wanyu/IteraTeR_v2 huggingface dataset:
Original:  Together with existing bounds due to Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch, we give the following essentially tight bounds on DS_s(n) for all s : DS_s(n) = n, for s=1, 2n-1, for s=2, Theta(n alpha(n)), for s=3, Theta(n 2^{alpha(n)), for s =4, Omega(nalpha(n) 2^{alpha(n)}) and O(nalpha^2(n) 2^{alpha(n)}), for s =5, n 2^{(1 \pm o(1)) alpha^t(n)/t!}, for s}%DIFDELCMD < \geq6 %%% and t = floor (s-2 ) /2 These bounds refute\le  conjectures due to Alon et al. , Nivasch, and Pettie .
Revised:  Together with existing bounds due to Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch, we give the following essentially tight bounds on DS_s(n) for all s : DS_s(n) = n, for s=1, 2n-1, for s=2, Theta(n alpha(n)), for s=3, Theta(n 2^{alpha(n)), for s =4, Omega(nalpha(n) 2^{alpha(n)}) and O(nalpha^2(n) 2^{alpha(n)}), for s =5, n 2^{(1 \pm o(1)) alpha^t(n)/t!}, for s}%DIFDELCMD < \geq6 %%% (s-2 ) /2 These bounds refute\le conjectures due to Alon et al. , Nivasch, and Pettie .
Edit intent: clarity
Original:  Together with existing bounds due to Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch, we give the following essentially tight bounds on DS_s(n) for all s : DS_s(n) = n, for s=1, 2n-1, for s=2, Theta(n alpha(n)), for s=3, Theta(n 2^{alpha(n)), for s =4, Omega(nalpha(n) 2^{alpha(n)}) and O(nalpha^2(n) 2^{alpha(n)}), for s =5, n 2^{(1 \pm o(1)) alpha^t(n)/t!}, for s}%DIFDELCMD < \geq6 %%% and t = floor (s-2 ) /2 These bounds refute\le  conjectures due to Alon et al. , Nivasch, and Pettie .
Revised:  Together with existing bounds due to Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch, we give the following essentially tight bounds on DS_s(n) for all s : DS_s(n) = n, for s=1, 2n-1, for s=2, Theta(n alpha(n)), for s=3, Theta(n 2^{alpha(n)), for s =4, Omega(nalpha(n) 2^{alpha(n)}) and O(nalpha^2(n) 2^{alpha(n)}), for s =5, n 2^{(1 \pm o(1)) alpha^t(n)/t!}, for s}%DIFDELCMD < \geq6 %%% and t = floor Klazar, and Nivasch ) /2 These bounds refute\le conjectures due to Alon et al. , Nivasch, and Pettie .
Edit intent: coherence
Original:  Together with existing bounds due to Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch, we give the following essentially tight bounds on DS_s(n) for all s : DS_s(n) = n, for s=1, 2n-1, for s=2, Theta(n alpha(n)), for s=3, Theta(n 2^{alpha(n)), for s =4, Omega(nalpha(n) 2^{alpha(n)}) and O(nalpha^2(n) 2^{alpha(n)}), for s =5, n 2^{(1 \pm o(1)) alpha^t(n)/t!}, for s}%DIFDELCMD < \geq6 %%% and t = floor (s-2 ) /2 These bounds refute\le  conjectures due to Alon et al. , Nivasch, and Pettie .
Revised:  Together with existing bounds due to Hart and Sharir, Agarwal, Sharir, and Shor, and Nivasch, we give the following essentially tight bounds on DS_s(n) for all s : DS_s(n) = n, for s=1, 2n-1, for s=2, Theta(n alpha(n)), for s=3, Theta(n 2^{alpha(n)), for s =4, Omega(nalpha(n) 2^{alpha(n)}) and O(nalpha^2(n) 2^{alpha(n)}), for s =5, n 2^{(1 \pm o(1)) alpha^t(n)/t!}, for s}%DIFDELCMD < \geq6 %%% and t = floor (s-2 ) These bounds refute\le conjectures due to Alon et al. , Nivasch, and Pettie .
Edit intent: clarity
Original: Approximating the length of the longest increasing sequence (LIS) of a data stream is a well-studied problem.
