wanyu/IteraTeR_v2 huggingface dataset:
Original:   Does there exist a function c () independent of n such that for every f: \Omega^n \to [0, 1] with E f(X^{(1) = \mu > 0: %DIFDELCMD < \begin{align*} %DIFDELCMD < E \left[ \prod_{j=1}^\ell f(X^{(j)}) \right] %DIFDELCMD < \ge c(\mu) > 0 \;
Revised:  - Does there exist a function c () independent of n such that for every f: \Omega^n \to [0, 1] with E f(X^{(1) = \mu > 0: %DIFDELCMD < \begin{align*} %DIFDELCMD < E \left[ \prod_{j=1}^\ell f(X^{(j)}) \right] %DIFDELCMD < \ge c(\mu) > 0 \;
Edit intent: fluency
Original:   Does there exist a function c () independent of n such that for every f: \Omega^n \to [0, 1] with E f(X^{(1) = \mu > 0: %DIFDELCMD < \begin{align*} %DIFDELCMD < E \left[ \prod_{j=1}^\ell f(X^{(j)}) \right] %DIFDELCMD < \ge c(\mu) > 0 \;
Revised:  Does there exist a function c () independent of n such that for every f: \Omega^n \to [0, 1] with f(X^{(1) = \mu > 0: %DIFDELCMD < \begin{align*} %DIFDELCMD < E \left[ \prod_{j=1}^\ell f(X^{(j)}) \right] %DIFDELCMD < \ge c(\mu) > 0 \;
Edit intent: coherence
Original:  ? \end{align*}%%% \begin{align*} \mathrm{E} \left[ \prod_{j=1}^\ell f(X^{(j)}) \right] \ge c(\mu) > 0 \, ? \end{align*} Instances of this question include the Finite Field Model version of Roth's and Szemeredi 's theorems as well as Borell's result about the optimality of noise stability of half-spaces.
Revised:  ? \end{align*}%%% \begin{align*} \mathrm{E} \left[ \prod_{j=1}^\ell f(X^{(j)}) \right] \ge c(\mu) > 0 \, ? \end{align*} Instances of this question include the finite field model version of Roth's and Szemeredi 's theorems as well as Borell's result about the optimality of noise stability of half-spaces.
Edit intent: fluency
Original:  ? \end{align*}%%% \begin{align*} \mathrm{E} \left[ \prod_{j=1}^\ell f(X^{(j)}) \right] \ge c(\mu) > 0 \, ? \end{align*} Instances of this question include the Finite Field Model version of Roth's and Szemeredi 's theorems as well as Borell's result about the optimality of noise stability of half-spaces.
Revised:  ? \end{align*}%%% \begin{align*} \mathrm{E} \left[ \prod_{j=1}^\ell f(X^{(j)}) \right] \ge c(\mu) > 0 \, ? \end{align*} Instances of this question include the Finite Field Model version of Roth's and Szemer\'edi 's theorems as well as Borell's result about the optimality of noise stability of half-spaces.
Edit intent: fluency
Original:  In particular, we settle the question for \ell = 2 and when \ell > 2 and P has bounded correlation \rho( P ) < 1. Under the same conditions we also characterize the %DIFDELCMD < {\em %DIFDELCMD < %%% obstructions for a similar lower bounds in the case of k different functions.
Revised:  In particular, we settle the question for \ell = 2 and when \ell > 2 and P has bounded correlation \rho( P ) < 1. Under the same conditions we also characterize the %DIFDELCMD < {\em %DIFDELCMD < %%% for a similar lower bounds in the case of k different functions.
Edit intent: clarity
Original:  In particular, we settle the question for \ell = 2 and when \ell > 2 and P has bounded correlation \rho( P ) < 1. Under the same conditions we also characterize the %DIFDELCMD < {\em %DIFDELCMD < %%% obstructions for a similar lower bounds in the case of k different functions.
Revised:  In particular, we settle the question for \ell = 2 and when \ell > 2 and P has bounded correlation \rho( P ) < 1. Under the same conditions we also characterize the %DIFDELCMD < {\em %DIFDELCMD < %%% obstructions _obstructions_ for similar lower bounds in the case of k different functions.
