wanyu/IteraTeR_v2 huggingface dataset:
Original:  The algorithm takes as input a short chain complex of free modules %DIFDELCMD < \[ F^2 \partial^2 F^1 \partial^1 %DIFDELCMD < F^0 \] %%%  such that M\cong \ker{\partial^1 / \mathrm{im{\partial^2}}  . It runs in time O( \sum_i | F^i |^3  ) and requires O( \sum_i | F^i |^2  ) memory, where | F^i | denotes the size of a basis of F^i.
Revised:  The algorithm takes as input a short chain complex of free modules %DIFDELCMD < \[ F^2 \partial^2 F^1 \partial^1 %DIFDELCMD < F^0 \] %%% such that M\cong \ker{\partial^1 / \mathrm{im{\partial^2}} . It runs in time O( \sum_i | F^i |^3 ) and requires O( | F^i |^2 ) memory, where | F^i | denotes the size of a basis of F^i.
Edit intent: fluency
Original:  We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, such as the height of the cavity.
Revised:  We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity.
Edit intent: fluency
Original:  We construct an a posteriori error bound estimator, based upon the Brezzi-Rapaz-Raviart theory, used in the greedy algorithm for the selection of the basis functions.
Revised:  We construct an a posterior error estimator, based upon the Brezzi-Rapaz-Raviart theory, used in the greedy algorithm for the selection of the basis functions.
Edit intent: clarity
Original:  We construct an a posteriori error bound estimator, based upon the Brezzi-Rapaz-Raviart theory, used in the greedy algorithm for the selection of the basis functions.
Revised:  We construct an a posteriori error bound estimator, based upon the Brezzi-Rappaz-Raviart theory, used in the greedy algorithm for the selection of the basis functions.
Edit intent: fluency
Original:  Finally we present several numerical tests for different configuration of the parameters .
Revised:  Finally we present several numerical tests for different parameter configuration .
Edit intent: clarity
Original: Progression of complex human diseases is associated with  transitions across dynamical regimes.
Revised: The progression of complex human diseases is associated with transitions across dynamical regimes.
Edit intent: fluency
Original: Homochirality is a generic and unique property of all biochemical life and the fractional circular polarization  it induces therefore constitutes an unambiguous biosignature.
Revised: Homochirality is a generic and unique property of all biochemical life and the fractional circular polarization it induces therefore constitutes a potentially unambiguous biosignature.
Edit intent: clarity
Original:  Our approach allows  to study and explore one of the most potential sources of reproducible cortical folding pattern: the specification of initial geometry of the brain.
Revised:  Our approach allows us to study and explore one of the most potential sources of reproducible cortical folding pattern: the specification of initial geometry of the brain.
Edit intent: fluency
Original: A crucial step in the analysis of persistent homology is  transformation of data into  a simplicial complex  . Modern packages for persistent homology often construct Vietoris--Rips or other distance-based simplicial complexes on point clouds because they are relatively easy to compute.
Revised: A crucial step in the analysis of persistent homology is the transformation of data into a simplicial complex . Modern packages for persistent homology often construct Vietoris--Rips or other distance-based simplicial complexes on point clouds because they are relatively easy to compute.
Edit intent: fluency
Original: A crucial step in the analysis of persistent homology is  transformation of data into  a simplicial complex  . Modern packages for persistent homology often construct Vietoris--Rips or other distance-based simplicial complexes on point clouds because they are relatively easy to compute.
Revised: A crucial step in the analysis of persistent homology is transformation of data into a simplicial complex ) . Modern packages for persistent homology often construct Vietoris--Rips or other distance-based simplicial complexes on point clouds because they are relatively easy to compute.
Edit intent: fluency
Original:  The regions with high coordination between the phosphate groups and the divalent cations are discernible  at very low  concentrations when the ribozyme does not form tertiary interactions.
Revised:  The regions with high coordination between the phosphate groups and the divalent cations are discernible even at very low concentrations when the ribozyme does not form tertiary interactions.
Edit intent: clarity
Original:  A massively parallel implementation of the algorithm is demonstrated on a GPU .
Revised:  A massively parallel implementation of the algorithm is demonstrated on a graphics processing unit (GPU) .
Edit intent: clarity
Original: In many machine learning applications, one needs to interactively select a sequence of items (e.g., recommending movies based on a user's feedback) or make sequential decisions in certain orders (e.g., guiding an agent through a series of states).
Revised: In many machine learning applications, one needs to interactively select a sequence of items (e.g., recommending movies based on a user's feedback) or make sequential decisions in a certain order (e.g., guiding an agent through a series of states).
Edit intent: fluency
Original:  Not only do sequences already pose a dauntingly large search space, but we must  take into account past observations, as well as the uncertainty of future outcomes.
Revised:  Not only do sequences already pose a dauntingly large search space, but we must also take into account past observations, as well as the uncertainty of future outcomes.
Edit intent: fluency
Original:  We show that once such dependencies are encoded by a directed graph, an adaptive greedy policy is guaranteed to achieve a constant factor approximation guarantee, where the constant naturally depends on the structural properties of the underlying graph . Additionally, to demonstrate the practical utility of our results, we run experiments on Amazon product recommendation and Wikipedia link prediction tasks.
Revised:  We show that once such dependencies are encoded by a directed graph, an adaptive greedy policy with strong theoretical guarantees . Additionally, to demonstrate the practical utility of our results, we run experiments on Amazon product recommendation and Wikipedia link prediction tasks.
Edit intent: clarity
Original:  We cover several related developments: achievability bounds for optimal decoding methods, efficient algorithms with practical storage and computation requirements,  and algorithm-independent converse bounds.
Revised:  We cover several related developments: efficient algorithms with practical storage and computation requirements, and algorithm-independent converse bounds.
