wanyu/IteraTeR_v2 huggingface dataset:
Original: We study the complexity of computing or approximating an equilibrium for a given finite n-player extensive form game of perfect recall (EFGPR), where n >= 3. Our results apply not only to Nash equilibrium (NE), but also to various important refinements of NE for EFGPRs, including : subgame-perfect equilibrium in behavior strategies, sequential equilibrium (SE), and extensive form trembling-hand perfect equilibrium (PE) . Of these, the most refined notion is PE .
Revised: We study the complexity of computing or approximating an equilibrium for a given finite n-player extensive form game of perfect recall (EFGPR), where n >= 3. Our results apply not only to Nash equilibrium (NE), but also to various important refinements of NE for EFGPRs, including : subgame-perfect equilibrium in behavior strategies, sequential equilibrium (SE), extensive-form perfect (PE) . Of these, the most refined notion is PE .
Edit intent: clarity
Original:  We also show that computing a " delta-almost subgame-perfect equilibrium " in behavior strategies for a given EFGPR and given delta > 0, is PPAD-complete.
Revised:  We also show that computing a " almost " in behavior strategies for a given EFGPR and given delta > 0, is PPAD-complete.
Edit intent: clarity
Original:  We also show that computing a " delta-almost subgame-perfect equilibrium " in behavior strategies for a given EFGPR and given delta > 0, is PPAD-complete.
Revised:  We also show that computing a " delta-almost subgame-perfect equilibrium " equilibrium notions, and show that they are PPAD-complete.
Edit intent: clarity
Original:  In doing so, we also define the more refined notions of "delta-almost epsilon-(quasi-)perfect" equilibrium, and show that computing one is PPAD-complete.
Revised:  In particular, we define "delta-almost epsilon-(quasi-)perfect" equilibrium, and show that computing one is PPAD-complete.
Edit intent: clarity
Original:  In doing so, we also define the more refined notions of "delta-almost epsilon-(quasi-)perfect" equilibrium, and show that computing one is PPAD-complete.
Revised:  In doing so, we also define the more refined notions of "delta-almost epsilon-(quasi-)perfect" equilibrium, and show computing one is PPAD-complete.
Edit intent: coherence
Original:  For 2-player EFGPRs, analogous complexity results follow from the algorithms of Koller, Megiddo, and von Stengel (1996), and von Stengel, van den Elzen, and Talman (2002) .
Revised:  For 2-player EFGPRs, analogous complexity results follow from the algorithms of Koller, Megiddo, and von Stengel (1996), von Stengel, van den Elzen, and Talman (2002) .
Edit intent: fluency
Original:  Greibach's Theorem applies to languages characterized by PRE . The decidability of universality of PRE over a single letter alphabet is linked to the existence of Fermat Primes  .
Revised:  Greibach's Theorem applies to languages characterized by PRE . The decidability of universality of PRE over a single letter alphabet is linked to the existence of Fermat Primes . Greibach's Theorem applies to languages characterized by PRE .
Edit intent: fluency
Original:  We exploit the causal structure of of citation networks to measure the dimension of the Minkowski space in which a these directed acyclic graphs can most easily be embedded explicitly taking time into account as one of the dimensions we are measuring.
Revised:  We exploit the causal structure of of citation networks to measure the dimension of the Minkowski space in which these directed acyclic graphs can most easily be embedded explicitly taking time into account as one of the dimensions we are measuring.
Edit intent: fluency
Original: We adapt and use methods from the causal set approach to quantum gravity to analyse the structure of citation networks from academic papers on the arXiv, supreme court judgements from the US , and patents .
Revised: We adapt and use methods from the causal set approach to quantum gravity to analyse the structure of citation networks from the arXiv, supreme court judgements from the US , and patents .
Edit intent: clarity
Original: We adapt and use methods from the causal set approach to quantum gravity to analyse the structure of citation networks from academic papers on the arXiv, supreme court judgements from the US , and patents .
Revised: We adapt and use methods from the causal set approach to quantum gravity to analyse the structure of citation networks from academic papers on the arXiv, supreme court judgements from the USA , and patents .
Edit intent: fluency
Original:  Our interpretation is that a high dimension corresponds to diverse citation behaviour while a low dimension indicates a narrow range of citations in a field .
