wanyu/IteraTeR_v2 huggingface dataset:
Original:  Such an improvement from time \Omega(n!) to polynomial time O(n^2) is possible because our algorithm works directly with the transposition tree on n vertices and does not require examining any of the permutations (only the proof requires examining the permutations).
Revised:  Such an improvement to polynomial time O(n^2) is possible because our algorithm works directly with the transposition tree on n vertices and does not require examining any of the permutations (only the proof requires examining the permutations).
Edit intent: clarity
Original:  It is also shown that the value computed by our algorithm is not necessarily unique, which is an important result because such trees are quite rare.
Revised:  It is also shown that the value computed by our algorithm is not necessarily unique, which is an important result because such examples are quite rare.
Edit intent: clarity
Original:  For all families of trees examined so far, the value computed by our algorithm happens to also be an upper bound on the diameter.
Revised:  For all families of trees we have investigated so far, the value computed by our algorithm happens to also be an upper bound on the diameter.
Edit intent: clarity
Original:  For all families of trees examined so far, the value computed by our algorithm happens to also be an upper bound on the diameter.
Revised:  For all families of trees examined so far, each of the possible values computed by our algorithm happens to also be an upper bound on the diameter.
Edit intent: clarity
Original:  In addition to providing a visual representation of the network topology at each time step, the sequence should preserve the mental map between layouts of consecutive time steps to allow a human to interpret the temporal evolution of the network.
Revised:  In addition to providing a visual representation of the network structure at each time step, the sequence should preserve the mental map between layouts of consecutive time steps to allow a human to interpret the temporal evolution of the network.
Edit intent: clarity
Original:  The proposed framework involves optimizing a modified cost function that augments the cost function of a static graph layout algorithm with a grouping penalty, which encourages nodes to stay close to other nodes belonging to the same group, and a temporal penalty, which encourages smooth movements of the nodes over time .
Revised:  The proposed framework creates regularized graph layouts by augmenting the cost function of a static graph layout algorithm with a grouping penalty, which encourages nodes to stay close to other nodes belonging to the same group, and a temporal penalty, which encourages smooth movements of the nodes over time .
Edit intent: clarity
Original:  We introduce two dynamic layout algorithms under this framework, namely dynamic multidimensional scaling (DMDS) and dynamic graph Laplacian layout (DGLL) , that are regularized versions of their static counterparts .
Revised:  We introduce two dynamic layout algorithms within the proposed framework, namely dynamic multidimensional scaling (DMDS) and dynamic graph Laplacian layout (DGLL) , that are regularized versions of their static counterparts .
Edit intent: clarity
Original:  We introduce two dynamic layout algorithms under this framework, namely dynamic multidimensional scaling (DMDS) and dynamic graph Laplacian layout (DGLL) , that are regularized versions of their static counterparts .
Revised:  We introduce two dynamic layout algorithms under this framework, namely dynamic multidimensional scaling (DMDS) and dynamic graph Laplacian layout (DGLL) .
Edit intent: coherence
Original:  We apply the proposed algorithms on several data sets to illustrate the importance of  regularization for producing interpretable visualizations of dynamic networks.
Revised:  We apply these algorithms on several data sets to illustrate the importance of regularization for producing interpretable visualizations of dynamic networks.
Edit intent: clarity
Original:  The question of the computational complexity of deciding what player has a winning strategy in an instance of this game was raised in the '90s by Goldstein and Reingold.
Revised:  The question of the computational complexity of deciding this game was raised in the '90s by Goldstein and Reingold.
Edit intent: clarity
Original: We prove the following results concerning the combinatorics of list decoding , motivated by the exponential gap between the known upper bound (of O(1/\gamma)) and lower bound (of \Omega_p(\log (1/\gamma))) for the list-size needed to decode up to radius p with rate \gamma away from capacity, i.e., 1-\h(p)-\gamma (here p\in (0,1/2) and \gamma > 0).
Revised: We study certain combinatorial aspects of list-decoding , motivated by the exponential gap between the known upper bound (of O(1/\gamma)) and lower bound (of \Omega_p(\log (1/\gamma))) for the list-size needed to decode up to radius p with rate \gamma away from capacity, i.e., 1-\h(p)-\gamma (here p\in (0,1/2) and \gamma > 0).
