wanyu/IteraTeR_v2 huggingface dataset:
Original: Every chordal graph G can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called tree model {\em  of G. The leafage \ell(G) of a connected chordal graph G is the minimum number of leaves of the host tree of a tree model of G and the vertex leafage vl\vl (G) is the smallest number k such that there exists a tree model of G in which every subtree has at most k leaves.
Revised: Every chordal graph G can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called {\em of G. The leafage \ell(G) of a connected chordal graph G is the minimum number of leaves of the host tree of a tree model of G and the vertex leafage vl\vl (G) is the smallest number k such that there exists a tree model of G in which every subtree has at most k leaves.
Edit intent: fluency
Original: Every chordal graph G can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called tree model {\em  of G. The leafage \ell(G) of a connected chordal graph G is the minimum number of leaves of the host tree of a tree model of G and the vertex leafage vl\vl (G) is the smallest number k such that there exists a tree model of G in which every subtree has at most k leaves.
Revised: Every chordal graph G can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called tree model {\em tree model of G. The leafage \ell(G) of a connected chordal graph G is the minimum number of leaves of the host tree of a tree model of G and the vertex leafage vl\vl (G) is the smallest number k such that there exists a tree model of G in which every subtree has at most k leaves.
Edit intent: fluency
Original: Every chordal graph G can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called tree model {\em  of G. The leafage \ell(G) of a connected chordal graph G is the minimum number of leaves of the host tree of a tree model of G and the vertex leafage vl\vl (G) is the smallest number k such that there exists a tree model of G in which every subtree has at most k leaves.
Revised: Every chordal graph G can be represented as the intersection graph of a collection of subtrees of a host tree, a so-called tree model {\em of G. The leafage \ell(G) of a connected chordal graph G is the minimum number of leaves of the host tree of a tree model of G . The vertex leafage\vl (G) is the smallest number k such that there exists a tree model of G in which every subtree has at most k leaves.
Edit intent: coherence
Original:   In this contribution, we  prove for every fixed k\geq 3 that deciding whether the vertex leafage of a  chordal graph is at most k is NP-complete .
Revised:  In this contribution, we study the vertex leafage. We prove for every fixed k\geq 3 that deciding whether the vertex leafage of a chordal graph is at most k is NP-complete .
Edit intent: coherence
Original:   In this contribution, we  prove for every fixed k\geq 3 that deciding whether the vertex leafage of a  chordal graph is at most k is NP-complete .
Revised:  In this contribution, we prove for every fixed k\geq 3 that deciding whether the vertex leafage of a given chordal graph is at most k is NP-complete .
Edit intent: clarity
Original:  In fact, our proof shows that the problem is NP -complete already on split graphs .
Revised:  In fact, our proof shows that the problem is NP-complete on split graphs .
Edit intent: clarity
Original:  We also show that for fixed leafage \ell, vertex leafage can be calculated in n O(\ell) time and for fixed vertex leafage k, leafage can be calculated.
Revised:  We also show that for fixed leafage \ell, vertex leafage can be calculated in time n^{O(\ell).
Edit intent: clarity
Original: Migraine patients with aura show a peculiar pattern of visual reactivity compared with those of migraine patients without aura and healthy subjects : an increased effective connectivity, connected to a reduced synchronization among EEG channels, for frequencies in the beta band.
Revised: Migraine patients with aura show a peculiar pattern of visual reactivity compared with those of migraine patients without aura : an increased effective connectivity, connected to a reduced synchronization among EEG channels, for frequencies in the beta band.
Edit intent: clarity
Original: The distance-3 cyclic lowest-density MDS array code (called the C-Code) is a good choice for RAID 6 because of its optimal storage efficiency, optimal update complexity, optimal length, and cyclic symmetry.
Revised: The distance-3 cyclic lowest-density MDS array code (called the C-Code) is a good candidate for RAID 6 because of its optimal storage efficiency, optimal update complexity, optimal length, and cyclic symmetry.
