wanyu/IteraTeR_v2 huggingface dataset:
Original:  Fifth, we show that the computation of the interleaving distance between two finitely presented multidimensional persistence modules M and N reduces to deciding the solvability of O(log m) systems of multivariate quadratic equations, each with O(m^2) variables and O(m^2) equations, where m is the total number of generators and relations in a minimal presentation for M and a minimal presentation for N.
Revised:  Fifth, we show that given finite presentations for two multi-D persistence modules M and N reduces to deciding the solvability of O(log m) systems of multivariate quadratic equations, each with O(m^2) variables and O(m^2) equations, where m is the total number of generators and relations in a minimal presentation for M and a minimal presentation for N.
Edit intent: clarity
Original:  Fifth, we show that the computation of the interleaving distance between two finitely presented multidimensional persistence modules M and N reduces to deciding the solvability of O(log m) systems of multivariate quadratic equations, each with O(m^2) variables and O(m^2) equations, where m is the total number of generators and relations in a minimal presentation for M and a minimal presentation for N.
Revised:  Fifth, we show that the computation of the interleaving distance between two finitely presented multidimensional persistence modules M and N reduces to deciding the solvability of O(log m) systems of multivariate quadratic equations, each with O(m^2) variables and O(m^2) equations, where m is the total number of generators and relations in the presentations for M and a minimal presentation for N.
Edit intent: fluency
Original:  Fifth, we show that the computation of the interleaving distance between two finitely presented multidimensional persistence modules M and N reduces to deciding the solvability of O(log m) systems of multivariate quadratic equations, each with O(m^2) variables and O(m^2) equations, where m is the total number of generators and relations in a minimal presentation for M and a minimal presentation for N.
Revised:  Fifth, we show that the computation of the interleaving distance between two finitely presented multidimensional persistence modules M and N reduces to deciding the solvability of O(log m) systems of multivariate quadratic equations, each with O(m^2) variables and O(m^2) equations, where m is the total number of generators and relations in a minimal presentation for M and N.
Edit intent: fluency
Original:  The aim of this of this paperis to develop the theory of interleavings and of d_I in the multi-D setting , with a view towards applications to multi-D persistent topological inference .
Revised:  In this paper, we develop the theory of interleavings and of d_I in the multi-D setting , with a view towards applications to multi-D persistent topological inference .
Edit intent: clarity
Original:  The aim of this of this paperis to develop the theory of interleavings and of d_I in the multi-D setting , with a view towards applications to multi-D persistent topological inference .
Revised:  The aim of this of this paperis to develop the theory of multidimensional interleavings , with a view towards applications to multi-D persistent topological inference .
Edit intent: clarity
Original:  The aim of this of this paperis to develop the theory of interleavings and of d_I in the multi-D setting , with a view towards applications to multi-D persistent topological inference .
Revised:  The aim of this of this paperis to develop the theory of interleavings and of d_I in the multi-D setting , with a view towards applications to topological data analysis .
Edit intent: clarity
Original:  This result yields a converse to the main result of 14%DIFDELCMD < ]%%% , answering a question posed in that paper . Second, we present a characterization of interleaved pairs of persistence moduleswhich expresses transparently the sense in which interleaved modules are algebraically similar.
Revised:  This result 14%DIFDELCMD < ]%%% , answering a question posed in that paper . Second, we present a characterization of interleaved pairs of persistence moduleswhich expresses transparently the sense in which interleaved modules are algebraically similar.
Edit intent: clarity
Original:  This result yields a converse to the main result of 14%DIFDELCMD < ]%%% , answering a question posed in that paper . Second, we present a characterization of interleaved pairs of persistence moduleswhich expresses transparently the sense in which interleaved modules are algebraically similar.
Revised:  This result yields a converse to the main result of %DIFDELCMD < ]%%% , answering a question posed in that paper . Second, we present a characterization of interleaved pairs of persistence moduleswhich expresses transparently the sense in which interleaved modules are algebraically similar.