Revised: Approximating the length of the longest increasing sequence (LIS) of an array is a well-studied problem.
Edit intent: clarity
Original:  There are many algorithms that estimate the size of the complement of the LIS, referred to as the\emph{distance to monotonicity , both in the streaming and property testing setting.
Revised:  \emph{distance to monotonicity , both in the streaming and property testing setting.
Edit intent: coherence
Original:  There are many algorithms that estimate the size of the complement of the LIS, referred to as the\emph{distance to monotonicity , both in the streaming and property testing setting.
Revised:  There are many algorithms that estimate the size of the complement of the LIS, referred to as the\emph{ , both in the streaming and property testing setting.
Edit intent: clarity
Original:  For any \delta > 0, our algorithm provides a (1+\delta)-multiplicative approximation for the distance  , and uses only O((\log^2 n)/\delta) space.
Revised:  For any \delta > 0, our algorithm provides a (1+\delta)-multiplicative approximation to the , and uses only O((\log^2 n)/\delta) space.
Edit intent: clarity
Original:  The previous best known approximation using poly-logarithmic space was a multiplicative 2-factor.
Revised:  The previous best known approximation using polylogarithmic space was a multiplicative 2-factor.
Edit intent: fluency
Original:  We are able to get a streaming algorithm for the longest common subsequence problem ( in the asymmetric setting introduced in \mbox{ AKO10 symbol appears very many times.
Revised:  We are able to get a streaming algorithm for the longest common subsequence problem ( in the asymmetric setting (inspired by \mbox{ AKO10 symbol appears very many times.
Edit intent: clarity
Original:  The size of the complement of the LCS is the edit distance between x and y with only insertions and deletions .
Revised:  The size of the complement of the LCS is the edit distance between x and y when insertions and deletions .
Edit intent: clarity
Original: -- the first with complexity sub-linear in q when t is fixed --- is that the nonzero roots in F_q can be partitioned into at most 2 t-1 q ^{(t-2)(t-1)} cosets of two subgroups S_1,S_2 of F^*_q, with S_1 in S_2. Another corollary is the first deterministic sub-linear algorithm for detecting common degree one factors of k-tuples of t-nomials in F_q[x] when k and t are fixed.
Revised: -- the first with complexity sub-linear in q when t is fixed --- is that the nonzero roots in F_q can be partitioned into at most 2 t-1 (q-1) ^{(t-2)(t-1)} cosets of two subgroups S_1,S_2 of F^*_q, with S_1 in S_2. Another corollary is the first deterministic sub-linear algorithm for detecting common degree one factors of k-tuples of t-nomials in F_q[x] when k and t are fixed.
Edit intent: clarity
Original:  If frames are restricted to the natural numbers or general linear orders, then satisfiability is known to be decidable, but of non-elementary complexity.
Revised:  If frames are restricted to N or general linear orders, then satisfiability is known to be decidable, but of non-elementary complexity.
Edit intent: clarity
Original:  In this paper, we consider monotone hybrid logics (i.e., the Boolean connectives are conjunction and disjunction only) over the natural numbers and general linear orders.
Revised:  In this paper, we consider monotone hybrid logics (i.e., the Boolean connectives are conjunction and disjunction only) over N and general linear orders.
Edit intent: clarity
Original:  We show that the satisfiability problem remains non-elementary over linear orders, but its complexity drops to PSPACE-completeness over natural numbers .
Revised:  We show that the satisfiability problem remains non-elementary over linear orders, but its complexity drops to PSPACE-completeness over N .
Edit intent: clarity
Original:  We categorize the strict fragments arising from different combinations of modal and hybrid operators into NP-complete and tractable , and show that the latter cases are complete for NC1 or LOGSPACE .
Revised:  We categorize the strict fragments arising from different combinations of modal and hybrid operators into NP-complete and tractable (i.e. complete for NC1or LOGSPACE) .
Edit intent: coherence
Original:  For the cases above NP, satisfiability over linear orders is harder than over natural numbers , while below NP it is at most as hard.
Revised:  For the cases above NP, satisfiability over linear orders is harder than over N , while below NP it is at most as hard.
