wanyu/IteraTeR_v2 huggingface dataset:
Original:  Our aim is to bridge the gap between the microscopic description of price formation (agent-based modeling) , and the Stochastic Differential Equations approach used classically to describe price evolution at macroscopic time scales .
Revised:  Our aim is to bridge the gap between the microscopic description of price formation (agent-based modeling) and the stochastic differential equations approach used classically to describe price evolution at macroscopic time scales .
Edit intent: fluency
Original:  We design the algorithms based on the work of Lee and Seung for the standard nonnegative matrix factorization [1], and derive the converged versions by utilizing ideas from the work of Lin [2].
Revised:  We design the algorithms based on the work of Lee and Seung [1], and derive the converged versions by utilizing ideas from the work of Lin [2].
Edit intent: coherence
Original:  For instance, we show that the case of honest n-variate (n+1)-nomials is doable in NP and, for p exceeding the constant term and the Newton polytope volume  , in constant time.
Revised:  For instance, we show that the case of honest n-variate (n+1)-nomials is doable in NP and, for p exceeding the Newton polytope volume , in constant time.
Edit intent: clarity
Original: In this work we study the following problem of scheduling with power control in wireless networks  : given a set of communication requests, one needs to assign the powers of the network nodes, and schedule the transmissions so that they can be done in a minimum time, taking into account the signal interference of parallelly transmitting nodes.
Revised: In this paper the problem of scheduling with power control in wireless networks : given a set of communication requests, one needs to assign the powers of the network nodes, and schedule the transmissions so that they can be done in a minimum time, taking into account the signal interference of parallelly transmitting nodes.
Edit intent: clarity
Original: In this work we study the following problem of scheduling with power control in wireless networks  : given a set of communication requests, one needs to assign the powers of the network nodes, and schedule the transmissions so that they can be done in a minimum time, taking into account the signal interference of parallelly transmitting nodes.
Revised: In this work we study the following problem of scheduling with power control in wireless networks is studied : given a set of communication requests, one needs to assign the powers of the network nodes, and schedule the transmissions so that they can be done in a minimum time, taking into account the signal interference of parallelly transmitting nodes.
Edit intent: clarity
Original: In this work we study the following problem of scheduling with power control in wireless networks  : given a set of communication requests, one needs to assign the powers of the network nodes, and schedule the transmissions so that they can be done in a minimum time, taking into account the signal interference of parallelly transmitting nodes.
Revised: In this work we study the following problem of scheduling with power control in wireless networks : given a set of communication requests, one needs to assign the powers of the network nodes, and schedule the transmissions so that they can be done in a minimum time, taking into account the signal interference of concurrently transmitting nodes.
Edit intent: fluency
Original:  We correct and complement one of the recent papers on this theme, by giving approximation algorithms for scheduling with power control for the case, when the nodes of the network are placed in a doubling metric space .
Revised:  Approximation algorithms are given for this problem, which use the mean power assignment. The problem of schduling with fixed mean power assignment is also considered, and approximation guarantees are proven .
Edit intent: coherence
Original:  We demonstrate that many of the length scales are minimized by using the epsilon-machine, compared to all of a process's alternative models.
Revised:  We demonstrate that many of the length scales are minimized by using the epsilon-machine, a process's alternative models.
Edit intent: clarity
Original:  We illustrate the results by presenting evidence that infinite Markov order and infinite crypticity are dominant properties in the space of finite-memory processes  .
Revised:  We illustrate the results by presenting evidence that infinite Markov and infinite cryptic orders are a dominant feature in the space of finite-memory processes .
Edit intent: clarity
Original: A vertex-coloring  of a graph G is said to be locally identifying  if for any pair (u,v) of adjacent vertices of G, with distinct closed neighborhood, the set of colors that appears in the closed neighborhoods of u and v are distinct.
Revised: A vertex-coloring of a graph G is said to be locally identifying , if for any pair (u,v) of adjacent vertices of G, with distinct closed neighborhood, the set of colors that appears in the closed neighborhoods of u and v are distinct.
Edit intent: fluency
Original: A vertex-coloring  of a graph G is said to be locally identifying  if for any pair (u,v) of adjacent vertices of G, with distinct closed neighborhood, the set of colors that appears in the closed neighborhoods of u and v are distinct.
