wanyu/IteraTeR_full_sent huggingface dataset:
Original:  Let \Delta = \{\delta _0,\delta _1,\delta _2,\ldots \} be a countable set of real numbers which has at least m+1 elements. Let f:R^m\rightarrow R be a C^2 function. Let \alpha >0. For each x\in \mathbb{R^m such that \nabla f(x)}%DIFDELCMD < \not%%%  = 0, let \delta (x) = \delta _j, where j is the smallest number so that \nabla ^2f(x)+\delta _j||\nabla f(x)||^{1+\alphaId is invertible.
Revised:  Let \Delta = ^m such that \nabla f(x)}%DIFDELCMD < \not%%% = 0, let \delta (x) = \delta _j, where j is the smallest number so that \nabla ^2f(x)+\delta _j||\nabla f(x)||^{1+\alphaId is invertible.
Edit intent: clarity
Original:  For each x\in \mathbb{R^m such that \nabla f(x)}%DIFDELCMD < \not%%%  = 0, let \delta (x) = \delta _j, where j is the smallest number so that \nabla ^2f(x)+\delta _j||\nabla f(x)||^{1+\alphaId is invertible.
Revised:  For each x\in \mathbb{R^m such that \nabla f(x)}%DIFDELCMD < \not%%% x_n-w_n where w_n = 0, let \delta (x) = \delta _j, where j is the smallest number so that \nabla ^2f(x)+\delta _j||\nabla f(x)||^{1+\alphaId is invertible.
Edit intent: meaning-changed
Original:  For each x\in \mathbb{R^m such that \nabla f(x)}%DIFDELCMD < \not%%%  = 0, let \delta (x) = \delta _j, where j is the smallest number so that \nabla ^2f(x)+\delta _j||\nabla f(x)||^{1+\alphaId is invertible.
Revised:  For each x\in \mathbb{R^m such that \nabla f(x)}%DIFDELCMD < \not%%% = pr_{A_n,+ = \delta _j, where j is the smallest number so that \nabla ^2f(x)+\delta _j||\nabla f(x)||^{1+\alphaId is invertible.
Edit intent: meaning-changed
Original:  For each x\in \mathbb{R^m such that \nabla f(x)}%DIFDELCMD < \not%%%  = 0, let \delta (x) = \delta _j, where j is the smallest number so that \nabla ^2f(x)+\delta _j||\nabla f(x)||^{1+\alphaId is invertible.
Revised:  For each x\in \mathbb{R^m such that \nabla f(x)}%DIFDELCMD < \not%%% = 0, let \delta (x) = Id is invertible.
Edit intent: meaning-changed
Original:  We define a sequence of x_n\in \mathbb{R}^m and invertible and symmetric m\times m matrices A_nas follows: A_n=} \nabla ^2f(x_n)+\delta (x_n) ||\nabla f(x_n)|| ^{1+\alpha = x_n-w_n, where w_n=pr_{A_n,+ A_n^{-1}  \nabla f(x_n).
Revised:  We define a sequence of x_n\in \mathbb{R}^m and invertible and symmetric m\times m matrices A_nas follows: A_n=} \nabla ^2f(x_n)+\delta _n ||\nabla f(x_n)|| ^{1+\alpha = x_n-w_n, where w_n=pr_{A_n,+ A_n^{-1} \nabla f(x_n).
Edit intent: fluency
Original:  We define a sequence of x_n\in \mathbb{R}^m and invertible and symmetric m\times m matrices A_nas follows: A_n=} \nabla ^2f(x_n)+\delta (x_n) ||\nabla f(x_n)|| ^{1+\alpha = x_n-w_n, where w_n=pr_{A_n,+ A_n^{-1}  \nabla f(x_n).
Revised:  We define a sequence of x_n\in \mathbb{R}^m and invertible and symmetric m\times m matrices A_nas follows: A_n=} \nabla ^2f(x_n)+\delta (x_n) ||\nabla f(x_n)|| ^2 .Id and v_n = x_n-w_n, where w_n=pr_{A_n,+ A_n^{-1} \nabla f(x_n).