Edit intent: clarity
Original:  In addition to the study of computational complexity, we examine model-theoretic properties of the fragments in question.
Revised:  In addition we examine model-theoretic properties of the fragments in question.
Edit intent: clarity
Original: We prove new lower bounds on the encoding length of Matching Vector (MV) codes.
Revised: A Matching Vector (MV) codes.
Edit intent: clarity
Original:  More precisely, families U=(u_1,...,u_t) , V=(v_1,...,v_t) with u_i,v_j \in %DIFDELCMD < \Z%%% _m^n such that =0 %DIFDELCMD < \mod %%% m for all i\in\ne \ne  [ t ] and \neq 0%DIFDELCMD < \mod %%% m for i \neq j.
Revised:  More precisely, families U=(u_1,...,u_t) and V=(v_1,...,v_t) with u_i,v_j \in %DIFDELCMD < \Z%%% _m^n such that =0 %DIFDELCMD < \mod %%% m for all i\in\ne \ne [ t ] and \neq 0%DIFDELCMD < \mod %%% m for i \neq j.
Edit intent: fluency
Original:  More precisely, families U=(u_1,...,u_t) , V=(v_1,...,v_t) with u_i,v_j \in %DIFDELCMD < \Z%%% _m^n such that =0 %DIFDELCMD < \mod %%% m for all i\in\ne \ne  [ t ] and \neq 0%DIFDELCMD < \mod %%% m for i \neq j.
Revised:  More precisely, families U=(u_1,...,u_t) , V=(v_1,...,v_t) where u_i,v_j \in %DIFDELCMD < \Z%%% _m^n such that =0 %DIFDELCMD < \mod %%% m for all i\in\ne \ne [ t ] and \neq 0%DIFDELCMD < \mod %%% m for i \neq j.
Edit intent: clarity
Original:  More precisely, families U=(u_1,...,u_t) , V=(v_1,...,v_t) with u_i,v_j \in %DIFDELCMD < \Z%%% _m^n such that =0 %DIFDELCMD < \mod %%% m for all i\in\ne \ne  [ t ] and \neq 0%DIFDELCMD < \mod %%% m for i \neq j.
Revised:  More precisely, families U=(u_1,...,u_t) , V=(v_1,...,v_t) with u_i,v_j \in %DIFDELCMD < \Z%%% _m^n such that =0 %DIFDELCMD < \mod %%% , and for any i\ne \ne [ t ] and \neq 0%DIFDELCMD < \mod %%% m for i \neq j.
Edit intent: clarity
Original:  More precisely, families U=(u_1,...,u_t) , V=(v_1,...,v_t) with u_i,v_j \in %DIFDELCMD < \Z%%% _m^n such that =0 %DIFDELCMD < \mod %%% m for all i\in\ne \ne  [ t ] and \neq 0%DIFDELCMD < \mod %%% m for i \neq j.
Revised:  More precisely, families U=(u_1,...,u_t) , V=(v_1,...,v_t) with u_i,v_j \in %DIFDELCMD < \Z%%% _m^n such that =0 %DIFDELCMD < \mod %%% m for all i\in\ne \ne [ t ] \neq 0%DIFDELCMD < \mod %%% m for i \neq j.
Edit intent: coherence
Original:  Our lower bounds for MV codes are obtained by improving the known%DIFDELCMD < {\em %%% upper bounds on such families-- a question that arises independently in combinatorics in the context of set systems with restricted modular intersections.
Revised:  Our lower bounds for MV codes are obtained by improving the known%DIFDELCMD < {\em %%% on such families-- a question that arises independently in combinatorics in the context of set systems with restricted modular intersections.
Edit intent: clarity
Original:  I derive asymptotic distribution of the bids/offers as a function of proportion between patient and impatient traders using my modification of Foucault, Kadan and Kandel dynamic Limit Order Book (LOB) model.
Revised:  I derive asymptotic distribution of the bids/offers as a function of the ratio of patient and impatient traders using my modification of Foucault, Kadan and Kandel dynamic Limit Order Book (LOB) model.
Edit intent: clarity
Original:  I derive asymptotic distribution of the bids/offers as a function of proportion between patient and impatient traders using my modification of Foucault, Kadan and Kandel dynamic Limit Order Book (LOB) model.