Edit intent: clarity
Original:  Supported by the well-known qualitative findings about economic crises, we argue that the theorem provides valuable quantitative insights into the mechanisms underlying the emergence of worldwide crises.
Edit intent: coherence
Original:  We use our results above to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables.
Revised:  We apply our results above to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables.
Edit intent: fluency
Original:  We use the bound 2 /3 m + \rho nand results on matching autarkies to obtain a new lower bound on the maximum number of clauses that can be satisfied by a truth assignment in any 3-satisfiable CNF formula.
Revised:  We use the bound 2 /3 + \rho nand results on matching autarkies to obtain a new lower bound on the maximum number of clauses that can be satisfied by a truth assignment in any 3-satisfiable CNF formula.
Edit intent: clarity
Original:  In %DIFDELCMD < {\sc %%% 3-Sat-SAT-AE , we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 2 /3 m + k clauses, where k is the parameter.
Revised:  In %DIFDELCMD < {\sc %%% 3-S-MAXSAT-AE , we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 2 /3 m + k clauses, where k is the parameter.
Edit intent: fluency
Original:  In %DIFDELCMD < {\sc %%% 3-Sat-SAT-AE , we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 2 /3 m + k clauses, where k is the parameter.
Revised:  In %DIFDELCMD < {\sc %%% 3-Sat-SAT-AE , we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 2m /3 m + k clauses, where k is the parameter.
Edit intent: fluency
Original:  In %DIFDELCMD < {\sc %%% 3-Sat-SAT-AE , we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 2 /3 m + k clauses, where k is the parameter.
Revised:  In %DIFDELCMD < {\sc %%% 3-Sat-SAT-AE , we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 2 /3 + k clauses, where k is the parameter.
Edit intent: clarity
Original:  We consider transmission of packets from one source to many receivers over a single-hop broadcast erasure channel , but the proposed scheme is applicable to more general lossy networks .
Revised:  We consider transmission of packets from one source to many receivers over a single-hop broadcast erasure channel .
Edit intent: coherence
Original:  In addition, the proposed NACK-based feedback mechanism enables all receivers to request  the number of retransmissions needed for successful decoding within a single time slot .
Revised:  In addition, our proposed NACK-based feedback mechanism enables all receivers to request the number of retransmissions needed for successful decoding within a single time slot .
Edit intent: fluency
Original:  In addition, the proposed NACK-based feedback mechanism enables all receivers to request  the number of retransmissions needed for successful decoding within a single time slot .
Revised:  In addition, the proposed NACK-based feedback mechanism enables all receivers to request the number of retransmissions needed for successful decoding .
Edit intent: clarity
Original:  We also show the robustness of this scheme to inaccuracies in the predictive model, including uncertainty in the number of receiving nodes , the packet erasure probability, as well as to losses of the feedback itself.
Revised:  We also show the robustness of this scheme to uncertainty in the predictive model, including uncertainty in the number of receiving nodes , the packet erasure probability, as well as to losses of the feedback itself.
Edit intent: clarity
Original:  We also show the robustness of this scheme to inaccuracies in the predictive model, including uncertainty in the number of receiving nodes , the packet erasure probability, as well as to losses of the feedback itself.
Revised:  We also show the robustness of this scheme to inaccuracies in the predictive model, including uncertainty in the number of receiving nodes and the packet erasure probability, as well as to losses of the feedback itself.
Edit intent: clarity
Original: This paper presents a detailed proof for the triality theorem in a class of global optimization problems .
Revised: Triality theory is proved for a general unconstrained global optimization problem .
Edit intent: clarity
Original:  The method is based on elementary linear algebra but it solves an open problem on the double-min duality left seven years ago .
Revised:  The method adopted is simple but mathematically rigorous .