Edit intent: clarity
Original:  By this we mean that d_I satisfies a stability property generalizing one which d_B is known to satisfy, and  in addition, that if d is any other pseudometric on multi-D persistence modules satisfying the same stability property  then d\leq d_I. Fourth, we show that the restriction of d_I to finitely presented persistence modules is a metric.
Revised:  By this we mean that d_I satisfies a stability property generalizing one which d_B is known to satisfy, and that in addition, that if d is any other pseudometric on multi-D persistence modules satisfying the same stability property then d\leq d_I. Fourth, we show that the restriction of d_I to finitely presented persistence modules is a metric.
Edit intent: fluency
Original:  By this we mean that d_I satisfies a stability property generalizing one which d_B is known to satisfy, and  in addition, that if d is any other pseudometric on multi-D persistence modules satisfying the same stability property  then d\leq d_I. Fourth, we show that the restriction of d_I to finitely presented persistence modules is a metric.
Revised:  By this we mean that d_I satisfies a stability property generalizing one which d_B is known to satisfy, and in addition, if d is any other pseudometric on multi-D persistence modules satisfying the same stability property then d\leq d_I. Fourth, we show that the restriction of d_I to finitely presented persistence modules is a metric.
Edit intent: coherence
Original:  By this we mean that d_I satisfies a stability property generalizing one which d_B is known to satisfy, and  in addition, that if d is any other pseudometric on multi-D persistence modules satisfying the same stability property  then d\leq d_I. Fourth, we show that the restriction of d_I to finitely presented persistence modules is a metric.
Revised:  By this we mean that d_I satisfies a stability property generalizing one which d_B is known to satisfy, and in addition, that if d is any other pseudometric on multidimensional persistence modules satisfying the same stability property then d\leq d_I. Fourth, we show that the restriction of d_I to finitely presented persistence modules is a metric.
Edit intent: clarity
Original:  By this we mean that d_I satisfies a stability property generalizing one which d_B is known to satisfy, and  in addition, that if d is any other pseudometric on multi-D persistence modules satisfying the same stability property  then d\leq d_I. Fourth, we show that the restriction of d_I to finitely presented persistence modules is a metric.
Revised:  By this we mean that d_I satisfies a stability property generalizing one which d_B is known to satisfy, and in addition, that if d is any other pseudometric on multi-D persistence modules satisfying the same stability property , then d\leq d_I. Fourth, we show that the restriction of d_I to finitely presented persistence modules is a metric.
Edit intent: fluency
Original: We employ evolutionary game theory to study topological conditions of scale-free networks for the evolution of cooperation.
Revised: Evolutionary game theory is employed to study topological conditions of scale-free networks for the evolution of cooperation.
Edit intent: clarity
Original:  We show that Apollonian Network (AN) is a perfect scale-free network , on which cooperation can spread to all individuals, even though there are initially only 3 or 4 hubs occupied by cooperators and all the others by defectors.
Revised:  We show that Apollonian Networks (ANs) are perfect scale-free network , on which cooperation can spread to all individuals, even though there are initially only 3 or 4 hubs occupied by cooperators and all the others by defectors.
Edit intent: clarity
Original:  We show that Apollonian Network (AN) is a perfect scale-free network , on which cooperation can spread to all individuals, even though there are initially only 3 or 4 hubs occupied by cooperators and all the others by defectors.
Revised:  We show that Apollonian Network (AN) is a perfect scale-free networks , on which cooperation can spread to all individuals, even though there are initially only 3 or 4 hubs occupied by cooperators and all the others by defectors.
Edit intent: fluency
Original:  Local topological features such as  clustering coefficient, degree, gradient as well as topology potential are adopted to analyze the advantage of ANs in cooperation enhancement.
Revised:  Local topological features such as degree, clustering coefficient, degree, gradient as well as topology potential are adopted to analyze the advantage of ANs in cooperation enhancement.
Edit intent: fluency
Original:  Local topological features such as  clustering coefficient, degree, gradient as well as topology potential are adopted to analyze the advantage of ANs in cooperation enhancement.
