wanyu/IteraTeR_full_sent huggingface dataset:
Original:  4 quaternion-based matrix, thus specifying the quaternion eigenvector corresponding to the optimal 3D rotation. Here we generalize this basic problem , sometimes referred to as the " Procrustes Problem, " and present algebraic solutions that exhibit properties that are inaccessible to traditional numerical methods.
Revised:  4 quaternion-based matrix, thus specifying the quaternion eigenvector corresponding to the optimal 3D point data, this " Procrustes Problem, " and present algebraic solutions that exhibit properties that are inaccessible to traditional numerical methods.
Edit intent: clarity
Original:  Here we generalize this basic problem , sometimes referred to as the " Procrustes Problem, " and present algebraic solutions that exhibit properties that are inaccessible to traditional numerical methods. We begin with the 4D data problem , a problem one dimension higher than the conventional 3D problem, but one that is also solvable by quaternion methods, we then study the 3D and 2D data problems as special cases.
Revised:  Here we generalize this basic problem , sometimes referred to as the " Procrustes Problem, " is often phrased in terms of minimizing a root-mean-square deviation or RMSD corresponding to a Euclidean distance measure relating the two sets of matched coordinates. We focus on quaternion eigensystem methods that have been exploited to solve this problem , a problem one dimension higher than the conventional 3D problem, but one that is also solvable by quaternion methods, we then study the 3D and 2D data problems as special cases.
Edit intent: meaning-changed
Original:  We begin with the 4D data problem , a problem one dimension higher than the conventional 3D problem, but one that is also solvable by quaternion methods, we then study the 3D and 2D data problems as special cases.
Revised:  We begin with the 4D data problem for at least five decades in several different bodies of scientific literature where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; we focus on these exact solutions to expose the structure of the entire eigensystem for the traditional 3D and 2D data problems as special cases.
Edit intent: meaning-changed
Original:  We begin with the 4D data problem , a problem one dimension higher than the conventional 3D problem, but one that is also solvable by quaternion methods, we then study the 3D and 2D data problems as special cases. In addition, we consider data that are themselves quaternions isomorphic to orthonormal triads describing 3 coordinate frames (amino acids in proteins possess such frames). Adopting a reasonable approximation to the exact quaternion-data minimization problem, we find a novel closed form "quaternion RMSD" (QRMSD) solution for the optimal rotation from a quaternion data set to a reference set.
Revised:  We begin with the 4D data problem , a problem one dimension higher than the conventional 3D problem, but one that is also solvable by quaternion methods, we then study the 3D spatial alignment problem. We then explore the structure of the less-studied orientation data context, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. We conclude with a brief discussion of the combined 3D translation-orientation data alignment problem. Appendices are devoted to a quaternion data set to a reference set.
Edit intent: meaning-changed
Original:  Adopting a reasonable approximation to the exact quaternion-data minimization problem, we find a novel closed form "quaternion RMSD" (QRMSD) solution for the optimal rotation from a quaternion data set to a reference set. We observe that composites of the RMSD and QRMSD measures, combined with problem-dependent parameters including scaling factors to make their incommensurate dimensions compatible, could be suitable for certain matching tasks .
Revised:  Adopting a reasonable approximation to the exact quaternion-data minimization problem, we find a novel closed form "quaternion RMSD" (QRMSD) solution for the optimal rotation from a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices, and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. Supplementary Material covers extensions of quaternion methods to the 4D problem .
Edit intent: meaning-changed
Original:  Unlike the induction of contractile flows, the maintenance of dynamic stability in highly labile yet internally pre-stressed active materials remains unknown. To this end , we synthesize a biomimetic active nematic liquid crystal from long semi-flexible actin filaments driven out-of-equilibrium by myosin motor activity.
Revised:  While myosin motors drive the actin cytoskeleton out-of-equilibrium, the role of myosin-driven active stresses in the accumulation and dissipation of mechanical energy is unclear. To investigate this , we synthesize a biomimetic active nematic liquid crystal from long semi-flexible actin filaments driven out-of-equilibrium by myosin motor activity.
Edit intent: clarity
Original:  To this end , we synthesize a biomimetic active nematic liquid crystal from long semi-flexible actin filaments driven out-of-equilibrium by myosin motor activity. We identify new actomyosin interactions that govern the dynamic architecture and mechanical response of the network to active stresses. These responses include dynamic steady states, in which myosin induces transverse plucking of actin filaments, whose curvatures show anomalous and strongly coupled fluctuations across a broad spectrum of filament bending modes. These fluctuations break detailed balance, yet promote dynamic stability in a network predisposed to instabilities. Furthermore, the actomyosin interactions that maintain dynamic stability are fundamentally distinct from those that drive contractile flows of actomyosin networks.