Revised: A vertex-coloring of a graph G is said to be locally identifying if for any adjacent vertices u and v with distinct closed neighborhood, the set of colors that appears in the closed neighborhoods of u and v are distinct.
Edit intent: clarity
Original: A vertex-coloring  of a graph G is said to be locally identifying  if for any pair (u,v) of adjacent vertices of G, with distinct closed neighborhood, the set of colors that appears in the closed neighborhoods of u and v are distinct.
Revised: A vertex-coloring of a graph G is said to be locally identifying if for any pair (u,v) of adjacent vertices of G, with distinct closed neighborhood, the sets of colors that appears in the closed neighborhoods of u and v are distinct.
Edit intent: fluency
Original: A vertex-coloring  of a graph G is said to be locally identifying  if for any pair (u,v) of adjacent vertices of G, with distinct closed neighborhood, the set of colors that appears in the closed neighborhoods of u and v are distinct.
Revised: A vertex-coloring of a graph G is said to be locally identifying if for any pair (u,v) of adjacent vertices of G, with distinct closed neighborhood, the set of colors that appear in the closed neighborhoods of u and v are distinct.
Edit intent: fluency
Original: A vertex-coloring  of a graph G is said to be locally identifying  if for any pair (u,v) of adjacent vertices of G, with distinct closed neighborhood, the set of colors that appears in the closed neighborhoods of u and v are distinct.
Revised: A vertex-coloring of a graph G is said to be locally identifying if for any pair (u,v) of adjacent vertices of G, with distinct closed neighborhood, the set of colors that appears in the closed neighborhood of u and v are distinct.
Edit intent: fluency
Original:   In this paper, we give several bounds on the minimum number of colors needed in such a coloring for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether  a subcubic bipartite graph  with large girth has a locally identifying coloring with 3 colors is an NP-complete problem.
Revised:  In this paper, we give several bounds on \chi_{lid for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether a subcubic bipartite graph with large girth has a locally identifying coloring with 3 colors is an NP-complete problem.
Edit intent: clarity
Original:   Lean clause-sets, defined as having no non-trivial autarkies, generalise minimally unsatisfiable clause-sets.
Revised:  - Lean clause-sets, defined as having no non-trivial autarkies, generalise minimally unsatisfiable clause-sets.
Edit intent: fluency
Original:   For the surplus we have surp(F) <= delta(F) = c(F) -
Revised:  - For the surplus we have surp(F) <= delta(F) = c(F) -
Edit intent: fluency
Original:  n(F), using the deficiency delta(F) of clause-sets, the difference between the number of clauses and the number of variables.  nM(k) is the k-th "non-Mersenne" number, skipping in the sequence of natural numbers all numbers of the form 2^n -
Revised:  n(F), using the deficiency delta(F) of clause-sets, the difference between the number of clauses and the number of variables. - nM(k) is the k-th "non-Mersenne" number, skipping in the sequence of natural numbers all numbers of the form 2^n -
Edit intent: fluency
Original:  The second parameterization is W[1]-complete, but the introduction of an additional parameter, the degeneracy of the hypergraph H=(V,{\cal F}), makes the problem not only fixed-parameter tractable, but also one with a polynomial-size kernel.
Revised:  The second parameterization is W[1]-complete, but the introduction of an additional parameter, the degeneracy of the hypergraph H=(V,{\cal F}), makes the problem not only fixed-parameter tractable, but also one with a linear kernel.