Revised:  We suggest that these differences can be interpreted in terms of the level of diversity or narrowness in citation behaviour .
Edit intent: clarity
Original: We introduce network L-cloning, a novel technique for creating ensembles of random networks from any given real-world or artificial network.
Revised: We introduce network L-cloning, a technique for creating ensembles of random networks from any given real-world or artificial network.
Edit intent: clarity
Original:  Each member of the ensemble is an " L-cloned network " constructed from L copies of the original network.
Revised:  Each member of the ensemble is an L-cloned network " constructed from L copies of the original network.
Edit intent: fluency
Original:  Each member of the ensemble is an " L-cloned network " constructed from L copies of the original network.
Revised:  Each member of the ensemble is an " L-cloned network constructed from L copies of the original network.
Edit intent: fluency
Original:  The density of triangles in an L-cloned \LC network, and hence its clustering coefficient, are reduced by a factor of L comparing to those of the original network.
Revised:  The density of triangles in an \LC network, and hence its clustering coefficient, are reduced by a factor of L comparing to those of the original network.
Edit intent: clarity
Original:  The density of triangles in an L-cloned \LC network, and hence its clustering coefficient, are reduced by a factor of L comparing to those of the original network.
Revised:  The density of triangles in an L-cloned \LC network, and hence its clustering coefficient, is reduced by a factor of L comparing to those of the original network.
Edit intent: fluency
Original:  The density of triangles in an L-cloned \LC network, and hence its clustering coefficient, are reduced by a factor of L comparing to those of the original network.
Revised:  The density of triangles in an L-cloned \LC network, and hence its clustering coefficient, are reduced by a factor of L compared to those of the original network.
Edit intent: fluency
Original:  Furthermore, the density of loops of any fixed length approaches zero for sufficiently large values of L.  As an application, we employ L-cloning to investigate the effect of short loops on dynamical processes running on networks and to inspect the accuracy of corresponding tree-based theories.
Revised:  Furthermore, the density of loops of any fixed length approaches zero for sufficiently large values of L. As an application, we employ these network cloning methods to investigate the effect of short loops on dynamical processes running on networks and to inspect the accuracy of corresponding tree-based theories.
Edit intent: clarity
Original:  We demonstrate that dynamics on L-cloned networks (with sufficiently large L) are accurately described by the so-called " adjacency tree-based theories ", which is a class of theoretical approaches for modeling various networked behaviors including percolation, SI epidemic spreading , and the Ising model.
Revised:  We demonstrate that dynamics on L-cloned networks (with sufficiently large L) are accurately described by the so-called adjacency tree-based theories ", which is a class of theoretical approaches for modeling various networked behaviors including percolation, SI epidemic spreading , and the Ising model.
Edit intent: fluency
Original:  We demonstrate that dynamics on L-cloned networks (with sufficiently large L) are accurately described by the so-called " adjacency tree-based theories ", which is a class of theoretical approaches for modeling various networked behaviors including percolation, SI epidemic spreading , and the Ising model.
Revised:  We demonstrate that dynamics on L-cloned networks (with sufficiently large L) are accurately described by the so-called " adjacency tree-based theories ", which is a class of theoretical approaches for modeling various networked behaviors including percolation, SI epidemics , and the Ising model.
Edit intent: clarity
Original:  In this study, we analyze the radical pair model form a microscopic spin transitional point of view and establish the role of nuclear and environmental decoherence in radical pair spin dynamics.
Revised:  In this study, the spin dynamics of the radical pair model form a microscopic spin transitional point of view and establish the role of nuclear and environmental decoherence in radical pair spin dynamics.
Edit intent: clarity
Original: The ellipsoid-infinity norm of a real m-by-n matrix A , denoted by \|A\|_{E\infty is the minimum \ell_\infty norm of an 0-centered ellipsoid E\subseteqR^m that contains all column vectors of A.
Revised: The ellipsoid-infinity norm of a real m\times n matrix A is the minimum \ell_\infty norm of an 0-centered ellipsoid E\subseteqR^m that contains all column vectors of A.
Edit intent: clarity
Original:  This quantity, introduced by the second author and Talwar in 2013, is polynomial-time computable and approximates the hereditary discrepancy herdisc ] \ } A as follows:  { \|A\|_{E\infty / O(\log m)} %DIFDELCMD < \le %DIFDELCMD < %%% \text{herdisc A }%DIFDELCMD < \le %%% \|A \|_{E\infty \cdot } \ A and herdisc\ A = } O(\sqrt{\log m} ) . Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality.