Edit intent: clarity
Original:  (The usual notion of list decoding corresponds to replacing%DIFDELCMD < {\em %%% average radius by the%DIFDELCMD < {\em %%% minimum radius of an enclosing Hamming ball. ) The remaining results are for the usual notion of list decoding : 1. We give a short simple proof, over all fixed alphabets, of the above-mentioned \Omega_p(\log ( 1/ \gamma)) lower bound due to Blinovsky.
Revised:  (The usual notion of list decoding corresponds to replacing%DIFDELCMD < {\em %%% radius by the%DIFDELCMD < {\em %%% minimum radius of an enclosing Hamming ball. ) The remaining results are for the usual notion of list decoding : 1. We give a short simple proof, over all fixed alphabets, of the above-mentioned \Omega_p(\log ( 1/ \gamma)) lower bound due to Blinovsky.
Edit intent: clarity
Original:  (The usual notion of list decoding corresponds to replacing%DIFDELCMD < {\em %%% average radius by the%DIFDELCMD < {\em %%% minimum radius of an enclosing Hamming ball. ) The remaining results are for the usual notion of list decoding : 1. We give a short simple proof, over all fixed alphabets, of the above-mentioned \Omega_p(\log ( 1/ \gamma)) lower bound due to Blinovsky.
Revised:  (The usual notion of list decoding corresponds to replacing%DIFDELCMD < {\em %%% average %DIFDELCMD < {\em %%% minimum radius of an enclosing Hamming ball. ) The remaining results are for the usual notion of list decoding : 1. We give a short simple proof, over all fixed alphabets, of the above-mentioned \Omega_p(\log ( 1/ \gamma)) lower bound due to Blinovsky.
Edit intent: clarity
Original:  (The usual notion of list decoding corresponds to replacing%DIFDELCMD < {\em %%% average radius by the%DIFDELCMD < {\em %%% minimum radius of an enclosing Hamming ball. ) The remaining results are for the usual notion of list decoding : 1. We give a short simple proof, over all fixed alphabets, of the above-mentioned \Omega_p(\log ( 1/ \gamma)) lower bound due to Blinovsky.
Revised:  (The usual notion of list decoding corresponds to replacing%DIFDELCMD < {\em %%% average radius by the%DIFDELCMD < {\em %%% radius of an enclosing Hamming ball. ) The remaining results are for the usual notion of list decoding : 1. We give a short simple proof, over all fixed alphabets, of the above-mentioned \Omega_p(\log ( 1/ \gamma)) lower bound due to Blinovsky.
Edit intent: clarity
Original:  (The usual notion of list decoding corresponds to replacing%DIFDELCMD < {\em %%% average radius by the%DIFDELCMD < {\em %%% minimum radius of an enclosing Hamming ball. ) The remaining results are for the usual notion of list decoding : 1. We give a short simple proof, over all fixed alphabets, of the above-mentioned \Omega_p(\log ( 1/ \gamma)) lower bound due to Blinovsky.
Revised:  (The usual notion of list decoding corresponds to replacing%DIFDELCMD < {\em %%% average radius by the%DIFDELCMD < {\em %%% minimum remaining results are for the usual notion of list decoding : 1. We give a short simple proof, over all fixed alphabets, of the above-mentioned \Omega_p(\log ( 1/ \gamma)) lower bound due to Blinovsky.
Edit intent: clarity
Original:  We prove that (a)  there are no positive constants \alpha and C such that every set K of  vectors with weight p contains a subset K' with |K'|\ge |K|^{\alpha} and {\rm dr}(K')\le C,  even when |K|\ge 2^p , (b) for a set K of vectors with weight p, and a constant C>2, there exists K'\subseteq K such that {\rm dr}(K  )\le C and K' \ge |K|^\alpha, where \alpha = 1/ \lceil \log(p/2)/\log(C/2) \rceil.