Edit intent: clarity
Original: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng , 2006 \mbox{ CD06 , Chen, Deng and Teng  2009 \mbox{ CDT09 . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant , 2005 \mbox{ AKV05 . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Revised: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng ( 2006 \mbox{ CD06 , Chen, Deng and Teng 2009 \mbox{ CDT09 . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant , 2005 \mbox{ AKV05 . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Edit intent: fluency
Original: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng , 2006 \mbox{ CD06 , Chen, Deng and Teng  2009 \mbox{ CDT09 . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant , 2005 \mbox{ AKV05 . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Revised: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng , 2006 ) , Chen, Deng and Teng 2009 \mbox{ CDT09 . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant , 2005 \mbox{ AKV05 . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Edit intent: coherence
Original: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng , 2006 \mbox{ CD06 , Chen, Deng and Teng  2009 \mbox{ CDT09 . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant , 2005 \mbox{ AKV05 . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Revised: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng , 2006 \mbox{ CD06 , Chen, Deng and Teng ( 2009 \mbox{ CDT09 . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant , 2005 \mbox{ AKV05 . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Edit intent: fluency
Original: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng , 2006 \mbox{ CD06 , Chen, Deng and Teng  2009 \mbox{ CDT09 . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant , 2005 \mbox{ AKV05 . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Revised: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng , 2006 \mbox{ CD06 , Chen, Deng and Teng 2009 )) . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant , 2005 \mbox{ AKV05 . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Edit intent: coherence
Original: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng , 2006 \mbox{ CD06 , Chen, Deng and Teng  2009 \mbox{ CDT09 . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant , 2005 \mbox{ AKV05 . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Revised: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng , 2006 \mbox{ CD06 , Chen, Deng and Teng 2009 \mbox{ CDT09 . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant ( 2005 \mbox{ AKV05 . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Edit intent: fluency
Original: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng , 2006 \mbox{ CD06 , Chen, Deng and Teng  2009 \mbox{ CDT09 . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant , 2005 \mbox{ AKV05 . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Revised: Determining a Nash equilibrium in a 2-player non-zero sum game is known to be PPAD-hard (Chen and Deng , 2006 \mbox{ CD06 , Chen, Deng and Teng 2009 \mbox{ CDT09 . The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant , 2005 )) . However, there do exist polynomial time tractable classes of win-lose bimatrix games -
Edit intent: clarity
Original:  such as, very sparse games (Codenotti, Leoncini and Resta , 2006 \mbox{ \cite{CLR06}\hspace{0pt} )}  and planar games (Addario-Berry, Olver and Vetta , 2007 \mbox{ \cite{AOV07}\hspace{0pt} }  ).
Revised:  such as, very sparse games (Codenotti, Leoncini and Resta ( 2006 \mbox{ \cite{CLR06}\hspace{0pt} )} and planar games (Addario-Berry, Olver and Vetta , 2007 \mbox{ \cite{AOV07}\hspace{0pt} } ).
Edit intent: fluency
Original:  such as, very sparse games (Codenotti, Leoncini and Resta , 2006 \mbox{ \cite{CLR06}\hspace{0pt} )}  and planar games (Addario-Berry, Olver and Vetta , 2007 \mbox{ \cite{AOV07}\hspace{0pt} }  ).
Revised:  such as, very sparse games (Codenotti, Leoncini and Resta , 2006 \mbox{ \cite{CLR06}\hspace{0pt} )} )) and planar games (Addario-Berry, Olver and Vetta , 2007 \mbox{ \cite{AOV07}\hspace{0pt} } ).
Edit intent: fluency
Original:  such as, very sparse games (Codenotti, Leoncini and Resta , 2006 \mbox{ \cite{CLR06}\hspace{0pt} )}  and planar games (Addario-Berry, Olver and Vetta , 2007 \mbox{ \cite{AOV07}\hspace{0pt} }  ).
Revised:  such as, very sparse games (Codenotti, Leoncini and Resta , 2006 \mbox{ \cite{CLR06}\hspace{0pt} )} and planar games (Addario-Berry, Olver and Vetta ( 2007 \mbox{ \cite{AOV07}\hspace{0pt} } ).
Edit intent: fluency
Original:  such as, very sparse games (Codenotti, Leoncini and Resta , 2006 \mbox{ \cite{CLR06}\hspace{0pt} )}  and planar games (Addario-Berry, Olver and Vetta , 2007 \mbox{ \cite{AOV07}\hspace{0pt} }  ).
Revised:  such as, very sparse games (Codenotti, Leoncini and Resta , 2006 \mbox{ \cite{CLR06}\hspace{0pt} )} and planar games (Addario-Berry, Olver and Vetta , 2007 \mbox{ \cite{AOV07}\hspace{0pt} } ) ).