Edit intent: coherence
Original:  Results show that the triality theory holds strongly in the tri-duality form if the primal problem and its canonical dual have the same dimension ;
Revised:  Results show that if the primal problem and its canonical dual have the same dimension ;
Edit intent: clarity
Original:  In every environment, the evolution of the pathogen is described by a version of the Fisher-Haldane-Wright equation while the switching between environments follows a Markov process with a given generator matrix .
Revised:  In every environment, the evolution of the pathogen is described by a version of the Fisher-Haldane-Wright equation while the switching between environments follows a Markov jump process .
Edit intent: clarity
Original:  In particular, we study the exchange of stability between monomorphic equilibria.
Revised:  In particular, we study the stability in probability of monomorphic equilibria.
Edit intent: clarity
Original:  Our results are consistent with the view that in a  fluctuating environment, the genotype with the highest mean fitness will eventually become fixed .
Revised:  We prove that in a fluctuating environment, the genotype with the highest mean fitness will eventually become fixed .
Edit intent: clarity
Original:  Makarychev found a new proof of G\'acs--K\"orner result by means of conditionally independent random variables.
Revised:  Makarychev found a new proof of G\'acs--K\"orner results by means of conditionally independent random variables.
Edit intent: fluency
Original:  The set of possible triples of lengths is called a profile of the b  pair (x, y) ;
Revised:  The set of possible triples of lengths is called a profile of the ,b pair (x, y) ;
Edit intent: fluency
Original:  The set of possible triples of lengths is called a profile of the b  pair (x, y) ;
Revised:  The set of possible triples of lengths is called a profile of the b, pair (x, y) ;
Edit intent: fluency
Original:  We also apply this framework to some other examples\eps  b .
Revised:  We also apply this framework to some other examples\eps ,b .
Edit intent: fluency
Original:  We also apply this framework to some other examples\eps  b .
Revised:  We also apply this framework to some other examples\eps b, .
Edit intent: fluency
Original:  Among the results, the experiment prompts an investigation into rates of convergence of decision procedures, the problem of extensionality and the definability of computable sets.
Revised:  Among the results, the experiment prompts an investigation into rates of convergence of decision procedures, the problem of extensionality and the decidability of computable sets.
Edit intent: clarity
Original:  This paper is an exploration of the conceptual issues that have arisen in the course of investigation speed-up and slowdown phenomena in small Turing machines.
Revised:  This paper is an exploration of the conceptual issues that have arisen in the course of investigating speed-up and slowdown phenomena in small Turing machines.
Edit intent: fluency
Original:  We present the results of a basic test that may spur others to further explore experimental approaches to theoretical results .
Revised:  We present the results of a test that may spur others to further explore experimental approaches to theoretical results .
Edit intent: clarity
Original:  We present the results of a basic test that may spur others to further explore experimental approaches to theoretical results .
Revised:  We present the results of a basic test that may spur experimental approaches to theoretical results .
Edit intent: clarity
Original:  The test involves an attempt to shortcut the computations of a (relatively) large set of small Turing machines  by means of  sequence prediction using a specialized function finder program.
Revised:  The test involves a systematic attempt to outrun the computation of a large number of small Turing machines by means of sequence prediction using a specialized function finder program.
Edit intent: clarity
Original:  Among the results, the experiment prompts an investigation into rates of convergence of decision procedures , the problem of extensionality and the decidability of computable sets .
Revised:  This massive experiment prompts an investigation into rates of convergence of decision procedures , the problem of extensionality and the decidability of computable sets .
Edit intent: clarity
Original:  Among the results, the experiment prompts an investigation into rates of convergence of decision procedures , the problem of extensionality and the decidability of computable sets .
Revised:  Among the results, the experiment prompts an investigation into rates of convergence of decision procedures and the decidability of computable sets .
Edit intent: clarity
Original:  These results show that the probability of stochastic tunneling is twice as large in the Wright-Fisher model as compared with the Moran model.