Revised:  I derive asymptotic distribution of the bids/offers as a function of proportion between patient and impatient traders using the dynamic version of the Foucault, Kadan and Kandel dynamic Limit Order Book (LOB) model.
Edit intent: clarity
Original:  In particular, dynamic LOB allows simulating the distribution of execution times and spreads from high-frequency quotes.
Revised:  In particular, dynamic LOB allows simulation of the distribution of execution times and spreads from high-frequency quotes.
Edit intent: fluency
Original:  Significant analytic progress is made towards future empirical study of trading as competition for immediacy of execution between traders.
Revised:  Significant analytic progress is made towards understanding of trading as competition for immediacy of execution between traders.
Edit intent: clarity
Original:  The results are qualitatively compared with empirical volumeat- price distribution of liquid stocks.
Revised:  The results are qualitatively compared with empirical volume-at-price distribution of highly liquid stocks.
Edit intent: clarity
Original:  In the noiseless scenario, we find that the learning times are distributed exponentially and the learning rates are given by ln {N(N-1)/[C + (N-1)^2]} in the case the Ntarget words are sampled randomly and by 1/N ln [(N-1)/C] in the case they follow a deterministic presentation sequence .
Revised:  In the noiseless scenario, we find that the learning times are distributed exponentially and the learning rates are given by ln {N(N-1)/[C + (N-1)^2]} in the case the N target words are sampled randomly and by 1/N ln [(N-1)/C] in the case they follow a deterministic presentation sequence .
Edit intent: fluency
Original:  In the noiseless scenario, we find that the learning times are distributed exponentially and the learning rates are given by ln {N(N-1)/[C + (N-1)^2]} in the case the Ntarget words are sampled randomly and by 1/N ln [(N-1)/C] in the case they follow a deterministic presentation sequence .
Revised:  In the noiseless scenario, we find that the learning times are distributed exponentially and the learning rates are given by ln {N(N-1)/[C + (N-1)^2]} in the case the Ntarget words are sampled randomly and by 1/N ln [(N-1)/C] in the case those words follow a deterministic presentation sequence .
Edit intent: clarity
Original:  We find that the learning times are distributed exponentially and the learning rates are given by ln N(N-1)/%DIFDELCMD < [%%% C + (N-1)^2 in the case the N target words are sampled randomly and by 1/N ln{N} \ln } [ (N-1)/C ] in the case those words follow a deterministic presentation sequence  .
Revised:  We find that the learning times are distributed exponentially and the learning rates are given by N(N-1)/%DIFDELCMD < [%%% C + (N-1)^2 in the case the N target words are sampled randomly and by 1/N ln{N} \ln } [ (N-1)/C ] in the case those words follow a deterministic presentation sequence .
Edit intent: coherence
Original:  We find that the learning times are distributed exponentially and the learning rates are given by ln N(N-1)/%DIFDELCMD < [%%% C + (N-1)^2 in the case the N target words are sampled randomly and by 1/N ln{N} \ln } [ (N-1)/C ] in the case those words follow a deterministic presentation sequence  .
Revised:  We find that the learning times are distributed exponentially and the learning rates are given by ln %DIFDELCMD < [%%% C + (N-1)^2 in the case the N target words are sampled randomly and by 1/N ln{N} \ln } [ (N-1)/C ] in the case those words follow a deterministic presentation sequence .
Edit intent: coherence
Original:  We find that the learning times are distributed exponentially and the learning rates are given by ln N(N-1)/%DIFDELCMD < [%%% C + (N-1)^2 in the case the N target words are sampled randomly and by 1/N ln{N} \ln } [ (N-1)/C ] in the case those words follow a deterministic presentation sequence  .
Revised:  We find that the learning times are distributed exponentially and the learning rates are given by ln N(N-1)/%DIFDELCMD < [%%% C + (N-1)^2 in the case the N target words are sampled randomly and by \frac{1{N} \ln } [ (N-1)/C ] in the case those words follow a deterministic presentation sequence .
Edit intent: coherence
Original:  We find that the learning times are distributed exponentially and the learning rates are given by ln N(N-1)/%DIFDELCMD < [%%% C + (N-1)^2 in the case the N target words are sampled randomly and by 1/N ln{N} \ln } [ (N-1)/C ] in the case those words follow a deterministic presentation sequence  .
