concentrations, c(x), as functions of the length x of the identically matching maximal segments in the genomes of a variety of species, typically present power-law tails extending to the largest scales, c(x) \propto x^{\alpha}, with similar or different negative \alphas (<-2 as observed so far ).
Edit intent: fluency
Original: Self-alignment concentrations, c(x), as functions of the length x of the identically matching maximal segments in the genomes of a variety of species, present typically power-law tails extending to the largest scales,  c(x) \propto x^{\alpha}, with similar or  different negative \alphas (<-2 as observed so far ).
Revised: Self-alignment concentrations, c(x), as functions of the length x of the identically matching maximal segments in the genomes of a variety of species, present typically power-law tails extending to the largest scales, i.e., c(x) \propto x^{\alpha}, with similar or different negative \alphas (<-2 as observed so far ).
Edit intent: coherence
Original: Self-alignment concentrations, c(x), as functions of the length x of the identically matching maximal segments in the genomes of a variety of species, present typically power-law tails extending to the largest scales,  c(x) \propto x^{\alpha}, with similar or  different negative \alphas (<-2 as observed so far ).
Revised: Self-alignment concentrations, c(x), as functions of the length x of the identically matching maximal segments in the genomes of a variety of species, present typically power-law tails extending to the largest scales, c(x) \propto x^{\alpha}, with similar or apparently different negative \alphas (<-2 as observed so far ).
Edit intent: clarity
Original:  Rev. Lett.,%DIFDELCMD < {\bf %%% 110 148101 (2012)%DIFDELCMD < ]%%% . Disorder is intrinsic and, applying quenching, we systematically show with decaying, steady-state and the general full time-dependent solutions that disorders in the configurations of a simple fragmentation system facilitate the various for x}\to  power-law tails.
Revised:  Rev. Lett.,%DIFDELCMD < {\bf %%% 148101 (2012)%DIFDELCMD < ]%%% . Disorder is intrinsic and, applying quenching, we systematically show with decaying, steady-state and the general full time-dependent solutions that disorders in the configurations of a simple fragmentation system facilitate the various for x}\to power-law tails.
Edit intent: clarity
Original:  Rev. Lett.,%DIFDELCMD < {\bf %%% 110 148101 (2012)%DIFDELCMD < ]%%% . Disorder is intrinsic and, applying quenching, we systematically show with decaying, steady-state and the general full time-dependent solutions that disorders in the configurations of a simple fragmentation system facilitate the various for x}\to  power-law tails.
Revised:  Rev. Lett.,%DIFDELCMD < {\bf %%% 110 %DIFDELCMD < ]%%% . Disorder is intrinsic and, applying quenching, we systematically show with decaying, steady-state and the general full time-dependent solutions that disorders in the configurations of a simple fragmentation system facilitate the various for x}\to power-law tails.
Edit intent: coherence
Original:  Rev. Lett.,%DIFDELCMD < {\bf %%% 110 148101 (2012)%DIFDELCMD < ]%%% . Disorder is intrinsic and, applying quenching, we systematically show with decaying, steady-state and the general full time-dependent solutions that disorders in the configurations of a simple fragmentation system facilitate the various for x}\to  power-law tails.
Revised:  Rev. Lett.,%DIFDELCMD < {\bf %%% 110 148101 (2012)%DIFDELCMD < ]%%% . However, disorder is intrinsic to the evolution system and, by freezing it in time (quenching) for the setup of a simple fragmentation model, we obtain decaying, steady-state and the general full time-dependent solutions that disorders in the configurations of a simple fragmentation system facilitate the various for x}\to power-law tails.
Edit intent: coherence
Original:  Suppose one is given a finite set E \subset R^d, a function f: E \rightarrow R, and possibly the gradients of f at the values of E as well.
Revised:  Suppose one is given a finite set E \subset R^d, a function f: E \rightarrow R, and possibly the gradients of f at the points of E as well.
Edit intent: clarity
Original:  Suppose one is given a finite set E \subset R^d, a function f: E \rightarrow R, and possibly the gradients of f at the points of E as well .