Revised:  These results show that the probability of stochastic tunneling is twice as large under the Wright-Fisher model as compared with the Moran model.
Edit intent: clarity
Original:  These results show that the probability of stochastic tunneling is twice as large in the Wright-Fisher model as compared with the Moran model.
Revised:  These results show that the probability of stochastic tunneling is twice as large in the Wright-Fisher model as it is under the Moran model.
Edit intent: clarity
Original:  This construction has the remarkable property, proved by Rost (1976) , that it minimises the variance of the stopping time among all solutions.
Revised:  This construction has the remarkable property, proved by Rost , that it minimises the variance of the stopping time among all solutions.
Edit intent: clarity
Original:  This construction has the remarkable property, proved by Rost (1976) , that it minimises the variance of the stopping time among all solutions.
Revised:  This construction has the remarkable property, proved by Rost (1976) , that it minimizes the variance of the stopping time among all solutions.
Edit intent: fluency
Original:  We show that, if the auxiliary probability distributions of the expansion are chosen to be a fixed point of the belief-propagation equation, the first term of the loop series expansion corresponds to the Bethe-Peierls free energy  , and corrections are contributed only by subgraphs that are free of dangling edges.
Revised:  If the auxiliary probability distributions of the expansion are chosen to be a fixed point of the belief-propagation equation, the first term of the loop series expansion corresponds to the Bethe-Peierls free energy , and corrections are contributed only by subgraphs that are free of dangling edges.
Edit intent: coherence
Original:  We show that, if the auxiliary probability distributions of the expansion are chosen to be a fixed point of the belief-propagation equation, the first term of the loop series expansion corresponds to the Bethe-Peierls free energy  , and corrections are contributed only by subgraphs that are free of dangling edges.
Revised:  We show that, if the auxiliary probability distributions of the expansion are chosen to be a fixed point of the belief-propagation equation, the first term of the loop series gives the Bethe-Peierls free energy , and corrections are contributed only by subgraphs that are free of dangling edges.
Edit intent: clarity
Original:  If the belief-propagation equation has multiple fixed points, a loop series expansion is performed for the grand partition function of the system .
Revised:  If the belief-propagation equation has multiple fixed points, a loop series expansion is performed for the grand partition function .
Edit intent: clarity
Original:  The first term of this series corresponds to the grand free energy of the mean-field first-step replica-symmetry-breaking ( 1RSB) cavity theory, and corrections again come only from subgraphs that are free of dangling edges, provided that the auxiliary probability distributions of the expansion are chosen to be a fixed point of the 1RSB belief-propagation equation.
Revised:  The first term of this series corresponds to the grand free energy of the mean-field first-step replica-symmetry-breaking ( 1RSB) cavity theory, and corrections again come only from subgraphs that are free of dangling edges, provided that the auxiliary probability distributions of the expansion are chosen to be a fixed point of the survey-propagation equation.
Edit intent: clarity
Original:  This approach leads to a deterministic algorithm that solves the closest vector problem for all \ell _p-norms, 1 < p < \infty, in time p ^{2n n^{O (n)} \log}  _2 (r)  and for all polyhedral norms in time m^{\mathcal{O (1)} n^{O (n)} \log_ 2 (r}  ) , where m is the number of constraints defining the polytope and r is an upper bound on the coefficients used to describe the convex body.
Revised:  This approach leads to a deterministic algorithm that solves the closest vector problem for all l _p-norms, 1 < p < \infty, in time p ^{2n n^{O (n)} \log} _2 (r) and for all polyhedral norms in time m^{\mathcal{O (1)} n^{O (n)} \log_ 2 (r} ) , where m is the number of constraints defining the polytope and r is an upper bound on the coefficients used to describe the convex body.
Edit intent: fluency
Original:  This approach leads to a deterministic algorithm that solves the closest vector problem for all \ell _p-norms, 1 < p < \infty, in time p ^{2n n^{O (n)} \log}  _2 (r)  and for all polyhedral norms in time m^{\mathcal{O (1)} n^{O (n)} \log_ 2 (r}  ) , where m is the number of constraints defining the polytope and r is an upper bound on the coefficients used to describe the convex body.