Edit intent: clarity
Original:  In particular, we settle the question for \ell = 2 and when \ell > 2 and P has bounded correlation \rho( P ) < 1. Under the same conditions we also characterize the %DIFDELCMD < {\em %DIFDELCMD < %%% obstructions for a similar lower bounds in the case of k different functions.
Revised:  In particular, we settle the question for \ell = 2 and when \ell > 2 and P has bounded correlation \rho( P ) < 1. Under the same conditions we also characterize the %DIFDELCMD < {\em %DIFDELCMD < %%% obstructions for a similar lower bounds in the case of \ell different functions.
Edit intent: clarity
Original:  Since there might exist exponentially many different reaction graph structures, we cannot expect to have a polynomial-time algorithm, but it can be shown that polynomial time is elapsed between displaying any two consecutive realizations.
Revised:  Since there might exist exponentially many different reaction graph structures, we cannot expect to have a polynomial-time algorithm, but we can organize the computation in such a way that polynomial time is elapsed between displaying any two consecutive realizations.
Edit intent: clarity
Original:  The correctness of the algorithm is proved, and the possibilities of parallel implementation are outlined .
Revised:  The correctness of the algorithm is proved, and possibilities of a parallel implementation are outlined .
Edit intent: fluency
Original:  The correctness of the algorithm is proved, and the possibilities of parallel implementation are outlined .
Revised:  The correctness of the algorithm is proved, and the possibilities of parallel implementation are discussed .
Edit intent: clarity
Original:  We use a refinement proof methodology to propagate the noninterference property of an abstract machine down to the concrete  level.
Revised:  We use a refinement proof methodology to propagate the noninterference property of the abstract machine down to the concrete level.
Edit intent: fluency
Original:  We use a refinement proof methodology to propagate the noninterference property of an abstract machine down to the concrete  level.
Revised:  We use a refinement proof methodology to propagate the noninterference property of an abstract machine down to the concrete machine level.
Edit intent: clarity
Original:  We define an intermediate layer in the refinement chain that helps factoring out the details of the information-flow control monitor .
Revised:  We use an intermediate layer in the refinement chain that helps factoring out the details of the information-flow control monitor .
Edit intent: clarity
Original:  We define an intermediate layer in the refinement chain that helps factoring out the details of the information-flow control monitor .
Revised:  We define an intermediate layer in the refinement chain that factors out the details of the information-flow control monitor .
Edit intent: clarity
Original:  Finally, we devise a dedicated Hoare logic to specify and prove the functional correctness of the low-level code used to implement the monitor .
Revised:  Finally, we devise a dedicated Hoare logic that abstracts from low-level code used to implement the monitor .
Edit intent: clarity
Original:  The seller can induce high degree consumers to adopt early by offering referral incentives - rewards to early adopters whose friends buy in the second period.
Revised:  The seller can induce high-degree consumers to adopt early by offering referral incentives - rewards to early adopters whose friends buy in the second period.
Edit intent: fluency
Original:  We show that referral incentives are optimal on certain networks while intertemporal price discrimination (i.e., a first-period price discount) is optimal on others  .
Revised:  We show that referral incentives are optimal on certain networks while inter-temporal price discrimination (i.e., a first-period price discount) is optimal on others .
Edit intent: fluency
Original:  The features of the proposed approach are studied for synthetic test cases and a collection of geometry processing applications .
Revised:  The features of the proposed approach are studied for synthetic test cases and a collection of geometry processing examples .
Edit intent: clarity
Original: Given a graph G=(V,E), the minimum branch vertices problem (MBV) consists in finding a spanning tree T=(V,E') of G minimizing the number of vertices with degree greater than two.
Revised: Given a graph G=(V,E), the minimum branch vertices problem consists in finding a spanning tree T=(V,E') of G minimizing the number of vertices with degree greater than two.