Revised:  Suppose one is given a finite set E \subset R^d, a function f: E \rightarrow R, and possibly the gradients of f at the points of E .
Edit intent: clarity
Original:  Under the null hypothesis, the observed graph is a realization of an Erd \H{o s-R%DIFDELCMD < {%%% \'e nyis-R\'enyi } random graph G(n,p).
Revised:  Under the null hypothesis, the observed graph is a realization of an Erd s-R%DIFDELCMD < {%%% \'e nyis-R\'enyi } random graph G(n,p).
Edit intent: fluency
Original:  Under the null hypothesis, the observed graph is a realization of an Erd \H{o s-R%DIFDELCMD < {%%% \'e nyis-R\'enyi } random graph G(n,p).
Revised:  Under the null hypothesis, the observed graph is a realization of an Erd \H{o s-R%DIFDELCMD < {%%% nyis-R\'enyi } random graph G(n,p).
Edit intent: fluency
Original:  Under the null hypothesis, the observed graph is a realization of an Erd \H{o s-R%DIFDELCMD < {%%% \'e nyis-R\'enyi } random graph G(n,p).
Revised:  Under the null hypothesis, the observed graph is a realization of an Erd \H{o s-R%DIFDELCMD < {%%% \'e \H{os-R\'enyi } random graph G(n,p).
Edit intent: fluency
Original:  We use basic tools from quantum computation to give a simple (and unconditional) proof that the  Witten-Reshetikhin-Turaev  invariant of 3-manifolds is \# P-hard to calculate.
Revised:  We use basic tools from quantum computation to give a proof that the Witten-Reshetikhin-Turaev invariant of 3-manifolds is \# P-hard to calculate.
Edit intent: coherence
Original:  We use basic tools from quantum computation to give a simple (and unconditional) proof that the  Witten-Reshetikhin-Turaev  invariant of 3-manifolds is \# P-hard to calculate.
Revised:  We use basic tools from quantum computation to give a simple (and unconditional) proof that the Witten-Reshetikhin-Turaev invariant of 3-manifolds is P-hard to calculate.
Edit intent: fluency
Original:  We then use this fact to show that, if NP \neq \# P\neqFP^NP} , then there exist infinitely many 3-manifold diagrams which cannot be made logarithmically "thin " (relative to their overall size) except perhaps by an exponentially large number of local moves.
Revised:  We then use this fact to show that, if \# P\neqFP^NP} , then there exist infinitely many 3-manifold diagrams which cannot be made logarithmically "thin " (relative to their overall size) except perhaps by an exponentially large number of local moves.
Edit intent: clarity
Original:  In other words, negative ties do not have a major role in community formation of signed networks.
Revised:  In other words, negative ties do not have a major role in detecting communities of studied signed networks.
Edit intent: clarity
Original: The very original concept of cognitive radio (CR) raised by Mitola targets at all the environment parameters, including those in physical layer, MAC layer, application layer as well as those obtained from reasoning, which is also referred to as "full cognitive radio".
Revised: The very original concept of cognitive radio (CR) raised by Mitola targets at all the environment parameters, including those in physical layer, MAC layer, application layer as well as the information extracted from reasoning. Hence the first CR is also referred to as "full cognitive radio".
Edit intent: clarity
Original:  Due to its difficult implementation, FCC and Simon Haykin separately proposed a much more simplified definition, in which CR mainly detects the spectrum holes via various sensing technologies, also called as "spectrum sensing cognitive radio".
Revised:  However, due to its difficult implementation, FCC and Simon Haykin separately proposed a much more simplified definition, in which CR mainly detects the spectrum holes via various sensing technologies, also called as "spectrum sensing cognitive radio".
Edit intent: coherence
Original:  With the rapid development of wireless communication, say the approaching of 5G cellular networks, the infrastructure of a wireless system becomes much more complicated , and hence, the functionality at every node is desired to be as intelligent as possible  .
Revised:  With the rapid development of wireless communication, the infrastructure of a wireless system becomes much more complicated , and hence, the functionality at every node is desired to be as intelligent as possible .