Revised:  To this end , we synthesize an actomyosin material in vitro whose active stress content can tune the network from stable to contractile. Each increment in activity determines a characteristic spectrum of actin filament fluctuations which is used to calculate the total mechanical work and the production of entropy in the material. We find that the balance of work and entropy does not increase monotonically and, surprisingly, the entropy production rate is maximized in the non-contractile, stable state. Our study provides evidence that the origins of system entropy production and activity-dependent dissipation arise from disorder in the molecular interactions between actin and myosin
Edit intent: clarity
Original: Let \Omega to be one of \X^{N+1},C[0,1],D[0,1]: a product of Polish spaces, space of continuous functions from [0,1] to a subset of R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to R^d  respectively.
Revised: Let \Omega be one of \X^{N+1},C[0,1],D[0,1]: a product of Polish spaces, space of continuous functions from [0,1] to a subset of R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to R^d respectively.
Edit intent: coherence
Original: Let \Omega to be one of \X^{N+1},C[0,1],D[0,1]: a product of Polish spaces, space of continuous functions from [0,1] to a subset of R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to R^d  respectively.
Revised: Let \Omega to be one of \X^{N+1},C[0,1],D[0,1]: product of Polish spaces, space of continuous functions from [0,1] to a subset of R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to R^d respectively.
Edit intent: coherence
Original: Let \Omega to be one of \X^{N+1},C[0,1],D[0,1]: a product of Polish spaces, space of continuous functions from [0,1] to a subset of R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to R^d  respectively.
Revised: Let \Omega to be one of \X^{N+1},C[0,1],D[0,1]: a product of Polish spaces, space of continuous functions from [0,1] to R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to R^d respectively.
Edit intent: clarity
Original: Let \Omega to be one of \X^{N+1},C[0,1],D[0,1]: a product of Polish spaces, space of continuous functions from [0,1] to a subset of R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to R^d  respectively.
Revised: Let \Omega to be one of \X^{N+1},C[0,1],D[0,1]: a product of Polish spaces, space of continuous functions from [0,1] to a subset of R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to R^d , respectively.
Edit intent: fluency
Original:  We  consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and satisfies some other convex transport constraints, which is given by \Gamma\kP\kP .
Revised:  We first consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and satisfies some other convex transport constraints, which is given by \Gamma\kP\kP .
Edit intent: coherence
Original:  We  consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and satisfies some other convex transport constraints, which is given by \Gamma\kP\kP .
Revised:  We consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and its disintegration P_x must be in some fixed \Gamma(x) \subset\kP\kP .
Edit intent: meaning-changed
Original:  We  consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and satisfies some other convex transport constraints, which is given by \Gamma\kP\kP .
Revised:  We consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and satisfies some other convex transport constraints, which is given by \Gamma\kP(\Omega), where\kP .
Edit intent: meaning-changed
Original:  We  consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and satisfies some other convex transport constraints, which is given by \Gamma\kP\kP .
Revised:  We consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and satisfies some other convex transport constraints, which is given by \Gamma\kP\kP(\Omega) is the set of probability measures on \Omega .
Edit intent: meaning-changed
Original:  We show that such probability measure exists if and only if the \alpha average of  so-called G-expectation of bounded uniformly continuous and bounded functions with respect to the measures in \Gamma is less than their \beta average.
Revised:  We show that such probability measure exists if and only if the \alpha average of the so-called G-expectation of bounded uniformly continuous and bounded functions with respect to the measures in \Gamma is less than their \beta average.
Edit intent: fluency
Original:  We show that such probability measure exists if and only if the \alpha average of  so-called G-expectation of bounded uniformly continuous and bounded functions with respect to the measures in \Gamma is less than their \beta average.
Revised:  We show that such probability measure exists if and only if the \alpha average of so-called G-expectation of bounded continuous functions with respect to the measures in \Gamma is less than their \beta average.
Edit intent: meaning-changed
Original:  As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times  .
Revised:  As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times . Second, we consider the optimal transport problem with constraints and obtain the Kantorovich duality. A corollary of this result is a monotonicity principle which gives us a geometric way of identifying the optimizer .
Edit intent: meaning-changed
Original: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Revised: %DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Edit intent: meaning-changed
Original: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Revised: Let \Omega be one of%DIFDELCMD < \X%%% 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Edit intent: meaning-changed
Original: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Revised: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 %DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Edit intent: fluency
Original: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Revised: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Edit intent: coherence
Original: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Revised: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D %DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Edit intent: fluency
Original: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Revised: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Edit intent: meaning-changed
Original: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Revised: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from %DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Edit intent: fluency
Original: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Revised: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Edit intent: meaning-changed
Original: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Revised: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R %DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Edit intent: fluency
Original: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively.
Revised: Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% ^d, respectively.
Edit intent: fluency
Original:  We first consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and its disintegration P_x must be in some fixed \Gamma(x) \subset }%DIFDELCMD < \kP%%% (\Omega), where%DIFDELCMD < \kP%%% (\Omega) is the set of probability measures on \Omega.