Revised:  This quantity, introduced by the second author and Talwar in 2013, is polynomial-time computable and approximates the hereditary discrepancy herdisc ]hereditary discrepancy \ } A as follows: { \|A\|_{E\infty / O(\log m)} %DIFDELCMD < \le %DIFDELCMD < %%% \text{herdisc A }%DIFDELCMD < \le %%% \|A \|_{E\infty \cdot } \ A and herdisc\ A = } O(\sqrt{\log m} ) . Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality.
Edit intent: fluency
Original:  This quantity, introduced by the second author and Talwar in 2013, is polynomial-time computable and approximates the hereditary discrepancy herdisc ] \ } A as follows:  { \|A\|_{E\infty / O(\log m)} %DIFDELCMD < \le %DIFDELCMD < %%% \text{herdisc A }%DIFDELCMD < \le %%% \|A \|_{E\infty \cdot } \ A and herdisc\ A = } O(\sqrt{\log m} ) . Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality.
Revised:  This quantity, introduced by the second author and Talwar in 2013, is polynomial-time computable and approximates the hereditary discrepancy herdisc ] \ } A as follows: { / O(\log m)} %DIFDELCMD < \le %DIFDELCMD < %%% \text{herdisc A }%DIFDELCMD < \le %%% \|A \|_{E\infty \cdot } \ A and herdisc\ A = } O(\sqrt{\log m} ) . Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality.
Edit intent: coherence
Original:  This quantity, introduced by the second author and Talwar in 2013, is polynomial-time computable and approximates the hereditary discrepancy herdisc ] \ } A as follows:  { \|A\|_{E\infty / O(\log m)} %DIFDELCMD < \le %DIFDELCMD < %%% \text{herdisc A }%DIFDELCMD < \le %%% \|A \|_{E\infty \cdot } \ A and herdisc\ A = } O(\sqrt{\log m} ) . Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality.
Revised:  This quantity, introduced by the second author and Talwar in 2013, is polynomial-time computable and approximates the hereditary discrepancy herdisc ] \ } A as follows: { \|A\|_{E\infty O(\log m)} %DIFDELCMD < \le %DIFDELCMD < %%% \text{herdisc A }%DIFDELCMD < \le %%% \|A \|_{E\infty \cdot } \ A and herdisc\ A = } O(\sqrt{\log m} ) . Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality.
Edit intent: clarity
Original:  This quantity, introduced by the second author and Talwar in 2013, is polynomial-time computable and approximates the hereditary discrepancy herdisc ] \ } A as follows:  { \|A\|_{E\infty / O(\log m)} %DIFDELCMD < \le %DIFDELCMD < %%% \text{herdisc A }%DIFDELCMD < \le %%% \|A \|_{E\infty \cdot } \ A and herdisc\ A = } O(\sqrt{\log m} ) . Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality.
Revised:  This quantity, introduced by the second author and Talwar in 2013, is polynomial-time computable and approximates the hereditary discrepancy herdisc ] \ } A as follows: { \|A\|_{E\infty / O(\log m)} %DIFDELCMD < \le %DIFDELCMD < %%% A }%DIFDELCMD < \le %%% \|A \|_{E\infty \cdot } \ A and herdisc\ A = } O(\sqrt{\log m} ) . Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality.
Edit intent: coherence
Original:  This quantity, introduced by the second author and Talwar in 2013, is polynomial-time computable and approximates the hereditary discrepancy herdisc ] \ } A as follows:  { \|A\|_{E\infty / O(\log m)} %DIFDELCMD < \le %DIFDELCMD < %%% \text{herdisc A }%DIFDELCMD < \le %%% \|A \|_{E\infty \cdot } \ A and herdisc\ A = } O(\sqrt{\log m} ) . Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality.
Revised:  This quantity, introduced by the second author and Talwar in 2013, is polynomial-time computable and approximates the hereditary discrepancy herdisc ] \ } A as follows: { \|A\|_{E\infty / O(\log m)} %DIFDELCMD < \le %DIFDELCMD < %%% \text{herdisc A }%DIFDELCMD < \le %%% \cdot } \ A and herdisc\ A = } O(\sqrt{\log m} ) . Here we show that both of the inequalities are asymptotically tight in the worst case, and we provide a simplified proof of the first inequality.