Revised:  Local topological features such as clustering coefficient, gradient as well as topology potential are adopted to analyze the advantage of ANs in cooperation enhancement.
Edit intent: clarity
Original:  Local topological features such as  clustering coefficient, degree, gradient as well as topology potential are adopted to analyze the advantage of ANs in cooperation enhancement.
Revised:  Local topological features such as clustering coefficient, degree, gradient as well as topology potential are adopted to analyze the advantages of ANs in cooperation enhancement.
Edit intent: fluency
Original:  However, advances in communication technology means that the 'lifetime' of an important piece of information is typically short.
Revised:  However, advances in communication technology means that the 'lifetime' of important information is typically short.
Edit intent: clarity
Original:  Hence, viewed as a tradable asset, information shares the characteristics of a nondurable commodity: while it can be stored and transmitted freely, its worth diminishes rapidly in time.
Revised:  Hence, viewed as a tradable asset, information shares the characteristics of a perishable commodity: while it can be stored and transmitted freely, its worth diminishes rapidly in time.
Edit intent: clarity
Original:  In view of recent developments where internet search engines and other information providers are offering valuable information to financial institutions, the problem of pricing information is becoming increasingly important.
Revised:  In view of recent developments where internet search engines and other information providers are offering information to financial institutions, the problem of pricing information is becoming increasingly important.
Edit intent: clarity
Original:  We show how dissipation fundamentally limits the lifetimes of these emergent structures, even though, on short timescales, the group properties may be enriched compared to their noiseless counterparts.
Revised:  We show how dissipation and irreversibility fundamentally limit the lifetimes of these emergent structures, even though, on short timescales, the group properties may be enriched compared to their noiseless counterparts.
Edit intent: clarity
Original:  We show that our  model contains a lot of models which was considered in the litterature.
Revised:  We show that our model contains a lot of models which were previously considered in the litterature.
Edit intent: clarity
Original:  Variants of tit-for-tat (repeating the previous action of the opponent) and the win-stay lose-shift strategy are known as strong competitors in the iterated social dilemma games.
Revised:  Variants of tit-for-tat (repeating the previous action of the opponent) and the win-stay lose-shift strategy are known as strong competitors in iterated social dilemma games.
Edit intent: fluency
Original:  We use process algebra to describe the behaviours produced by instruction sequences under execution and to describe two protocols implementing these behaviours in the case of remote processing .
Revised:  We use process algebra to describe the behaviours produced by instruction sequences under execution and to describe two protocols implementing these behaviours in the case where the processing of instructions takes place remotely .
Edit intent: clarity
Original: We derive tight expressions for the maximum values of the number of k-faces, 0kd-1, of the Minkowski sum  P_1P_2  of two d-dimensional convex polytopes P_1 and P_2, as a function of the number of vertices of the polytopes.
Revised: We derive tight expressions for the maximum number of k-faces, 0kd-1, of the Minkowski sum P_1P_2 of two d-dimensional convex polytopes P_1 and P_2, as a function of the number of vertices of the polytopes.
Edit intent: clarity
Original: We derive tight expressions for the maximum values of the number of k-faces, 0kd-1, of the Minkowski sum  P_1P_2  of two d-dimensional convex polytopes P_1 and P_2, as a function of the number of vertices of the polytopes.
Revised: We derive tight expressions for the maximum values of the number of k-faces, 0kd-1, of the Minkowski sum , P_1P_2 of two d-dimensional convex polytopes P_1 and P_2, as a function of the number of vertices of the polytopes.
Edit intent: fluency
Original: We derive tight expressions for the maximum values of the number of k-faces, 0kd-1, of the Minkowski sum  P_1P_2  of two d-dimensional convex polytopes P_1 and P_2, as a function of the number of vertices of the polytopes.
Revised: We derive tight expressions for the maximum values of the number of k-faces, 0kd-1, of the Minkowski sum P_1P_2 , of two d-dimensional convex polytopes P_1 and P_2, as a function of the number of vertices of the polytopes.