Revised:  This approach leads to a deterministic algorithm that solves the closest vector problem for all \ell _p-norms, 1 < p < \infty, in time p n^{O (n)} \log} _2 (r) and for all polyhedral norms in time m^{\mathcal{O (1)} n^{O (n)} \log_ 2 (r} ) , where m is the number of constraints defining the polytope and r is an upper bound on the coefficients used to describe the convex body.
Edit intent: clarity
Original:  This approach leads to a deterministic algorithm that solves the closest vector problem for all \ell _p-norms, 1 < p < \infty, in time p ^{2n n^{O (n)} \log}  _2 (r)  and for all polyhedral norms in time m^{\mathcal{O (1)} n^{O (n)} \log_ 2 (r}  ) , where m is the number of constraints defining the polytope and r is an upper bound on the coefficients used to describe the convex body.
Revised:  This approach leads to a deterministic algorithm that solves the closest vector problem for all \ell _p-norms, 1 < p < \infty, in time p ^{2n n^{O (n)} \log} log _2 (r) and for all polyhedral norms in time m^{\mathcal{O (1)} n^{O (n)} \log_ 2 (r} ) , where m is the number of constraints defining the polytope and r is an upper bound on the coefficients used to describe the convex body.
Edit intent: fluency
Original:  This approach leads to a deterministic algorithm that solves the closest vector problem for all \ell _p-norms, 1 < p < \infty, in time p ^{2n n^{O (n)} \log}  _2 (r)  and for all polyhedral norms in time m^{\mathcal{O (1)} n^{O (n)} \log_ 2 (r}  ) , where m is the number of constraints defining the polytope and r is an upper bound on the coefficients used to describe the convex body.
Revised:  This approach leads to a deterministic algorithm that solves the closest vector problem for all \ell _p-norms, 1 < p < \infty, in time p ^{2n n^{O (n)} \log} _2 (r) ^{O (1) and for all polyhedral norms in time m^{\mathcal{O (1)} n^{O (n)} \log_ 2 (r} ) , where m is the number of constraints defining the polytope and r is an upper bound on the coefficients used to describe the convex body.
Edit intent: coherence
Original:  Thus the message of " giving you the ads that interest you " does not tell the whole story of behavioral targeting, and the resulting differentiation in user experience may, or may not, work to the advantage of the user.
Revised:  Thus the message of " fair affirmative action, " does not tell the whole story of behavioral targeting, and the resulting differentiation in user experience may, or may not, work to the advantage of the user.
Edit intent: clarity
Original:  When the ion channel currents have a linear current-voltage relationship, we show that the there is at most one steady state, and that the steady state is always asymptotically stable.
Revised:  When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is always asymptotically stable.
Edit intent: fluency
Original:  When the ion channel currents have a linear current-voltage relationship, we show that the there is at most one steady state, and that the steady state is always asymptotically stable.
Revised:  When the ion channel currents have a linear current-voltage relationship, we show that the there is at most one steady state, and that the steady state is globally asymptotically stable.
Edit intent: clarity
Original:  The key tool in our argument is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.
Revised:  The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.
Edit intent: clarity
Original:  We show that network learning and generalization are optimized near criticality, given task complexity and the amount of information provided surpass threshold values.
Revised:  We show that network learning and generalization are optimized near criticality, given task complexity and the amount of information provided threshold values.
Edit intent: clarity
Original:  We hypothesize that this diversity supports efficient exploration and robustness of solutions , also reflected in our observation  that the variance of the fitness values is maximal in critical network populations.
Revised:  We hypothesize that this supports efficient exploration and robustness of solutions , also reflected in our observation that the variance of the fitness values is maximal in critical network populations.
Edit intent: clarity