Edit intent: clarity
Original:  We consider a simple combinatorial lower bound for the problem  from which we propose a decomposition approach  to break down the problem into several smaller subproblems that are more tractable computationally  and then recombine the obtained solutions to generate a solution to the original larger problem.
Revised:  We consider a simple combinatorial lower bound for the problem , from which we propose a decomposition approach to break down the problem into several smaller subproblems that are more tractable computationally and then recombine the obtained solutions to generate a solution to the original larger problem.
Edit intent: fluency
Original:  We consider a simple combinatorial lower bound for the problem  from which we propose a decomposition approach  to break down the problem into several smaller subproblems that are more tractable computationally  and then recombine the obtained solutions to generate a solution to the original larger problem.
Revised:  We consider a simple combinatorial lower bound for the problem from which we propose a decomposition approach . The motivation is to break down the problem into several smaller subproblems that are more tractable computationally and then recombine the obtained solutions to generate a solution to the original larger problem.
Edit intent: coherence
Original:  We consider a simple combinatorial lower bound for the problem  from which we propose a decomposition approach  to break down the problem into several smaller subproblems that are more tractable computationally  and then recombine the obtained solutions to generate a solution to the original larger problem.
Revised:  We consider a simple combinatorial lower bound for the problem from which we propose a decomposition approach to break down the problem into several smaller subproblems which are more tractable computationally and then recombine the obtained solutions to generate a solution to the original larger problem.
Edit intent: fluency
Original:  We consider a simple combinatorial lower bound for the problem  from which we propose a decomposition approach  to break down the problem into several smaller subproblems that are more tractable computationally  and then recombine the obtained solutions to generate a solution to the original larger problem.
Revised:  We consider a simple combinatorial lower bound for the problem from which we propose a decomposition approach to break down the problem into several smaller subproblems that are more tractable computationally , and then recombine the obtained solutions to generate a solution to the original larger problem.
Edit intent: fluency
Original:  We consider a simple combinatorial lower bound for the problem  from which we propose a decomposition approach  to break down the problem into several smaller subproblems that are more tractable computationally  and then recombine the obtained solutions to generate a solution to the original larger problem.
Revised:  We consider a simple combinatorial lower bound for the problem from which we propose a decomposition approach to break down the problem into several smaller subproblems that are more tractable computationally and then recombine the obtained solutions to generate a solution to the original problem.
Edit intent: clarity
Original:  We also propose effective constructive heuristics to the problem which take into consideration the problem  structure in order to obtain good feasible solutions.
Revised:  We also propose effective constructive heuristics to the problem which take into consideration the problem 's structure in order to obtain good feasible solutions.
Edit intent: fluency
Original:  Computational results show that our decomposition approach is very fast and can drastically reduce the size of the subproblems that have to be solved , allowing a branch-and-cut algorithm to perform much better than when used over the full original problem.
Revised:  Computational results show that our decomposition approach is very fast and can drastically reduce the size of the subproblems to be solved , allowing a branch-and-cut algorithm to perform much better than when used over the full original problem.
Edit intent: coherence
Original:  Computational results show that our decomposition approach is very fast and can drastically reduce the size of the subproblems that have to be solved , allowing a branch-and-cut algorithm to perform much better than when used over the full original problem.
Revised:  Computational results show that our decomposition approach is very fast and can drastically reduce the size of the subproblems that have to be solved . This allows a branch and cut algorithm to perform much better than when used over the full original problem.
Edit intent: coherence
Original:  The results also show that the proposed constructive heuristics are very rapid and generate very good quality solutions, outperforming other heuristics available in the literature in several situations.
Revised:  The results also show that the proposed constructive heuristics are highly efficient and generate very good quality solutions, outperforming other heuristics available in the literature in several situations.
Edit intent: clarity
Original:  While many existing works focus on what contents to cache at each SBS, an equally important but much less investigated problem is what contents to deliver  given the cache status and user requests .
Revised:  While many existing works focus on what contents to cache at each SBS, an equally important problem is what contents to deliver given the cache status and user requests .
Edit intent: clarity
Original:  While many existing works focus on what contents to cache at each SBS, an equally important but much less investigated problem is what contents to deliver  given the cache status and user requests .