Edit intent: fluency
Original:  For even dimensions d2, the maximum values are attained if P_1 and P_2 are cyclic d-polytopes with disjoint vertex sets.
Revised:  For even dimensions d2, the maximum values are attained when P_1 and P_2 are cyclic d-polytopes with disjoint vertex sets.
Edit intent: fluency
Original:  For odd dimensions d3, the maximum values are attained if P_1 and P_2 are \lfloord{2}\rfloor-neighborly d-polytopes, whose vertex sets are chosen appropriately from two distinct d-dimensional moment-like curves.
Revised:  For odd dimensions d3, the maximum values are attained when P_1 and P_2 are \lfloord{2}\rfloor-neighborly d-polytopes, whose vertex sets are chosen appropriately from two distinct d-dimensional moment-like curves.
Edit intent: fluency
Original:  However , it was also recently pointed out that this condition is not sufficient;
Revised:  However , it was also recently shown that this condition is not sufficient;
Edit intent: coherence
Original:  in contrast, a necessary and sufficient condition is a global stability property of the associated ordinary differential equation.
Revised:  in contrast, a sufficient condition is a global stability property of the associated ordinary differential equation.
Edit intent: clarity
Original:  Such a property was established only for a specific case, namely for a homogeneous system (all nodes have the same parameters) and when the number of backoff stages is either two or infinite and with other restrictive conditions.
Edit intent: coherence
Original:  It is shown that the uniqueness of a solution to the associated fixed point equation is not sufficient;
Revised:  We show that the uniqueness of a solution to the associated fixed point equation is not sufficient;
Edit intent: coherence
Original: Using the probability theory-based approach, this paper reveals the equivalence of an arbitrary NP-complete problem to a problem of checking whether a level hypersurface of a specifically constructed harmonic cost function (with all diagonal entries of its Hessian matrix equal to zero) intersects with a unit hypercube in many-dimensional Euclidean space.
Revised: Using the probability theory-based approach, this paper reveals the equivalence of an arbitrary NP-complete problem to a problem of checking whether a level set of a specifically constructed harmonic cost function (with all diagonal entries of its Hessian matrix equal to zero) intersects with a unit hypercube in many-dimensional Euclidean space.
Edit intent: clarity
Original: We give a new generalization of the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in monogenous algebras over finite fields through interpolation on algebraic curves.
Revised: We give new improvements to the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in monogenous algebras over finite fields through interpolation on algebraic curves.
Edit intent: clarity
Original: We give a new generalization of the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in monogenous algebras over finite fields through interpolation on algebraic curves.
Revised: We give a new generalization of the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in extensions of finite fields through interpolation on algebraic curves.
Edit intent: clarity
Original:  This allows us to fix errors in, improve, and generalize, previous works of Shparlinski-Tsfasman-Vladut, Ballet, and Cenk-\"Ozbudak .
Revised:  This allows us to fix errors in, improve, and generalize, previous works of Shparlinski-Tsfasman-Vladut, Ballet, and Cenk-Ozbudak .
Edit intent: fluency
Original:  Besides, generalities on interpolation schemes , as well as an alternative method that solves certain constructiveness issues , are also discussed.
Revised:  Besides, generalities on interpolation systems , as well as an alternative method that solves certain constructiveness issues , are also discussed.
Edit intent: clarity
Original: We develop a scheme for estimating state-dependent drift and diffusion coefficients in a stochastic differential equation from a time-series data.
Revised: A scheme is developed for estimating state-dependent drift and diffusion coefficients in a stochastic differential equation from a time-series data.
Edit intent: clarity
Original: We develop a scheme for estimating state-dependent drift and diffusion coefficients in a stochastic differential equation from a time-series data.
Revised: We develop a scheme for estimating state-dependent drift and diffusion coefficients in a stochastic differential equation from time-series data.
Edit intent: fluency
Original:  the scheme does not require to specify parametric forms for the drift and diffusion coefficients  .
Revised:  the scheme does not require to specify parametric forms for the drift and diffusion coefficients in advance .