Edit intent: meaning-changed
Original:  We define a sequence of x_n\in \mathbb{R}^m and invertible and symmetric m\times m matrices A_nas follows: A_n=} \nabla ^2f(x_n)+\delta (x_n) ||\nabla f(x_n)|| ^{1+\alpha = x_n-w_n, where w_n=pr_{A_n,+ A_n^{-1}  \nabla f(x_n).
Revised:  We define a sequence of x_n\in \mathbb{R}^m and invertible and symmetric m\times m matrices A_nas follows: A_n=} \nabla ^2f(x_n)+\delta (x_n) ||\nabla f(x_n)|| ^{1+\alpha = A_n^{-1} \nabla f(x_n).
Edit intent: meaning-changed
Original:  We define a sequence of x_n\in \mathbb{R}^m and invertible and symmetric m\times m matrices A_nas follows: A_n=} \nabla ^2f(x_n)+\delta (x_n) ||\nabla f(x_n)|| ^{1+\alpha = x_n-w_n, where w_n=pr_{A_n,+ A_n^{-1}  \nabla f(x_n).
Revised:  We define a sequence of x_n\in \mathbb{R}^m and invertible and symmetric m\times m matrices A_nas follows: A_n=} \nabla ^2f(x_n)+\delta (x_n) ||\nabla f(x_n)|| ^{1+\alpha = x_n-w_n, where w_n=pr_{A_n,+ A_n^{-1} . \nabla f(x_n).
Edit intent: fluency
Original:   The main result of this paper roughly says that if f is C^3 and a sequence \{x_n\}, constructed by the New Q-Newton's method from a random initial point x_0, {\bf converges}, then the limit point is  not a saddle point, and the convergence rate is the same as that of Newton's method  .
Revised:  Here \delta _n is an appropriate real number so that A_n is invertible, and pr_{A_n,\pm The main result of this paper roughly says that if f is C^3 and a sequence \{x_n\}, constructed by the New Q-Newton's method from a random initial point x_0, {\bf converges}, then the limit point is not a saddle point, and the convergence rate is the same as that of Newton's method .
Edit intent: meaning-changed
Original:   The main result of this paper roughly says that if f is C^3 and a sequence \{x_n\}, constructed by the New Q-Newton's method from a random initial point x_0, {\bf converges}, then the limit point is  not a saddle point, and the convergence rate is the same as that of Newton's method  .
Revised:  The main result of this paper roughly says that if f is C^3 and a sequence \{x_n\}, constructed by the New Q-Newton's method from a random initial point x_0, {\bf converges}, then the limit point is a critical point and is not a saddle point, and the convergence rate is the same as that of Newton's method .
Edit intent: meaning-changed
Original:   The main result of this paper roughly says that if f is C^3 and a sequence \{x_n\}, constructed by the New Q-Newton's method from a random initial point x_0, {\bf converges}, then the limit point is  not a saddle point, and the convergence rate is the same as that of Newton's method  .
Revised:  The main result of this paper roughly says that if f is C^3 and a sequence \{x_n\}, constructed by the New Q-Newton's method from a random initial point x_0, {\bf converges}, then the limit point is not a saddle point, and the convergence rate is the same as that of Newton's method . At the end of the paper, we present some issues (saddle points and convergence) one faces when implementing Newton's method and modifications into Deep Neural Networks. In the appendix, we test the good performance of New Q-Newton's method on various benchmark test functions such as Rastrigin, Askley, Rosenbroch and many other, against algorithms such as Newton's method, BFGS, Adaptive Cubic Regularization, Random damping Newton's method and Inertial Newton's method, as well as Unbounded Two-way Backtracking Gradient Descent. The experiments demonstrate in particular that the assumption that f is C^3 is necessary for some conclusions in the main theoretical results .
Edit intent: meaning-changed
Original: This tutorial shows how to build, fit, and criticize disease transmission models in Stan, and should be useful to researchers interested in modeling the COVID-19 outbreak and doing Bayesian inference . Bayesian modeling provides a principled way to quantify uncertainty and incorporate  prior knowledge into the model .
Revised: This tutorial shows how to build, fit, and criticize disease transmission models in Stan, and should be useful to researchers interested in modeling the SARS-CoV-2 pandemic and other infectious diseases in a Bayesian framework . Bayesian modeling provides a principled way to quantify uncertainty and incorporate prior knowledge into the model .