Revised:  We find that the learning times are distributed exponentially and the learning rates are given by ln N(N-1)/%DIFDELCMD < [%%% C + (N-1)^2 in the case the N target words are sampled randomly and by 1/N ln{N} \ln } [ \frac{N-1 ] in the case those words follow a deterministic presentation sequence .
Edit intent: clarity
Original:  We find that the learning times are distributed exponentially and the learning rates are given by ln N(N-1)/%DIFDELCMD < [%%% C + (N-1)^2 in the case the N target words are sampled randomly and by 1/N ln{N} \ln } [ (N-1)/C ] in the case those words follow a deterministic presentation sequence  .
Revised:  We find that the learning times are distributed exponentially and the learning rates are given by ln N(N-1)/%DIFDELCMD < [%%% C + (N-1)^2 in the case the N target words are sampled randomly and by 1/N ln{N} \ln } [ (N-1)/C ] in the case they follow a deterministic presentation sequence .
Edit intent: clarity
Original:  As a consequence of our result, it is impossible to efficiently test for RIP provided NP%DIFDELCMD < \not%%% \subseteq BPP, an assumption which is slightly stronger than P \neq NP.
Revised:  As a consequence of our result, it is impossible to efficiently test for RIP provided %DIFDELCMD < \not%%% \subseteq BPP, an assumption which is slightly stronger than P \neq NP.
Edit intent: clarity
Original:  As a consequence of our result, it is impossible to efficiently test for RIP provided NP%DIFDELCMD < \not%%% \subseteq BPP, an assumption which is slightly stronger than P \neq NP.
Revised:  As a consequence of our result, it is impossible to efficiently test for RIP provided NP%DIFDELCMD < \not%%% P \neq NP.
Edit intent: coherence
Original:  We present a computational and theoretical study of  a many-body Brownian ratchet, in which a "gel " of multiple, stiff polymerizing filaments pushes a diffusing obstacle.
Revised:  We present a computational and theoretical study of a many-body Brownian ratchet, in the specific form of a growing polymer gel that pushes a diffusing obstacle.
Edit intent: clarity
Original:  Hence, it is quite clear the connection between  self-similar solution and the traveling wave solution from these results.
Revised:  Hence, it is quite clear the connection between the self-similar solution and the traveling wave solution from these results.
Edit intent: fluency
Original:  The theorem states that the (c,s)-approximation NP-hardness of UGP implies the ((\rho+1)c-\rho,(\rho+1)s-\rho)-approximation NP-hardness of GUGP(\rho) for any \rho>0 , which confirms that the region of (c,s)-approximation NP-hardness of GUGP(\rho) is significantly larger than the known region of (c,s)-approximation NP-hardness of UGP.
Revised:  The theorem states that the (c,s)-approximation NP-hardness of UGP implies the ((\rho+1)c-\rho,(\rho+1)s-\rho)-approximation NP-hardness of GUGP(\rho) for any \rho>0 The theorem confirms that the region of (c,s)-approximation NP-hardness of GUGP(\rho) is significantly larger than the known region of (c,s)-approximation NP-hardness of UGP.
Edit intent: coherence
Original:  In addition , the author presents another gap amplification theorem assuming a to-be-proven parallel repetition lemma, which claims harder approximation of GUGP(\rho) than the first theorem from the (c, s)-approximation NP-hardness of UGP satisfying 1-c<\alpha(1-s), where \alpha>0 is a universal constant.
Revised:  As a corollary , the author presents another gap amplification theorem assuming a to-be-proven parallel repetition lemma, which claims harder approximation of GUGP(\rho) than the first theorem from the (c, s)-approximation NP-hardness of UGP satisfying 1-c<\alpha(1-s), where \alpha>0 is a universal constant.
Edit intent: coherence
Original:  In addition , the author presents another gap amplification theorem assuming a to-be-proven parallel repetition lemma, which claims harder approximation of GUGP(\rho) than the first theorem from the (c, s)-approximation NP-hardness of UGP satisfying 1-c<\alpha(1-s), where \alpha>0 is a universal constant.