Revised:  We prove that (a) there are no positive constants \alpha and C such that every set K of vectors with weight p contains a subset K' with |K'|\ge |K|^{\alpha} and {\rm dr}(K')\le C, even when |K|\ge 2^p , (b) For a set K of vectors with weight p, and a constant C>2, there exists K'\subseteq K such that {\rm dr}(K )\le C and K' \ge |K|^\alpha, where \alpha = 1/ \lceil \log(p/2)/\log(C/2) \rceil.
Edit intent: fluency
Original:  We prove that (a)  there are no positive constants \alpha and C such that every set K of  vectors with weight p contains a subset K' with |K'|\ge |K|^{\alpha} and {\rm dr}(K')\le C,  even when |K|\ge 2^p , (b) for a set K of vectors with weight p, and a constant C>2, there exists K'\subseteq K such that {\rm dr}(K  )\le C and K' \ge |K|^\alpha, where \alpha = 1/ \lceil \log(p/2)/\log(C/2) \rceil.
Revised:  We prove that (a) there are no positive constants \alpha and C such that every set K of vectors with weight p contains a subset K' with |K'|\ge |K|^{\alpha} and {\rm dr}(K')\le C, even when |K|\ge 2^p , (b) for a set K of vectors with weight p, and a constant C>2, there exists K'\subseteq K such that {\rm dr}(K ' )\le C and K' \ge |K|^\alpha, where \alpha = 1/ \lceil \log(p/2)/\log(C/2) \rceil.
Edit intent: fluency
Original: The effects of bubbles on the cerebral bloodflow are difficult to quantify.
Revised: Bubbles introduced to the arterial circulation during invasive medical procedures can have devastating consequences for brain function but their effects are currently difficult to quantify.
Edit intent: clarity
Original:  The model takes into account realistic adhesion forces,  fluid dynamical considerations, a realistic parameterisaton for the rate of bubble dissolutionand the effects of buoyancy .
Revised:  The model takes into account realistic adhesion forces, fluid dynamical considerations, and bubble dissolution. This allows investigation of the effects of buoyancy .
Edit intent: clarity
Original: In this paper we extend our previous investigations on systemic features of the immune system, based on a statistical mechanics approach .
Revised: In this work we adopt a statistical mechanics approach .
Edit intent: clarity
Original:  Let F be a differential field of characteristic 0 and \mathcal{F be the field of differential rational functions generated by a single indeterminate u.
Revised:  Let F be a differential field of characteristic 0 and \mathcal{F be the field of differential rational functions generated by a single indeterminate u.
Edit intent: fluency
Original:  Let be given non constant rational functions v_1,...  v_n\in \mathcal{F subfield G\subseteq \mathcal{F .
Revised:  Let be given non constant rational functions v_1,... , v_n\in \mathcal{F subfield G\subseteq \mathcal{F .
Edit intent: fluency
Original:  Let be given non constant rational functions v_1,...  v_n\in \mathcal{F subfield G\subseteq \mathcal{F .
Revised:  Let be given non constant rational functions v_1,... v_n\in \mathcal{F subfield G\subseteq \mathcal{F .
Edit intent: fluency
Original:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j \text{ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \text{ord \textrm{ (v_j) and d:=\max_j \text{deg \textrm{ (v_j).
Revised:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j \text{ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \text{ord \textrm{ (v_j) and d:=\max_j \text{deg \textrm{ (v_j).
Edit intent: fluency
Original:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j \text{ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \text{ord \textrm{ (v_j) and d:=\max_j \text{deg \textrm{ (v_j).
Revised:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \text{ord \textrm{ (v_j) and d:=\max_j \text{deg \textrm{ (v_j).
Edit intent: fluency
Original:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j \text{ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \text{ord \textrm{ (v_j) and d:=\max_j \text{deg \textrm{ (v_j).
Revised:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j \text{ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \textrm{ (v_j) and d:=\max_j \text{deg \textrm{ (v_j).
Edit intent: fluency
Original:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j \text{ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \text{ord \textrm{ (v_j) and d:=\max_j \text{deg \textrm{ (v_j).
Revised:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j \text{ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \text{ord \textrm{ord (v_j) and d:=\max_j \text{deg \textrm{ (v_j).
Edit intent: fluency
Original:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j \text{ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \text{ord \textrm{ (v_j) and d:=\max_j \text{deg \textrm{ (v_j).