Edit intent: clarity
Original: This tutorial shows how to build, fit, and criticize disease transmission models in Stan, and should be useful to researchers interested in modeling the COVID-19 outbreak and doing Bayesian inference . Bayesian modeling provides a principled way to quantify uncertainty and incorporate  prior knowledge into the model .
Revised: This tutorial shows how to build, fit, and criticize disease transmission models in Stan, and should be useful to researchers interested in modeling the COVID-19 outbreak and doing Bayesian inference . Bayesian modeling provides a principled way to quantify uncertainty and incorporate both data and prior knowledge into the model .
Edit intent: clarity
Original: This tutorial shows how to build, fit, and criticize disease transmission models in Stan, and should be useful to researchers interested in modeling the COVID-19 outbreak and doing Bayesian inference . Bayesian modeling provides a principled way to quantify uncertainty and incorporate  prior knowledge into the model . What is more, Stan 's main inference engine, Hamiltonian Monte Carlo sampling, is amiable to diagnostics, which means we can verify whether our inference is reliable. Stan is an expressive probabilistic programing language that abstracts the inference and allows users to focus on the modeling.
Revised: This tutorial shows how to build, fit, and criticize disease transmission models in Stan, and should be useful to researchers interested in modeling the COVID-19 outbreak and doing Bayesian inference . Bayesian modeling provides a principled way to quantify uncertainty and incorporate prior knowledge into the model estimates. Stan is an expressive probabilistic programing language that abstracts the inference and allows users to focus on the modeling.
Edit intent: clarity
Original:  In this tutorial, we demonstrate with a simple Susceptible-Infected-Recovered (SIR) model how to formulate, fit, and diagnose a compartmental  model in Stan  .
Revised:  In this tutorial, we demonstrate with a simple Susceptible-Infected-Recovered (SIR) model how to formulate, fit, and diagnose a compartmental model in Stan , first with a simple Susceptible-Infected-Recovered (SIR) model, then with a more elaborate transmission model used during the SARS-CoV-2 pandemic .
Edit intent: meaning-changed
Original: Difficulty algorithms are a fundamental component of Proof-of-Work blockchains, aimed at maintaining stable block production times by dynamically adjusting the network difficulty in response to the miners' constantly changing computational power.
Revised: Difficulty algorithms are a fundamental component of Proof-of-Work blockchains, difficulty algorithms serve the crucial purpose of maintaining a stable transaction throughput by dynamically adjusting the network difficulty in response to the miners' constantly changing computational power.
Edit intent: clarity
Original:  However, without careful design this could create vulnerabilities that incentivize miners to engage in coin-hopping strategies which yield an unreliable system due to  unstable processing of transactions.
Revised:  However, without careful design this could create vulnerabilities that incentivize miners to engage in coin-hopping strategies to manipulate the block difficulty for profit. Such miner behavior results in an unreliable system due to unstable processing of transactions.
Edit intent: meaning-changed
Original:  We compare the described algorithm to that of Bitcoin Cash in a simulated mining environment and verify that the former would eliminate the severe oscillations in block solve times. Lastly, we outline how this model can more generally replace difficulty algorithms in other Proof-of-Work blockchains .
Revised:  We compare the described algorithm to that of Bitcoin Cash in a simulated mining environment and verify that the former would eliminate the severe oscillations in transaction throughput .
Edit intent: clarity
Original: The combination of neuroscience-oriented and computer-science-oriented approaches is the most promising method to develop artificial general intelligence (AGI) that can learn general tasks similar to humans. Currently, two main routes of learning exist, including neuroscience-inspired methods , represented by local synaptic plasticity, and machine-learning methods, represented by backpropagation .
Revised: Two main routes of learning exist, including neuroscience-inspired methods , represented by local synaptic plasticity, and machine-learning methods, represented by backpropagation .
Edit intent: coherence
Original:   Here, we report a hybrid spiking neural network model that integrates the two approaches by introducing a  meta-local module and a two-phase causality modellingmethod.
Revised:  On the other hand, spiking neural network embodies "computation" in spatiotemporal domain with unique features of rich coding scheme and threshold switching, which is very suitable for low power and high parallel neuromorphic computing. Here, we report a hybrid spiking neural network model that integrates the two approaches by introducing a meta-local module and a two-phase causality modellingmethod.