Revised:  In addition , the author shows that UGC on GUGP(\rho) than the first theorem from the (c, s)-approximation NP-hardness of UGP satisfying 1-c<\alpha(1-s), where \alpha>0 is a universal constant.
Edit intent: clarity
Original: In the second paper of this series , the author introduces GUGP(\rho) model and defines the counterpart of UGC on GUGP(\rho), which is a generalization of UGP where the weights of edges are allowed to be negative, and the magnitude of the total weight of all satisfied negative edges is no more than \rho times the total weight of all positive edges.
Revised: In this paper , the author introduces GUGP(\rho) model and defines the counterpart of UGC on GUGP(\rho), which is a generalization of UGP where the weights of edges are allowed to be negative, and the magnitude of the total weight of all satisfied negative edges is no more than \rho times the total weight of all positive edges.
Edit intent: clarity
Original:  the author  strengthens the previous known hardness for approximating Minimum Unique Game Problem ,
Revised:  the author strengthens the previous known hardness for approximating Minimum Unique Game ,
Edit intent: clarity
Original:  5/4-\epsilon, by proving that Min 2Lin-2 is NP-hard to be approximated within 2-\epsilon and strengthens the previous known hardness for approximating Small Set Expansion Problem , 4/3-\epsilon, by proving that Min Bisection is NP-hard to be approximated within 3-\epsilon , assuming a variation of Feige's Hypothesis, which claims that refuting Unbalanced Biased Max 3XOR is NP-hard on average on a natural distribution .
Revised:  5/4-\epsilon, by proving that Min 2-Lin-2 is hard to within 3/ 2-\epsilon and strengthens the previous known hardness for approximating Small Set Expansion Problem , 4/3-\epsilon, by proving that Min Bisection is NP-hard to be approximated within 3-\epsilon , assuming a variation of Feige's Hypothesis, which claims that refuting Unbalanced Biased Max 3XOR is NP-hard on average on a natural distribution .
Edit intent: clarity
Original:  5/4-\epsilon, by proving that Min 2Lin-2 is NP-hard to be approximated within 2-\epsilon and strengthens the previous known hardness for approximating Small Set Expansion Problem , 4/3-\epsilon, by proving that Min Bisection is NP-hard to be approximated within 3-\epsilon , assuming a variation of Feige's Hypothesis, which claims that refuting Unbalanced Biased Max 3XOR is NP-hard on average on a natural distribution .
Revised:  5/4-\epsilon, by proving that Min 2Lin-2 is NP-hard to be approximated within 2-\epsilon and strengthens the previous known hardness for approximating Small Set Expansion , 4/3-\epsilon, by proving that Min Bisection is NP-hard to be approximated within 3-\epsilon , assuming a variation of Feige's Hypothesis, which claims that refuting Unbalanced Biased Max 3XOR is NP-hard on average on a natural distribution .
Edit intent: clarity
Original:  5/4-\epsilon, by proving that Min 2Lin-2 is NP-hard to be approximated within 2-\epsilon and strengthens the previous known hardness for approximating Small Set Expansion Problem , 4/3-\epsilon, by proving that Min Bisection is NP-hard to be approximated within 3-\epsilon , assuming a variation of Feige's Hypothesis, which claims that refuting Unbalanced Biased Max 3XOR is NP-hard on average on a natural distribution .
Revised:  5/4-\epsilon, by proving that Min 2Lin-2 is NP-hard to be approximated within 2-\epsilon and strengthens the previous known hardness for approximating Small Set Expansion Problem , 4/3-\epsilon, by proving that Min Bisection is hard to approximate within 3-\epsilon , assuming a variation of Feige's Hypothesis, which claims that refuting Unbalanced Biased Max 3XOR is NP-hard on average on a natural distribution .
Edit intent: clarity
Original: In this paper, the author puts forward a variation of Feige's Hypothesis, which claims that it is hard on average refuting Unbalanced Max 3-XOR against biased assignments on a natural distribution.
Revised: In this paper, the author puts forward a variation of Feige's Hypothesis, which claims that it is hard on average refuting Unbalanced Max 3-XOR under biased assignments on a natural distribution.
Edit intent: clarity