Revised:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j \text{ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \text{ord \textrm{ (v_j) and d:=\max_j \textrm{ (v_j).
Edit intent: clarity
Original:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j \text{ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \text{ord \textrm{ (v_j) and d:=\max_j \text{deg \textrm{ (v_j).
Revised:  The differential L\"uroth's theorem proved by Ritt in 1932 states that there exists v\in \mathcal G such that G= \mathcal{F . Here we prove that the total order and degree of a generator v are bounded by \min_j \text{ord (v_j) and (nd(e+1)+1)^{2e+1}, respectively, where e:=\max_j \text{ord \textrm{ (v_j) and d:=\max_j \text{deg \textrm{deg (v_j).
Edit intent: fluency
Original:   Furthermore, we show that saving the maximum number of lemmings is APX-hard, even when only Climber skills are available.
Revised:  Furthermore, we show that saving the maximum number of lemmings is APX-hard, even when only Climbers are available.
Edit intent: clarity
Original:  This contrasts with the membership in PO of the same optimization problem restricted to landscapes with no "deadly areas" (such as water or traps) and only Climber and Floater skills , as previously established by Cormode.
Revised:  This contrasts with the membership in P of the decision problem restricted to landscapes with no "deadly areas" (such as water or traps) and only Climber and Floater skills , as previously established by Cormode.
Edit intent: clarity
Original:  This contrasts with the membership in PO of the same optimization problem restricted to landscapes with no "deadly areas" (such as water or traps) and only Climber and Floater skills , as previously established by Cormode.
Revised:  This contrasts with the membership in PO of the same optimization problem restricted to levels with no "deadly areas" (such as water or traps) and only Climber and Floater skills , as previously established by Cormode.
Edit intent: clarity
Original:  This contrasts with the membership in PO of the same optimization problem restricted to landscapes with no "deadly areas" (such as water or traps) and only Climber and Floater skills , as previously established by Cormode.
Revised:  This contrasts with the membership in PO of the same optimization problem restricted to landscapes with no "deadly areas" (such as water or traps) and only Climbers and Floaters , as previously established by Cormode.
Edit intent: clarity
Original:  However, if we restrict to levels with only polynomially many available Builders, Bashers, and Miners, then the game is solvable  NP.
Revised:  However, if we restrict to levels with only polynomially many available Builders, Bashers, and Miners, then the game is solvable in NP.
Edit intent: fluency
Original:  Furthermore, we show that saving the maximum number of lemmings is APX-hard, even when only Climbers are available.
Revised:  Furthermore, we show that saving the maximum number of lemmings is APX-hard, even when only Climber skills are available.
Edit intent: clarity
Original:  This contrasts with the membership in P of the decision problem restricted to levels with no "deadly areas" (such as water or traps) and only Climbers and Floaters , as previously established by Cormode.
Revised:  This contrasts with the membership in P of the decision problem restricted to levels with no "deadly areas" (such as water or traps) and only Climber and Floater skills , as previously established by Cormode.
Edit intent: clarity
Original:  However  , if we restrict the game to levels in which the available Builder , Basher, and Miner skills are only polynomially many (and there is any number of other skills), then the game is solvable in NP.
Revised:  However we also prove that , if we restrict the game to levels in which the available Builder , Basher, and Miner skills are only polynomially many (and there is any number of other skills), then the game is solvable in NP.
Edit intent: coherence
Original:  Robert Solovay 's showed 10 that infinitely many strings x have maximal plain complexity but not maximal prefix-free complexity (among the strings of the same length) ;
Revised:  Robert Solovay 10 that infinitely many strings x have maximal plain complexity but not maximal prefix-free complexity (among the strings of the same length) ;
Edit intent: clarity
Original:  Robert Solovay 's showed 10 that infinitely many strings x have maximal plain complexity but not maximal prefix-free complexity (among the strings of the same length) ;
Revised:  Robert Solovay 's showed 10 that infinitely many strings x have maximal plain complexity but not maximal prefix complexity (among the strings of the same length) ;
Edit intent: clarity
Original:  i.e. for some c and infinitely many x : |x| - C(x) \leq c and |x| + K(|x|) - K(x) \geq loglog |x| - c log log log |x|.