Edit intent: meaning-changed
Original:   Here, we report a hybrid spiking neural network model that integrates the two approaches by introducing a  meta-local module and a two-phase causality modellingmethod. The model can not only optimize local plasticityrules, but also receive top-down supervision information .
Revised:  Here, we report a hybrid spiking neural network model that integrates the two approaches by introducing a meta-local module and a two-phase parametric modelling. The hybrid model can meta-learn general local plasticity, and receive top-down supervision information .
Edit intent: meaning-changed
Original:  The model can not only optimize local plasticityrules, but also receive top-down supervision information . In addition to flexibly supporting multiple spike-based coding schemes, we demonstrate that this  model facilitates learning of many general tasks, including fault-tolerance learning, few-shot learning and multiple-task learning , and show its efficiency on the Tianjic neuromorphic platform .
Revised:  The model can not only optimize local plasticityrules, but also receive top-down supervision information for multi-scale learning. We demonstrate that this model facilitates learning of many general tasks, including fault-tolerance learning, few-shot learning and multiple-task learning , and show its efficiency on the Tianjic neuromorphic platform .
Edit intent: clarity
Original:  In addition to flexibly supporting multiple spike-based coding schemes, we demonstrate that this  model facilitates learning of many general tasks, including fault-tolerance learning, few-shot learning and multiple-task learning , and show its efficiency on the Tianjic neuromorphic platform .
Revised:  In addition to flexibly supporting multiple spike-based coding schemes, we demonstrate that this hybrid model facilitates learning of many general tasks, including fault-tolerance learning, few-shot learning and multiple-task learning , and show its efficiency on the Tianjic neuromorphic platform .
Edit intent: meaning-changed
Original:  In addition to flexibly supporting multiple spike-based coding schemes, we demonstrate that this  model facilitates learning of many general tasks, including fault-tolerance learning, few-shot learning and multiple-task learning , and show its efficiency on the Tianjic neuromorphic platform .
Revised:  In addition to flexibly supporting multiple spike-based coding schemes, we demonstrate that this model facilitates learning of many general tasks, including fault-tolerance learning, few-shot learning and multiple-task learning . Furthermore, the implementation of the hybrid model on the Tianjic neuromorphic platform .
Edit intent: meaning-changed
Original:  In addition to flexibly supporting multiple spike-based coding schemes, we demonstrate that this  model facilitates learning of many general tasks, including fault-tolerance learning, few-shot learning and multiple-task learning , and show its efficiency on the Tianjic neuromorphic platform . This work provides a new route for brain-inspired computing and facilitates AGI development .
Revised:  In addition to flexibly supporting multiple spike-based coding schemes, we demonstrate that this model facilitates learning of many general tasks, including fault-tolerance learning, few-shot learning and multiple-task learning , and show its efficiency on the Tianjic neuromorphic platform proves that it can fully utilize the advantages of neuromorphic hardware architecture and promote energy-efficient on-chip applications .
Edit intent: meaning-changed
Original: In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM), and its local kernel variants, to approximate second-order differential operators defined on smooth manifolds with boundary that naturally arise in elliptic PDE models.
Revised: In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM), and its local kernel variants, to approximate second-order differential operators defined on smooth manifolds with boundaries that naturally arise in elliptic PDE models.
Edit intent: fluency
Original:  To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points  specified along the  tangential direction at the boundary of the original manifold .
Revised:  To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points , specified along the tangential direction at the boundary of the original manifold .
Edit intent: fluency
Original:  To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points  specified along the  tangential direction at the boundary of the original manifold .
Revised:  To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points specified along the estimated tangential direction at the boundary of the original manifold .
Edit intent: meaning-changed
Original:  To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points  specified along the  tangential direction at the boundary of the original manifold .
Revised:  To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points specified along the tangential direction at the sampled points at the boundary .
Edit intent: clarity
Original:  As opposed to the classical DM which diverges near the boundary, the proposed GPDM estimator converges pointwise even near the boundary , assuming that the extended manifold is smooth . Applying the consistent GPDM estimator to solve the well-posed elliptic PDEs with classical boundary conditions (Dirichlet, Neumann, and Robin), we establish the convergence of the approximate solution under appropriate smoothness assumptions.