Revised:  While many existing works focus on what contents to cache at each SBS, an equally important but much less investigated problem is what contents to deliver given the cache status .
Edit intent: clarity
Original:  In this paper, we study the optimal content delivery strategy in cache-enabled HetNets by taking into account the inherent multicast capability of wireless medium.
Revised:  In this paper, we study the optimal content delivery strategy in cache-enabled HetNets by taking into account the inherent multicast nature of wireless medium.
Edit intent: clarity
Original:  This stochastic optimization problem is an infinite horizon average cost Markov decision process (MDP) , which is well known to be challenging .
Revised:  This problem is an infinite horizon average cost Markov decision process (MDP) , which is well known to be challenging .
Edit intent: coherence
Original:  This stochastic optimization problem is an infinite horizon average cost Markov decision process (MDP) , which is well known to be challenging .
Revised:  This stochastic optimization problem is an infinite horizon average cost Markov decision process (MDP) .
Edit intent: clarity
Original:  By using relative value iteration algorithm and the special properties of the request queue dynamics, we characterize some properties of the value function of the MDP. Based on these properties, we show that the optimal multicast scheduling policy , which is adaptive to the request queue state, is of the threshold type.
Revised:  By using relative value iteration and the special properties of the request queue dynamics, we characterize some properties of the value function of the MDP. Based on these properties, we show that the optimal multicast scheduling policy , which is adaptive to the request queue state, is of the threshold type.
Edit intent: clarity
Original:  By using relative value iteration algorithm and the special properties of the request queue dynamics, we characterize some properties of the value function of the MDP. Based on these properties, we show that the optimal multicast scheduling policy , which is adaptive to the request queue state, is of the threshold type.
Revised:  By using relative value iteration algorithm and the special properties of the request queue dynamics, we characterize some properties of the value function of the MDP. Based on these properties, we show that the optimal multicast scheduling policy is of the threshold type.
Edit intent: clarity
Original:  Finally , we propose a low complexity optimal algorithm by exploiting the structural properties of the optimal policy.
Revised:  Then , we propose a low complexity optimal algorithm by exploiting the structural properties of the optimal policy.
Edit intent: fluency
Original:  While many existing works focus on what contents to cache at each SBS, an equally important problem is what contents to deliver  to satisfy dynamic user demands given the cache status.
Revised:  While many existing works focus on what contents to cache at each SBS, an equally important problem is what contents to deliver so as to satisfy dynamic user demands given the cache status.
Edit intent: clarity
Original:  In this paper, we study the optimal content delivery strategy in cache-enabled HetNets by taking into account the inherent multicast nature of wireless medium.
Revised:  In this paper, we study optimal content delivery strategy in cache-enabled HetNets by taking into account the inherent multicast nature of wireless medium.
Edit intent: fluency
Original:  In this paper, we study the optimal content delivery strategy in cache-enabled HetNets by taking into account the inherent multicast nature of wireless medium.
Revised:  In this paper, we study the optimal content delivery in cache-enabled HetNets by taking into account the inherent multicast nature of wireless medium.
Edit intent: clarity
Original:  In this paper, we study the optimal content delivery strategy in cache-enabled HetNets by taking into account the inherent multicast nature of wireless medium.
Revised:  In this paper, we study the optimal content delivery strategy in cache-enabled HetNets by taking into account the inherent multicast capability of wireless medium.
Edit intent: clarity
Original:  We establish a content-centric request queue model and then formulate a stochastic multicast scheduling problem to jointly minimize the average network delay and power costs .
Revised:  We consider stochastic content multicast scheduling to jointly minimize the average network delay and power costs .
Edit intent: clarity
Original:  By using relative value iteration and the  special properties of the request queue dynamics, we characterize some properties of the value function of the MDP. Based on these properties, we show that the optimal multicast scheduling policy is of the threshold type.
Revised:  By using relative value iteration special properties of the request queue dynamics, we characterize some properties of the value function of the MDP. Based on these properties, we show that the optimal multicast scheduling policy is of the threshold type.