Edit intent: fluency
Original: In an undirected graph G =(V, E), a subset C\subseteq V  such that C is a dominating set of G, and each vertex in V  is dominated by a distinct subset of vertices from C, is called an identifying code of G. The concept of identifying codes was introduced by Karpovsky et al. in 1998.
Revised: In an undirected graph G , a subset C\subseteq V such that C is a dominating set of G, and each vertex in V is dominated by a distinct subset of vertices from C, is called an identifying code of G. The concept of identifying codes was introduced by Karpovsky et al. in 1998.
Edit intent: clarity
Original:  Because of the variety of its applications, for example for fault-detection in networks or the location of fires in facilities, it has since been widely studied.
Edit intent: coherence
Original:  For a given  graph G, let \M(G) be the minimum cardinality of an identifying code in G. In this paper, we show that for any connected  triangle-free graph G on n vertices having maximum degree \Delta %DIFDELCMD < \ge %%% 2 , \M(G)\le n- \frac{n{3(\Delta+1)}} n{\Delta+o(\Delta)}n{\Delta-1+o(1)}\M\le n{\Delta} .
Revised:  For a given identifiable graph G, let \M(G) be the minimum cardinality of an identifying code in G. In this paper, we show that for any connected triangle-free graph G on n vertices having maximum degree \Delta %DIFDELCMD < \ge %%% 2 , \M(G)\le n- \frac{n{3(\Delta+1)}} n{\Delta+o(\Delta)}n{\Delta-1+o(1)}\M\le n{\Delta} .
Edit intent: fluency
Original:  For a given  graph G, let \M(G) be the minimum cardinality of an identifying code in G. In this paper, we show that for any connected  triangle-free graph G on n vertices having maximum degree \Delta %DIFDELCMD < \ge %%% 2 , \M(G)\le n- \frac{n{3(\Delta+1)}} n{\Delta+o(\Delta)}n{\Delta-1+o(1)}\M\le n{\Delta} .
Revised:  For a given graph G, let \M(G) be the minimum cardinality of an identifying code in G. In this paper, we show that for any connected triangle-free graph G on n vertices having maximum degree \Delta %DIFDELCMD < \ge %%% 2 , \M(G)\le n- \frac{n{3(\Delta+1)}} n{\Delta+o(\Delta)}n{\Delta-1+o(1)}\M(G)\le n{\Delta} .
Edit intent: fluency
Original:  By looking at the Galois group of programs, we study the the intermediate types of algorithms possible .
Revised:  By looking at the Galois group of programs, we study the intermediate types of algorithms possible .
Edit intent: fluency
Original: We introduce a new method to isolate the real roots of a square-free polynomial F=i=0^n A_i x^i with real coefficients A_i, where |A_n|%DIFDELCMD < \ge %%% 1 and |A_i|%DIFDELCMD < \le %%% 2^{\tau .
Revised: We introduce a new approach to isolate the real roots of a square-free polynomial F=i=0^n A_i x^i with real coefficients A_i, where |A_n|%DIFDELCMD < \ge %%% 1 and |A_i|%DIFDELCMD < \le %%% 2^{\tau .
Edit intent: clarity
Original: We introduce a new method to isolate the real roots of a square-free polynomial F=i=0^n A_i x^i with real coefficients A_i, where |A_n|%DIFDELCMD < \ge %%% 1 and |A_i|%DIFDELCMD < \le %%% 2^{\tau .
Revised: We introduce a new method to isolate the real roots of a square-free polynomial F=i=0^n A_i x^i with real coefficients %DIFDELCMD < \ge %%% 1 and |A_i|%DIFDELCMD < \le %%% 2^{\tau .
Edit intent: coherence
Original: We introduce a new method to isolate the real roots of a square-free polynomial F=i=0^n A_i x^i with real coefficients A_i, where |A_n|%DIFDELCMD < \ge %%% 1 and |A_i|%DIFDELCMD < \le %%% 2^{\tau .
Revised: We introduce a new method to isolate the real roots of a square-free polynomial F=i=0^n A_i x^i with real coefficients A_i, where |A_n|%DIFDELCMD < \ge %%% %DIFDELCMD < \le %%% 2^{\tau .
