wanyu/IteraTeR_v2 huggingface dataset:
Original: In this paper, we present a method to determine if a lift-and-project cut for a mixed-integer linear program is regular , in which case the cut is equivalent to an intersection cut from some basis of the linear relaxation.
Revised: In this paper, we present a method to determine if a lift-and-project cut for a mixed-integer linear program is regular , in which case the cut is equivalent to an intersection cut from the bases of the linear relaxation.
Edit intent: clarity
Original:  While it is known since Balas and Perregaard (2003) that such equivalence always holds for lift-and-project cuts from split disjunctions  , Balas and Kis (2016) have recently shown that there is a necessary and sufficient condition in the case of arbitrary disjunctions: a lift-and-project cut is regular if, and only if, it corresponds to a regular  solution of the Cut Generating Linear Program (CGLP).
Revised:  While it is known since Balas and Perregaard (2003) that lift-and-project cuts from split disjunctions , Balas and Kis (2016) have recently shown that there is a necessary and sufficient condition in the case of arbitrary disjunctions: a lift-and-project cut is regular if, and only if, it corresponds to a regular solution of the Cut Generating Linear Program (CGLP).
Edit intent: clarity
Original:  Third, we describe a numerical procedure that verifies if a lift-and-project cut is not regular .
Revised:  Third, we describe a numerical procedure based on such formulation that identifies irregular lift-and-project cut is not regular .
Edit intent: clarity
Original:  Third, we describe a numerical procedure that verifies if a lift-and-project cut is not regular .
Revised:  Third, we describe a numerical procedure that verifies if a lift-and-project cuts .
Edit intent: clarity
Original:  Recent studies have found trends in the evolution of music styles, but little is known about quantitative laws and theories .
Revised:  Recent studies have found trends in the evolution of music styles, but little is known about their relations to the evolution theory .
Edit intent: clarity
Original: The greedy leaf removal (GLR) procedure on a graph is the iterative removal of any leaf ( vertex with degree one ) with its nearest neighbor (root) , which is adopted heuristically to find approximate solutions in some combinatorial optimization problems, such as the minimal vertex cover (MVC) problem and the maximal matching (MM) problem.
Revised: The greedy leaf removal (GLR) procedure on a graph is an iterative removal of any leaf ( vertex with degree one ) with its nearest neighbor (root) , which is adopted heuristically to find approximate solutions in some combinatorial optimization problems, such as the minimal vertex cover (MVC) problem and the maximal matching (MM) problem.
Edit intent: fluency
Original: The greedy leaf removal (GLR) procedure on a graph is the iterative removal of any leaf ( vertex with degree one ) with its nearest neighbor (root) , which is adopted heuristically to find approximate solutions in some combinatorial optimization problems, such as the minimal vertex cover (MVC) problem and the maximal matching (MM) problem.
Revised: The greedy leaf removal (GLR) procedure on a graph is the iterative removal of any vertex with degree one ) with its nearest neighbor (root) , which is adopted heuristically to find approximate solutions in some combinatorial optimization problems, such as the minimal vertex cover (MVC) problem and the maximal matching (MM) problem.
Edit intent: fluency
Original: The greedy leaf removal (GLR) procedure on a graph is the iterative removal of any leaf ( vertex with degree one ) with its nearest neighbor (root) , which is adopted heuristically to find approximate solutions in some combinatorial optimization problems, such as the minimal vertex cover (MVC) problem and the maximal matching (MM) problem.
Revised: The greedy leaf removal (GLR) procedure on a graph is the iterative removal of any leaf ( vertex with degree one (leaf) along with its nearest neighbor (root) , which is adopted heuristically to find approximate solutions in some combinatorial optimization problems, such as the minimal vertex cover (MVC) problem and the maximal matching (MM) problem.
Edit intent: clarity
Original:  The result of the GLR procedure has two faces , the residual subgraph (core) and the collection of removed roots.
Revised:  The result of the GLR procedure has two faces : a residual subgraph as a core, and a set of removed roots.
Edit intent: clarity
Original:  While the emergence of cores (core percolation) on general random graphs has been solved analytically, a theoretical analysis of roots is ignored except in the case of Erd\"{o}s-R\'{e}nyi random graphs.
Revised:  While the emergence of cores (core percolation) on general random graphs has been solved analytically, a theory for roots is ignored except in the case of Erd\"{o}s-R\'{e}nyi random graphs.
Edit intent: clarity
Original:  To explore the lesser studied face, we extend the theory of cores to study roots on uncorrelated random graphs.
Revised:  Here we analytically study roots on uncorrelated random graphs.
Edit intent: clarity
Original:  To explore the lesser studied face, we extend the theory of cores to study roots on uncorrelated random graphs.