Revised:  As opposed to the classical DM which diverges near the boundary, the proposed GPDM estimator converges pointwise even near the boundary . Applying the consistent GPDM estimator to solve the well-posed elliptic PDEs with classical boundary conditions (Dirichlet, Neumann, and Robin), we establish the convergence of the approximate solution under appropriate smoothness assumptions.
Edit intent: clarity
Original:  We  validate the proposed mesh-free PDE solver with supporting numerical examples on various problems , defined on simple manifolds embedded in 2D to 5D Euclidean spaces as well as on an unknown manifold.
Revised:  We numerically validate the proposed mesh-free PDE solver with supporting numerical examples on various problems , defined on simple manifolds embedded in 2D to 5D Euclidean spaces as well as on an unknown manifold.
Edit intent: clarity
Original:  We  validate the proposed mesh-free PDE solver with supporting numerical examples on various problems , defined on simple manifolds embedded in 2D to 5D Euclidean spaces as well as on an unknown manifold.
Revised:  We validate the proposed mesh-free PDE solver on various problems , defined on simple manifolds embedded in 2D to 5D Euclidean spaces as well as on an unknown manifold.
Edit intent: clarity
Original:  We  validate the proposed mesh-free PDE solver with supporting numerical examples on various problems , defined on simple manifolds embedded in 2D to 5D Euclidean spaces as well as on an unknown manifold.
Revised:  We validate the proposed mesh-free PDE solver with supporting numerical examples on various problems defined on simple manifolds embedded in 2D to 5D Euclidean spaces as well as on an unknown manifold.
Edit intent: fluency
Original:  We  validate the proposed mesh-free PDE solver with supporting numerical examples on various problems , defined on simple manifolds embedded in 2D to 5D Euclidean spaces as well as on an unknown manifold.
Revised:  We validate the proposed mesh-free PDE solver with supporting numerical examples on various problems , defined on simple sub-manifolds embedded in Euclidean spaces as well as on an unknown manifold.
Edit intent: meaning-changed
Original:  The coefficients in the model are evaluated by fitting to empirical data for six US states: California, Louisiana, New Jersey, New York State, Texas, and Washington State .
Revised:  The coefficients in the model are evaluated by fitting to empirical data for six US states: California, Louisiana, New Jersey, New York State, and Texas. Together these states make up 43\% of the US population; some of these states appear to have handled their initial outbreaks well, while others appear to be emerging hotspots .
Edit intent: meaning-changed
Original:  Should  stay-at-home orders be revoked , most states may experience oscillating yearly infections. Even if additional lockdowns are applied in Texas, and then released according to White House guidelines of 14 days of decreasing cases, a similar endemic situation may occur. Additionally, if lockdowns had been instituted one to three weeks sooner , the number of Covid-19 deaths in New York could have been significantly reduced, but surprisingly not in Texas (perhaps due to aversion to lockdown and absence of a decree).
Revised:  Should stay-at-home orders be revoked , most states may experience a secondary peak in 2021. If lockdowns had been instituted one to three weeks sooner , the number of Covid-19 deaths in New York could have been significantly reduced, but surprisingly not in Texas (perhaps due to aversion to lockdown and absence of a decree).
Edit intent: meaning-changed
Original:  Additionally, if lockdowns had been instituted one to three weeks sooner , the number of Covid-19 deaths in New York could have been significantly reduced, but surprisingly not in Texas (perhaps due to aversion to lockdown and absence of a decree).
Revised:  Additionally, if lockdowns had been kept in place , the number of Covid-19 deaths in New York could have been significantly reduced, but surprisingly not in Texas (perhaps due to aversion to lockdown and absence of a decree).
Edit intent: clarity
Original:  Additionally, if lockdowns had been instituted one to three weeks sooner , the number of Covid-19 deaths in New York could have been significantly reduced, but surprisingly not in Texas (perhaps due to aversion to lockdown and absence of a decree).
Revised:  Additionally, if lockdowns had been instituted one to three weeks sooner , the number of Covid-19 deaths so far could have been significantly reduced, but surprisingly not in Texas (perhaps due to aversion to lockdown and absence of a decree).
Edit intent: meaning-changed