Edit intent: clarity
Original:  With the rapid development of wireless communication, say the approaching of 5G cellular networks, the infrastructure of a wireless system becomes much more complicated , and hence, the functionality at every node is desired to be as intelligent as possible  .
Revised:  With the rapid development of wireless communication, say the approaching of 5G cellular networks, the infrastructure of a wireless system becomes much more complicated while the functionality at every node is desired to be as intelligent as possible .
Edit intent: coherence
Original:  It is then essential to look back into Mitola's CR definition and ask whether one could, besides obtaining the "on/off" status of the licensed user only, achieve more parameters in an cognitive way.
Revised:  It is then interesting to re-look into Mitola's CR definition and ask whether one could, besides obtaining the "on/off" status of the licensed user only, achieve more parameters in an cognitive way.
Edit intent: clarity
Original:  It is then essential to look back into Mitola's CR definition and ask whether one could, besides obtaining the "on/off" status of the licensed user only, achieve more parameters in an cognitive way.
Revised:  It is then essential to look back into Mitola's definition and think whether one could, besides obtaining the "on/off" status of the licensed user only, achieve more parameters in an cognitive way.
Edit intent: clarity
Original:  It is then essential to look back into Mitola's CR definition and ask whether one could, besides obtaining the "on/off" status of the licensed user only, achieve more parameters in an cognitive way.
Revised:  It is then essential to look back into Mitola's CR definition and ask whether one could, besides obtaining the "on/off" status of the licensed user only, achieve more parameters in a cognitive way.
Edit intent: fluency
Original:  Three representative examples are provided on the basis of recent research progress to illustrate the feasibility as well as the validity of the proposed architecture .
Revised:  Three representative examples are provided based on the recent research progress to illustrate the feasibility as well as the validity of the proposed architecture .
Edit intent: fluency
Original: Resource Description Framework (RDF) has become a prevalent standard model for data integration and interchange on the Web and SPARQL has been the predominant query language over RDF graphs .
Revised: Although SPARQL has been the predominant query language over RDF graphs .
Edit intent: coherence
Original:  However , some query meanings may not be well captured by only using SPARQL  . On the other hand, the keyword search enjoys widespread usage because of its intuitive way of specifying information needs but suffers from the problem of low precision.
Revised:  However , some query intentions cannot be well captured by only using SPARQL . On the other hand, the keyword search enjoys widespread usage because of its intuitive way of specifying information needs but suffers from the problem of low precision.
Edit intent: clarity
Original:  We evaluate our method in three large real RDF graphs and experiments demonstrate both effectiveness and efficiency of our method .
Revised:  We evaluate our method in large real RDF graphs and experiments demonstrate both effectiveness and efficiency of our method .
Edit intent: clarity
Original: Suppose q is a prime power and f  is a univariate polynomial (over the finite field F_q) with exactly t monomial terms and degree <q-1.
Revised: Suppose q is a prime power and f is a univariate polynomial with exactly t monomial terms and degree <q-1.
Edit intent: clarity
Original:  To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas  recently proved an upper bound of 2(q-1)^{(t-2)/(t-1)} on the number of cosets of F^*_q needed to cover the roots of f in F^*_q. Here, we exhibit explicit f with root structure approaching this bound: we present f vanishing on q^{(t-2)/t}+...
Revised:  To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas recently proved an upper bound of 2(q-1)^{(t-2)/(t-1)} on the number of cosets of F^*_q needed to cover the roots of f in F^*_q. Here, we give explicit f with root structure approaching this bound: we present f vanishing on q^{(t-2)/t}+...
Edit intent: clarity
Original: +q^{1/t}+1 distinct cosets of F^*_q. For p prime , we give some computational evidence that the maximal number of rootsin F^*_p for a trinomial a+bx^{e_2+x^{e_3}, with \gcd(e_2,e_3)=1, may grow more slowly than p^{1/3}.
Revised: +q^{1/t}+1 distinct cosets of F^*_q. +x^{e_3}, with \gcd(e_2,e_3)=1, may grow more slowly than p^{1/3}.
Edit intent: clarity