Edit intent: clarity
Original:  In order to perform the nonparametric estimation, a maximum likelihood method are combined with a concept based on a kernel density estimation.
Revised:  In order to perform the nonparametric estimation, a maximum likelihood method is combined with a concept based on a kernel density estimation.
Edit intent: fluency
Original:  In order to deal with discrete observation or sparsity of the time-series data, we employ a local linearization method  .
Revised:  In order to deal with discrete observation or sparsity of the time-series data, a local linearization method .
Edit intent: clarity
Original: We consider the minimum Manhattan network problem, which is defined as follows.
Revised: We study the minimum Manhattan network problem, which is defined as follows.
Edit intent: clarity
Original:  We present, for any fixed dimension d and any \epsilon \eps >0, an O(n ^\epsilon )-approximation  .
Revised:  We present, for any fixed dimension d and any \eps >0, an O(n ^\epsilon )-approximation .
Edit intent: fluency
Original:  We present, for any fixed dimension d and any \epsilon \eps >0, an O(n ^\epsilon )-approximation  .
Revised:  We present, for any fixed dimension d and any \epsilon \eps >0, an O(n ^\eps )-approximation .
Edit intent: fluency
Original:  We present, for any fixed dimension d and any \epsilon \eps >0, an O(n ^\epsilon )-approximation  .
Revised:  We present, for any fixed dimension d and any \epsilon \eps >0, an O(n ^\epsilon )-approximation algorithm .
Edit intent: clarity
Original:  For 3D, we also give a 4(k-1)-approximation  for the case that the terminals are contained in the union of k \ge 2 parallel planes.
Revised:  For 3D, we also give a 4(k-1)-approximation algorithm for the case that the terminals are contained in the union of k \ge 2 parallel planes.
Edit intent: clarity
Original:  The resulting Core Lucid language would inherit the properties of generalization attempts of GIPL ( 1999-2011 ) and TransLucid ( 2008-2011 ) for all future and recent Lucid-implementing systems to follow.
Revised:  The resulting Core Lucid language would inherit the properties of generalization attempts of GIPL ( 1999-2011 ) and TransLucid ( 2008-2011 ) for all future and recent Lucid implementing systems to follow.
Edit intent: fluency
Original:  In turn, the sum of reciprocals of individual node structural centralities, i1/C^*(i) = i l^+_{ii}, i.e. the trace of \bb L^+, yields the well-known Kirchoff index (K), an overall structural descriptor for the network.
Revised:  In turn, the sum of reciprocals of individual node structural centralities, i1/C^*(i) = i l^+_{ii}, i.e. the trace of \bb L^+, yields the well-known Kirchhoff index (K), an overall structural descriptor for the network.
Edit intent: fluency
Original:  These interpretations respectively help establish the structural centrality (C^{*}(i))  as a measure of both the overall position as well as the overall connectedness of node i in the network which in turn reflects the robustness of node i to random multiple edge failures.
Revised:  These interpretations respectively help establish the structural centrality (C^{*}(i)) of node i as a measure of both the overall position as well as the overall connectedness of node i in the network which in turn reflects the robustness of node i to random multiple edge failures.
Edit intent: fluency
Original:  These interpretations respectively help establish the structural centrality (C^{*}(i))  as a measure of both the overall position as well as the overall connectedness of node i in the network which in turn reflects the robustness of node i to random multiple edge failures.
Revised:  These interpretations respectively help establish the structural centrality (C^{*}(i)) as a measure of its overall position as well as the overall connectedness of node i in the network which in turn reflects the robustness of node i to random multiple edge failures.
Edit intent: clarity
Original:  These interpretations respectively help establish the structural centrality (C^{*}(i))  as a measure of both the overall position as well as the overall connectedness of node i in the network which in turn reflects the robustness of node i to random multiple edge failures.
Revised:  These interpretations respectively help establish the structural centrality (C^{*}(i)) as a measure of both the overall position as well as its overall connectedness in the network which in turn reflects the robustness of node i to random multiple edge failures.
Edit intent: clarity