Revised:  To explore the lesser studied face, we extend the theory of cores to study roots on random graphs.
Edit intent: clarity
Original:  We further show that our result leads to a general analytical framework to estimate MVCs on random graphs without cores and MMs on random graphs .
Revised:  We further show that our result leads to a general analytical framework to estimate MVCs on random graphs with or without cores .
Edit intent: clarity
Original: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance  to the previous collision s . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Revised: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-sections \Sigma_t(s) depends on the distance to the previous collision s . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Edit intent: fluency
Original: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance  to the previous collision s . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Revised: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance s to the previous collision s . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Edit intent: fluency
Original: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance  to the previous collision s . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Revised: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance to the previous collision or boundary . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Edit intent: clarity
Original: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance  to the previous collision s . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Revised: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance to the previous collision s . We generalize collision and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Edit intent: clarity
Original: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance  to the previous collision s . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Revised: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance to the previous collision s . We generalize collision , track-length, and track-length estimators, and other related Monte Carlo methods, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Edit intent: clarity
Original: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance  to the previous collision s . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Revised: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance to the previous collision s . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or collision rates in generalized transport theory .
Edit intent: clarity
Original: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance  to the previous collision s . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized transport theory .
Revised: When lifting the classical assumption of independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance to the previous collision s . We generalize collision , track-length, and expected-value Monte Carlo estimators, and some of their common variations, to support unbiased estimation of either flux integrals or reaction rates in generalized radiative transfer (GRT) .
Edit intent: clarity
Original:  We consider collision estimators with additional ficticious scattering which , in the limit of infinite ficticious density, become track-length estimators.
Revised:  We consider collision estimators with additional fictitious scattering that , in the limit of infinite ficticious density, become track-length estimators.
Edit intent: fluency
Original:  We consider collision estimators with additional ficticious scattering which , in the limit of infinite ficticious density, become track-length estimators.
Revised:  We consider collision estimators with additional ficticious scattering which , in the limit of infinite fictitious density, become track-length estimators.
Edit intent: fluency
Original:  The generalization includes support for the distinction between correlated- and uncorrelated-origin free-path distributions , which are both required to imbed non-exponential transport in bounded scenes in a reciprocal manner.
Revised:  The generalizations support distinct correlated- and uncorrelated-origin free-path distributions , which are both required to imbed non-exponential transport in bounded scenes in a reciprocal manner.
Edit intent: clarity
Original:  The generalization includes support for the distinction between correlated- and uncorrelated-origin free-path distributions , which are both required to imbed non-exponential transport in bounded scenes in a reciprocal manner.
Revised:  The generalization includes support for the distinction between correlated- and uncorrelated-origin free-path statistics , which are both required to imbed non-exponential transport in bounded scenes in a reciprocal manner.
Edit intent: clarity
Original:  The generalization includes support for the distinction between correlated- and uncorrelated-origin free-path distributions , which are both required to imbed non-exponential transport in bounded scenes in a reciprocal manner.
Revised:  The generalization includes support for the distinction between correlated- and uncorrelated-origin free-path distributions , which are required to imbed non-exponential transport in bounded scenes in a reciprocal manner.
Edit intent: clarity
Original:  To numerically evaluate the estimators we derive new Green's functions for the isotropic point source in infinite media with isotropic scattering , derived using a Fourier transform approach . Additionally, new moment-preserving diffusion approximations for non-exponential random media are derivedfor both correlated and uncorrelated emission , and their accuracy is evaluated for Gamma and power-law free-path distributions.
Revised:  To numerically evaluate the estimators we derive new Green's functions for the isotropic point source in infinite media with isotropic scattering . Additionally, new moment-preserving diffusion approximations for non-exponential random media are derivedfor both correlated and uncorrelated emission , and their accuracy is evaluated for Gamma and power-law free-path distributions.
Edit intent: clarity
Original:  To numerically evaluate the estimators we derive new Green's functions for the isotropic point source in infinite media with isotropic scattering , derived using a Fourier transform approach . Additionally, new moment-preserving diffusion approximations for non-exponential random media are derivedfor both correlated and uncorrelated emission , and their accuracy is evaluated for Gamma and power-law free-path distributions.
Revised:  To numerically evaluate the estimators we derive new Green's functions for the isotropic point source in infinite media with isotropic scattering , derived using a Fourier transform approach . Additionally, new moment-preserving diffusion approximations for these Green's functions are derived , and their accuracy is evaluated for Gamma and power-law free-path distributions.
Edit intent: clarity
Original: When lifting the classical assumption of spatially-independent scattering centers in  linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-sections \Sigma_t(s) depends on the distance s to the previous collision or boundary.