Revised:  We first consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and its disintegration P_x must be in some fixed \Gamma(x) \subset }%DIFDELCMD < \kP%%% %DIFDELCMD < \kP%%% (\Omega) is the set of probability measures on \Omega.
Edit intent: coherence
Original:  We first consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and its disintegration P_x must be in some fixed \Gamma(x) \subset }%DIFDELCMD < \kP%%% (\Omega), where%DIFDELCMD < \kP%%% (\Omega) is the set of probability measures on \Omega. The main application we have in mind is the martingale optimal transport problem when the martingales are assumed to have bounded volatility/ quadratic variation.
Revised:  We first consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and its disintegration P_x must be in some fixed \Gamma(x) \subset }%DIFDELCMD < \kP%%% (\Omega), where%DIFDELCMD < \kP%%% This paper focuses on martingale optimal transport problem when the martingales are assumed to have bounded volatility/ quadratic variation.
Edit intent: clarity
Original:  We show that such probability measure exists if and only if the \alpha average of the so-called G-expectation of bounded continuous functions with respect to the measures in \Gamma is less than their \beta average. As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times. Second, we consider the optimal transport problem with constraints and obtain the Kantorovich duality.
Revised:  First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Several applications are provided: martingale measures with volatility uncertainty, optimal transport with capacity constraints, and Skorokhod embedding with bounded times. Next, we extend this result to multi-marginal constraints. Finally, we consider an optimal transport problem with constraints and obtain the Kantorovich duality.
Edit intent: meaning-changed
Original:  Moreover, because graphlets can deal only with unweighted PSNs, and because accounting for edge weights when constructing PSNs could improve PSC accuracy, we also propose a deep learning framework that automatically learns network features from the weighted PSNs. When evaluated on a large set of  9, 509 CATH and 11,451 SCOP protein domains  , our proposed approaches are superior to existing PSC approaches in terms of both accuracyand running time.
Revised:  Moreover, because graphlets can deal only with unweighted PSNs, and because accounting for edge weights when constructing PSNs could improve PSC accuracy, we also propose a deep learning framework that automatically learns network features from the weighted PSNs. When evaluated on a large set of ~ 9, 509 CATH and 11,451 SCOP protein domains , our proposed approaches are superior to existing PSC approaches in terms of both accuracyand running time.
Edit intent: fluency
Original:  Moreover, because graphlets can deal only with unweighted PSNs, and because accounting for edge weights when constructing PSNs could improve PSC accuracy, we also propose a deep learning framework that automatically learns network features from the weighted PSNs. When evaluated on a large set of  9, 509 CATH and 11,451 SCOP protein domains  , our proposed approaches are superior to existing PSC approaches in terms of both accuracyand running time.
Revised:  Moreover, because graphlets can deal only with unweighted PSNs, and because accounting for edge weights when constructing PSNs could improve PSC accuracy, we also propose a deep learning framework that automatically learns network features from the weighted PSNs. When evaluated on a large set of 9, 400 CATH and ~12,800 SCOP protein domains , our proposed approaches are superior to existing PSC approaches in terms of both accuracyand running time.
Edit intent: meaning-changed
Original:  Moreover, because graphlets can deal only with unweighted PSNs, and because accounting for edge weights when constructing PSNs could improve PSC accuracy, we also propose a deep learning framework that automatically learns network features from the weighted PSNs. When evaluated on a large set of  9, 509 CATH and 11,451 SCOP protein domains  , our proposed approaches are superior to existing PSC approaches in terms of both accuracyand running time.
Revised:  Moreover, because graphlets can deal only with unweighted PSNs, and because accounting for edge weights when constructing PSNs could improve PSC accuracy, we also propose a deep learning framework that automatically learns network features from the weighted PSNs. When evaluated on a large set of 9, 509 CATH and 11,451 SCOP protein domains (spanning 36 PSN sets) , our proposed approaches are superior to existing PSC approaches in terms of both accuracyand running time.
Edit intent: meaning-changed
Original:  Moreover, because graphlets can deal only with unweighted PSNs, and because accounting for edge weights when constructing PSNs could improve PSC accuracy, we also propose a deep learning framework that automatically learns network features from the weighted PSNs. When evaluated on a large set of  9, 509 CATH and 11,451 SCOP protein domains  , our proposed approaches are superior to existing PSC approaches in terms of both accuracyand running time.
Revised:  Moreover, because graphlets can deal only with unweighted PSNs, and because accounting for edge weights when constructing PSNs could improve PSC accuracy, we also propose a deep learning framework that automatically learns network features from the weighted PSNs. When evaluated on a large set of 9, 509 CATH and 11,451 SCOP protein domains , our proposed approaches are superior to existing PSC approaches in terms of accuracy, with comparable running time.
Edit intent: clarity