Edit intent: coherence
Original:  By using relative value iteration and the  special properties of the request queue dynamics, we characterize some properties of the value function of the MDP. Based on these properties, we show that the optimal multicast scheduling policy is of the threshold type.
Revised:  By using relative value iteration and the and special properties of the request queue dynamics, we characterize some properties of the value function of the MDP. Based on these properties, we show that the optimal multicast scheduling policy is of the threshold type.
Edit intent: fluency
Original:  By using relative value iteration and the  special properties of the request queue dynamics, we characterize some properties of the value function of the MDP. Based on these properties, we show that the optimal multicast scheduling policy is of the threshold type.
Revised:  By using relative value iteration and the special properties of the request queue dynamics, we characterize some properties of the value function of the MDP. Based on these properties, we show that the optimal multicast scheduling policy is of threshold type.
Edit intent: fluency
Original:  To further reduce the complexity, we propose a low complexity suboptimal policy, which possesses similar structural properties to the optimal policy, and develop a low complexity algorithm to obtain this policy.
Revised:  We also propose a low-complexity suboptimal policy, which possesses similar structural properties to the optimal policy, and develop a low complexity algorithm to obtain this policy.
Edit intent: coherence
Original:  To further reduce the complexity, we propose a low complexity suboptimal policy, which possesses similar structural properties to the optimal policy, and develop a low complexity algorithm to obtain this policy.
Revised:  To further reduce the complexity, we propose a low complexity suboptimal policy, which possesses similar structural properties to the optimal policy, and develop a low-complexity algorithm to obtain this policy.
Edit intent: fluency
Original: Historical economic growth is analysedusing the method of reciprocal values .
Revised: Data describing historical economic growth are analysed .
Edit intent: clarity
Original:  The analysis demonstrates that the natural tendency for the historical economic growth was to increase hyperbolically .
Revised:  The analysis demonstrates that the natural tendency for the historical economic growth and suggests new lines of research .
Edit intent: clarity
Original:  Its higher implementation complexity, however , might make it difficult to use in legacy codes  .
Revised:  The higher implementation complexity, however , might make it difficult to use in legacy codes .
Edit intent: fluency
Original:  Its higher implementation complexity, however , might make it difficult to use in legacy codes  .
Revised:  Its higher implementation complexity, however , might make it difficult to use in legacy codes , though .
Edit intent: coherence
Original:  The paper introduces an OpenMP implementation of pipelined Parareal and compares it to a standard MPI-based implementation .
Revised: For the parallel-in-time integration method Parareal, pipelining can be used to hide some of the cost of the serial correction step and improve its efficiency. The paper introduces an OpenMP implementation of pipelined Parareal and compares it to a standard MPI-based implementation .
Edit intent: coherence
Original:  The paper introduces an OpenMP implementation of pipelined Parareal and compares it to a standard MPI-based implementation .
Revised:  The paper introduces a basic OpenMP implementation of pipelined Parareal and compares it to a standard MPI-based implementation .
Edit intent: clarity
Original:  The paper introduces an OpenMP implementation of pipelined Parareal and compares it to a standard MPI-based implementation .
Revised:  The paper introduces an OpenMP implementation of pipelined Parareal and compares it to a standard MPI-based variant .
Edit intent: clarity
Original:  Both versions yield essentially identical runtimes, but, depending on the compiler, the OpenMP variant consumes about 7\% less energy .
Revised:  Both versions yield almost identical runtimes, but, depending on the compiler, the OpenMP variant consumes about 7\% less energy .
Edit intent: clarity
Original:  The higher implementation complexity , including manual control of locks, might make it difficult to use in legacy codes , though .
Revised:  However, its higher implementation complexity , including manual control of locks, might make it difficult to use in legacy codes , though .
Edit intent: coherence
Original:  The higher implementation complexity , including manual control of locks, might make it difficult to use in legacy codes , though .
Revised:  The higher implementation complexity might make it difficult to use in legacy codes , though .
Edit intent: clarity