Revised:  i.e. for some c there exist infinitely many x : |x| - C(x) \leq c and |x| + K(|x|) - K(x) \geq loglog |x| - c log log log |x|.
Edit intent: clarity
Original:  i.e. for some c and infinitely many x : |x| - C(x) \leq c and |x| + K(|x|) - K(x) \geq loglog |x| - c log log log |x|.
Revised:  i.e. for some c and infinitely many x such that |x| - C(x) \leq c and |x| + K(|x|) - K(x) \geq loglog |x| - c log log log |x|.
Edit intent: clarity
Original:   Using the result above, we provide a short proof of Solovay's result.
Revised:  Using the result above, we provide a short proof for Solovay's result.
Edit intent: fluency
Original:  We also generalize it by showing that for some c and for all n there are strings x of length n with n-C (x) \leq c , and n + K(n) - K(x) \geq K(K(n)|n) - 3 K(K(K(n)|n)|n) - c. This is very close to the upperbound K(K(n)|n) + O(1) proved by Solovay .
Revised:  We also generalize it by showing that for some c and for all n there are strings x of length n with n - C (x) \leq c , and n + K(n) - K(x) \geq K(K(n)|n) - 3 K(K(K(n)|n)|n) - c. This is very close to the upperbound K(K(n)|n) + O(1) proved by Solovay .
Edit intent: fluency
Original:  We also generalize it by showing that for some c and for all n there are strings x of length n with n-C (x) \leq c , and n + K(n) - K(x) \geq K(K(n)|n) - 3 K(K(K(n)|n)|n) - c. This is very close to the upperbound K(K(n)|n) + O(1) proved by Solovay .
Revised:  We also generalize it by showing that for some c and for all n there are strings x of length n with n-C (x) \leq c , and n + K(n) - K(x) > K(K(n)|n) - 3 K(K(K(n)|n)|n) - c. This is very close to the upperbound K(K(n)|n) + O(1) proved by Solovay .
Edit intent: clarity
Original:  We also generalize it by showing that for some c and for all n there are strings x of length n with n-C (x) \leq c , and n + K(n) - K(x) \geq K(K(n)|n) - 3 K(K(K(n)|n)|n) - c. This is very close to the upperbound K(K(n)|n) + O(1) proved by Solovay .
Revised:  We also generalize it by showing that for some c and for all n there are strings x of length n with n-C (x) \leq c , and n + K(n) - K(x) \geq K(K(n)|n) - 3 K(K(K(n)|n)|n) - c. We also prove a close upper bound K(K(n)|n) + O(1) proved by Solovay .
Edit intent: coherence
Original: Was the spread of agropastoralism from the Fertile Crescent throughout Europe influenced by rapid climatic shifts ?
Revised: Was the spread of agropastoralism from the Fertile Crescent throughout Europe influenced by extreme climate events, or was it independent of climate ?
Edit intent: clarity
Original:  Our study shows that the inclusion of climate events improves the simulation of typical lags between cultural complexes, but that the overall difference to a model without climate events is not significant.
Revised:  Our study shows that the consideration of climate events improves the simulation of typical lags between cultural complexes, but that the overall difference to a model without climate events is not significant.
Edit intent: clarity
Original:  We introduce a framework that models \emph{online voter control in sequential elections .
Revised:  We introduce a framework that models \emph{online voter control in sequential elections .
Edit intent: fluency
Original:  In addition, we obtain  completeness for coNP in the deleting/adding cases with a bounded deletion/addition limit, and  for NP in the partition cases with only one candidate.
Revised:  In addition, we obtain completeness for coNP in the deleting/adding cases with a bounded deletion/addition limit, and we obtain completeness for NP in the partition cases with only one candidate.
Edit intent: coherence
Original:  A checking test for a read-once function f over B depending essentially on all its variables is a set of input vectors distinguishing f from all other read-once functions of the same variables.
Revised:  A checking test for a read-once function f over B depending on all its variables is a set of input vectors distinguishing f from all other read-once functions of the same variables.
Edit intent: clarity