Revised: When lifting the assumption of spatially-independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-sections \Sigma_t(s) depends on the distance s to the previous collision or boundary.
Edit intent: clarity
Original: When lifting the classical assumption of spatially-independent scattering centers in  linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-sections \Sigma_t(s) depends on the distance s to the previous collision or boundary.
Revised: When lifting the classical assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-sections \Sigma_t(s) depends on the distance s to the previous collision or boundary.
Edit intent: fluency
Original: When lifting the classical assumption of spatially-independent scattering centers in  linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-sections \Sigma_t(s) depends on the distance s to the previous collision or boundary.
Revised: When lifting the classical assumption of spatially-independent scattering centers in linear transport theory, collision rate is no longer proportional to flux because the macroscopic cross-section \Sigma_t(s) depends on the distance s to the previous collision or boundary.
Edit intent: fluency
Original:  We generalize collision and track-length estimators, and other related Monte Carlo methods , to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT).
Revised:  We generalize collision and track-length estimators, and other related Monte Carlo methods . We generalize collision and track-length estimators to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT).
Edit intent: fluency
Original:  This creates a nonlocal relationship between collision rate and flux and requires revising a number of familiar deterministic and Monte Carlo methods.
Edit intent: coherence
Original:  Additionally, new moment-preserving diffusion approximations for these Green's functions are derived and reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.
Revised:  Additionally, new moment-preserving diffusion approximations for these Green's functions are derived , which reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.
Edit intent: coherence
Original: When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to vector flux / radiance because the macroscopic cross-section \Sigma_t(s) depends on the distance s to the previous collision or boundary.
Revised: When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to angular flux / radiance because the macroscopic cross-section \Sigma_t(s) depends on the distance s to the previous collision or boundary.
Edit intent: clarity
Original: A polygonal mesh representation provides an efficient approximation for 3D shapes.
Revised: Polygonal meshes provide an efficient representation for 3D shapes.
Edit intent: clarity
Original:  It explicitly captures both shape surface and topology, and leverages non-uniformity to represent large flat regions as well as sharp, intricate features.
Revised:  They explicitly capture both shape surface and topology, and leverages non-uniformity to represent large flat regions as well as sharp, intricate features.
Edit intent: fluency
Original:  It explicitly captures both shape surface and topology, and leverages non-uniformity to represent large flat regions as well as sharp, intricate features.
Revised:  It explicitly captures both shape surface and topology, and leverage non-uniformity to represent large flat regions as well as sharp, intricate features.
Edit intent: fluency
Original: We compare two approaches for modelling imperfect information in infinite games on finite-state systems.
Revised: We compare two approaches for modelling imperfect information in infinite games by using finite-state systems.
Edit intent: clarity
Original:  As a new approach, we consider a model based on indistinguishability relations described by synchronous two-tape automata.
Revised:  In contrast, the second approach features indistinguishability relations described by synchronous two-tape automata.
Edit intent: clarity
Original:  We point out that the indistinguishability-relation model is strictly more expressive than the one based on observations.
Revised:  The indistinguishability-relation model is strictly more expressive than the one based on observations.
Edit intent: coherence
Original:  We point out that the indistinguishability-relation model is strictly more expressive than the one based on observations.
Revised:  We point out that the indistinguishability-relation model turns out to be strictly more expressive than the one based on observations.
Edit intent: clarity
Original:  Further, we give an abstract characterisation of indistinguishablity relations that admit an equivalent representation by a finite-state observation function.
Revised:  Further, we give an abstract characterisation of indistinguishablity relations that admit a representation as a finite-state observation function.
Edit intent: clarity
Original:  Finally, we present an algorithm for constructing an observation function equivalent to a given indistinguishability relation, whenever this is possible .
Revised:  We show that the characterisation is decidable, and give a procedure to construct a corresponding Mealy machine whenever one exists .
Edit intent: clarity
Original:  As it bridges information on the neural support of mental processes, this representation improves decoding performance for 80\% of the 35 widely-different functional imaging studiesthat we consider.
Revised:  As it bridges information on the neural support of mental processes, this representation improves decoding performance for 80\% of 35 widely-different functional imaging studiesthat we consider.
Edit intent: fluency
Original:  However, since the earliest days of the field, some scholars have questioned whether all citations should value the same, and have gone on to weight them by a variety of factors.
Revised:  However, since the earliest days of the field, some scholars have questioned whether all citations should be worth the same, and have gone on to weight them by a variety of factors.
Edit intent: clarity
Original:  This work takes an alternate approach to resolving the underlying problem: the proposal is to value citations by the impact of the citing articles  .
Revised:  This work takes an alternative approach to resolving the underlying problem: the proposal is to value citations by the impact of the citing articles .
Edit intent: fluency