Edit intent: fluency
Original:  Most notably, we prove a new lower bound of \Omega(\log^{d-1} n) for the d-dimensional Tusnady problem \emph{ , asking for the combinatorial discrepancy of an n-point set in \mathbb{R}^d with respect to axis-parallel boxes.
Revised:  Most notably, we prove a new lower bound of \Omega(\log^{d-1} n) for the \emph{ , asking for the combinatorial discrepancy of an n-point set in \mathbb{R}^d with respect to axis-parallel boxes.
Edit intent: clarity
Original: We give a discrete analogue of the Aleksandrov theory of the  Monge-Ampere equation for nonlinear difference equations.
Revised: We give the Monge-Ampere equation for nonlinear difference equations.
Edit intent: coherence
Original:  these properties have been shown to correlate with the patho-physiological state of the cells in view of the development of novel early-diagnostic strategies.
Revised:  these properties have been shown to correlate with cells' fate and patho-physiological state of the cells in view of the development of novel early-diagnostic strategies.
Edit intent: clarity
Original:  these properties have been shown to correlate with the patho-physiological state of the cells in view of the development of novel early-diagnostic strategies.
Revised:  these properties have been shown to correlate with the patho-physiological state in view of the development of novel early-diagnostic strategies.
Edit intent: clarity
Original:  We report several case studies performed during the school and systematize our experiences.
Edit intent: coherence
Original:  finally, we are trying to describe what we have achieved during the week of the school in a way that others may repeat (and hopefully improve) our experiments.
Revised:  finally, we try to describe what we have achieved during the week of the school in a way that others may repeat (and hopefully improve) our experiments.
Edit intent: coherence
Original:  finally, we are trying to describe what we have achieved during the week of the school in a way that others may repeat (and hopefully improve) our experiments.
Revised:  finally, we are trying to describe what we have achieved during the week of the school in a way that others may reproduce (and hopefully improve) our experiments.
Edit intent: fluency
Original:  Furthermore, our proofs can be used to show that the labeled variant (where each robot is assigned with a specific target position), again, for unit-square robots, is PSPACE-hard as well, which sets another precedence, as previous hardness results require the robots to be of different shape .
Revised:  Furthermore, our proofs can be used to show that the labeled variant (where each robot is assigned with a specific target position), again, for unit-square robots, is PSPACE-hard as well, which sets another precedence, as previous hardness results require the robots to be of different shapes .
Edit intent: fluency
Original:  To handle such challenges, in this paper we propose a tracking approach where the object is modelled as a continuously updated set of affine subspaces, with each subspace constructed from the object's appearance over several consecutive frames.
Revised:  To handle such challenges, in this paper we propose a tracking approach where the object is modelled as a continuously updated bag of affine subspaces, with each subspace constructed from the object's appearance over several consecutive frames.
Edit intent: fluency
Original:  Furthermore, during the search for the object 's location in a new frame, we propose to represent the candidate regions also as affine subspaces , by including the immediate tracking history over several frames . Distances between affine subspaces from the object model and candidate regions are obtained by exploiting the non-Euclidean geometry of Grassmann manifolds.
Revised:  Furthermore, during the search for the object in a new frame, we propose to represent the candidate regions also as affine subspaces , by including the immediate tracking history over several frames . Distances between affine subspaces from the object model and candidate regions are obtained by exploiting the non-Euclidean geometry of Grassmann manifolds.
Edit intent: clarity
Original:  Furthermore, during the search for the object 's location in a new frame, we propose to represent the candidate regions also as affine subspaces , by including the immediate tracking history over several frames . Distances between affine subspaces from the object model and candidate regions are obtained by exploiting the non-Euclidean geometry of Grassmann manifolds.
Revised:  Furthermore, during the search for the object 's location in a new frame, we propose to represent the candidate regions also as affine subspaces . Distances between affine subspaces from the object model and candidate regions are obtained by exploiting the non-Euclidean geometry of Grassmann manifolds.