Edit intent: coherence
Original:  In particular, we show that the hardness of isolating the real roots of F is essentially determined by the geometry of the roots and not by the complexity  of the coefficients.
Revised:  It is further shown that the hardness of isolating the real roots of F is essentially determined by the geometry of the roots and not by the complexity of the coefficients.
Edit intent: coherence
Original:  In particular, we show that the hardness of isolating the real roots of F is essentially determined by the geometry of the roots and not by the complexity  of the coefficients.
Revised:  In particular, we show that the hardness of isolating the real roots of F is exclusively determined by the geometry of the roots and not by the complexity of the coefficients.
Edit intent: clarity
Original:  More precisely, our new algorithm demands for%DIFDELCMD < \Otilde%%% (n(\Sigma(F)+n\log\Gamma+\tau)(\Sigma(F)+n\log\Gamma)) bit operations, where \sigma_i denotes the separation of the i-th (complex) root of F , \Sigma(F):=i=1^n\log\sigma_i^{-1, and \Gamma constitutes a bound on the modulus of all roots.
Revised:  %DIFDELCMD < \Otilde%%% (n(\Sigma(F)+n\log\Gamma+\tau)(\Sigma(F)+n\log\Gamma)) bit operations, where \sigma_i denotes the separation of the i-th (complex) root of F , \Sigma(F):=i=1^n\log\sigma_i^{-1, and \Gamma constitutes a bound on the modulus of all roots.
Edit intent: clarity
Original:  More precisely, our new algorithm demands for%DIFDELCMD < \Otilde%%% (n(\Sigma(F)+n\log\Gamma+\tau)(\Sigma(F)+n\log\Gamma)) bit operations, where \sigma_i denotes the separation of the i-th (complex) root of F , \Sigma(F):=i=1^n\log\sigma_i^{-1, and \Gamma constitutes a bound on the modulus of all roots.
Revised:  More precisely, our new algorithm demands for%DIFDELCMD < \Otilde%%% , and \Gamma constitutes a bound on the modulus of all roots.
Edit intent: coherence
Original:  In this paper, following a series of new results regarding the existence of empty pseudo-triangles in point sets with triangular convex hulls, we determine the exact values of E(k, 5)
Revised:  In this paper, following a series of new results about the existence of empty pseudo-triangles in point sets with triangular convex hulls, we determine the exact values of E(k, 5)
Edit intent: fluency
Original:  and E(5, \ell), and prove new bounds on E(k, 6) and E(6, \ell), for k, \ell\geq 3. By dropping the emptiness condition from E(k, \ell), we get another related quantity F(k, \ell), which is the smallest integer such that any set of at least F(k, \ell) points in the plane, no three on a line, contains a convex polygon with k vertices or a pseudo-triangle with \ell vertices.
Revised:  and E(5, \ell), and prove bounds on E(k, 6) and E(6, \ell), for k, \ell\geq 3. By dropping the emptiness condition from E(k, \ell), we get another related quantity F(k, \ell), which is the smallest integer such that any set of at least F(k, \ell) points in the plane, no three on a line, contains a convex polygon with k vertices or a pseudo-triangle with \ell vertices.
Edit intent: clarity
Original:  and E(5, \ell), and prove new bounds on E(k, 6) and E(6, \ell), for k, \ell\geq 3. By dropping the emptiness condition from E(k, \ell), we get another related quantity F(k, \ell), which is the smallest integer such that any set of at least F(k, \ell) points in the plane, no three on a line, contains a convex polygon with k vertices or a pseudo-triangle with \ell vertices.
Revised:  and E(5, \ell), and prove new bounds on E(k, 6) and E(6, \ell), for k, \ell\geq 3. By dropping the emptiness condition , we define another related quantity F(k, \ell), which is the smallest integer such that any set of at least F(k, \ell) points in the plane, no three on a line, contains a convex polygon with k vertices or a pseudo-triangle with \ell vertices.
Edit intent: clarity
Original: Rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of  vertices is connected by at least one path in which no two edges are coloured the same.
Revised: The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same.
Edit intent: fluency
Original: Rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of  vertices is connected by at least one path in which no two edges are coloured the same.
Revised: Rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same.
Edit intent: fluency