Edit intent: coherence
Original:  Furthermore, during the search for the object 's location in a new frame, we propose to represent the candidate regions also as affine subspaces , by including the immediate tracking history over several frames . Distances between affine subspaces from the object model and candidate regions are obtained by exploiting the non-Euclidean geometry of Grassmann manifolds.
Revised:  Furthermore, during the search for the object 's location in a new frame, we propose to represent the candidate regions also as affine subspaces , by including the immediate tracking history over several frames . Distances between subspaces are then obtained by exploiting the non-Euclidean geometry of Grassmann manifolds.
Edit intent: clarity
Original:  Quantitative evaluations on challenging image sequences indicate that the proposed approach obtains considerably better performance than several recent state-of-the-art methods such as Tracking-Learning-Detection and Multiple Instance Learning Tracking .
Revised:  Quantitative evaluations on challenging image sequences indicate that the proposed method achieves higher tracking accuracy than several recent state-of-the-art methods such as Tracking-Learning-Detection and Multiple Instance Learning Tracking .
Edit intent: clarity
Original:  Quantitative evaluations on challenging image sequences indicate that the proposed approach obtains considerably better performance than several recent state-of-the-art methods such as Tracking-Learning-Detection and Multiple Instance Learning Tracking .
Revised:  Quantitative evaluations on challenging image sequences indicate that the proposed approach obtains considerably better performance than several recent discriminative trackers .
Edit intent: clarity
Original:  In this article, we provide an overview of state-of-the-art RF-enabled WET technology and it applications to wireless communications, with highlights on the key design challenges, solutions, and opportunities ahead.
Revised:  In this article, we provide an overview of state-of-the-art RF-enabled WET technologies and their applications to wireless communications, with highlights on the key design challenges, solutions, and opportunities ahead.
Edit intent: clarity
Original:  \lceil \rceil
Revised:  \lceil \rceil.
Edit intent: fluency
Original:  The algorithms (modified in an obvious way to work with the complex numbers) from the present paper can be used to evaluate complex Riemann zeta function \zeta(s) on lines \sigma+it, \sigma>1, in polynomial time and linear space in  \lceil \log_2(t)\rceil  .
Revised:  The algorithm (modified in an obvious way to work with the complex numbers) from the present paper can be used to evaluate complex Riemann zeta function \zeta(s) on lines \sigma+it, \sigma>1, in polynomial time and linear space in \lceil \log_2(t)\rceil .
Edit intent: fluency
Original:  The algorithms (modified in an obvious way to work with the complex numbers) from the present paper can be used to evaluate complex Riemann zeta function \zeta(s) on lines \sigma+it, \sigma>1, in polynomial time and linear space in  \lceil \log_2(t)\rceil  .
Revised:  The algorithms (modified in an obvious way to work with the complex numbers) from the present paper can be used to evaluate complex Riemann zeta function \zeta(s) for s= \sigma+it, \sigma>1, in polynomial time and linear space in \lceil \log_2(t)\rceil .
Edit intent: clarity
Original:  The algorithms (modified in an obvious way to work with the complex numbers) from the present paper can be used to evaluate complex Riemann zeta function \zeta(s) on lines \sigma+it, \sigma>1, in polynomial time and linear space in  \lceil \log_2(t)\rceil  .
Revised:  The algorithms (modified in an obvious way to work with the complex numbers) from the present paper can be used to evaluate complex Riemann zeta function \zeta(s) on lines \sigma+it, \sigma>1, in polynomial time and linear space in \lceil \log_2(t)\rceil ) .
Edit intent: fluency
Original:  The algorithm ( modified in an obvious way to work with the complex numbers ) from the present paper can be used to evaluate complex Riemann zeta function \zeta(s) for s=\sigma+it, \sigma > \ne 1 , in polynomial time and linear space in n wherein 2^{-n} is a precision of the computation (but the modified algorithm will be exponential in time and space in \lceil \log_2(t)\rceil )\lceil \rceil .
Revised:  The algorithm from the present paper modified in an obvious way to work with the complex numbers ) from the present paper can be used to evaluate complex Riemann zeta function \zeta(s) for s=\sigma+it, \sigma > \ne 1 , in polynomial time and linear space in n wherein 2^{-n} is a precision of the computation (but the modified algorithm will be exponential in time and space in \lceil \log_2(t)\rceil )\lceil \rceil .
Edit intent: fluency