Edit intent: clarity
Original:  We cover several related developments: achievability bounds for optimal decoding methods, efficient algorithms with practical storage and computation requirements,  and algorithm-independent converse bounds.
Revised:  We cover several related developments: achievability bounds for optimal decoding methods, efficient algorithms with practical storage and computation requirements, achievability bounds for optimal decoding methods, and algorithm-independent converse bounds.
Edit intent: fluency
Original:  We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the rate and capacity of group testing, indicating the amount of information learned per test.
Revised:  We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the "rate" of group testing, indicating the amount of information learned per test.
Edit intent: clarity
Original:  We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the "rate" {\em  of group testing, indicating the amount of information learned per test.
Revised:  We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the {\em of group testing, indicating the amount of information learned per test.
Edit intent: clarity
Original:  We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the "rate" {\em  of group testing, indicating the amount of information learned per test.
Revised:  We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the "rate" {\em rate of group testing, indicating the amount of information learned per test.
Edit intent: clarity
Original: Programming digital contracts comes with unique challenges, which include describing and enforcing protocols of interaction, controlling resource usage, and tracking linear assets.
Revised: Programming digital contracts comes with unique challenges, which include expressing and enforcing protocols of interaction, controlling resource usage, and tracking linear assets.
Edit intent: clarity
Original:  This article presents the type-theoretic foundation of Nomos, a programming language for digital contracts whose strong static guarantees match the domain-specific requirements and facilitate contract development.
Revised:  This article presents the type-theoretic foundation of Nomos, a programming language for digital contracts whose type system addresses the aforementioned domain-specific requirements and facilitate contract development.
Edit intent: clarity
Original:  To control resource usage, Nomos uses resource-aware session types and automatic amortized resource analysis, a type based technique for inferring resource bounds.
Revised:  To control resource usage, Nomos uses resource-aware session types and automatic amortized resource analysis, a type-based technique for inferring resource bounds.
Edit intent: fluency
Original:  To demonstrate the practicability of Nomos' session-type--based design, we implemented three digital contracts  in an existing prototype language .
Revised:  To demonstrate the practicability of Nomos' session-type--based design, we report on the performance of three digital contracts in an existing prototype language .
Edit intent: clarity
Original:  To demonstrate the practicability of Nomos' session-type--based design, we implemented three digital contracts  in an existing prototype language .
Revised:  To demonstrate the practicability of Nomos' session-type--based design, we implemented three digital contracts implemented in an existing prototype language .
Edit intent: clarity
Original:  This article presents the  type-theoretic foundation of Nomos, a programming language for digital contracts whose type system addresses the aforementioned domain-specific requirements .
Revised:  This article presents the design and type-theoretic foundation of Nomos, a programming language for digital contracts whose type system addresses the aforementioned domain-specific requirements .
Edit intent: clarity
Original:  This article presents the  type-theoretic foundation of Nomos, a programming language for digital contracts whose type system addresses the aforementioned domain-specific requirements .
Revised:  This article presents the type-theoretic foundation of Nomos, a programming language for digital contracts that addresses these challenges .
Edit intent: clarity
Original:  To express and enforce protocols, Nomos is based on shared binary session types rooted in linear logic .
Revised:  To express and enforce protocols, Nomos is based on shared binary session types .
Edit intent: clarity
Original:  To control resource usage, Nomos uses resource-aware session types and automatic amortized resource analysis , a type-based technique for inferring resource bounds.
Revised:  To control resource usage, Nomos employs automatic amortized resource analysis , a type-based technique for inferring resource bounds.
Edit intent: coherence
Original:  To track linear assets, Nomos employs a linear type system that prevents assets from being duplicated or discarded.
Revised:  To track linear assets, Nomos uses a linear type system that prevents assets from being duplicated or discarded.
Edit intent: fluency
Original: Light field cameras provide intriguing possibilities, such as post-capture refocus or the ability to look behind an object .
Revised: Light field cameras provide intriguing possibilities, such as post-capture refocus or the ability to synthesize images from novel viewpoints .
Edit intent: clarity
Original:  To avoid this, we introduce a sheared local Fourier slice equation that allows for refocusing directly from a compressed light field, either to obtain an image or a compressed representation of it.
Revised:  To avoid this, we introduce local Fourier slice equation that allows for refocusing directly from a compressed light field, either to obtain an image or a compressed representation of it.
Edit intent: clarity
Original:  To avoid this, we introduce a sheared local Fourier slice equation that allows for refocusing directly from a compressed light field, either to obtain an image or a compressed representation of it.
Revised:  To avoid this, we introduce a sheared local Fourier slice photography that allows for refocusing directly from a compressed light field, either to obtain an image or a compressed representation of it.
Edit intent: clarity
Original:  That research is complemented in this paper where the differentiability of reservoir filters is fully characterized for very general classes of discrete-time deterministic inputs.
Revised:  That research is complemented in this paper with the characterization of the differentiability of reservoir filters is fully characterized for very general classes of discrete-time deterministic inputs.
Edit intent: clarity
Original:  That research is complemented in this paper where the differentiability of reservoir filters is fully characterized for very general classes of discrete-time deterministic inputs.
Revised:  That research is complemented in this paper where the differentiability of reservoir filters for very general classes of discrete-time deterministic inputs.
Edit intent: clarity
Original:  The local nature of the differential allows the formulation of conditions that ensure both the local and global existence of differentiable and, in passing, fading memory solutions, which links to existing research on the input-dependent nature of the echo state property .
Revised:  The local nature of the differential allows the formulation of conditions that ensure both the local and global existence of differentiable and, in particular, links to existing research on the input-dependent nature of the echo state property .
Edit intent: clarity
Original:  The concepts presented in this article are based on an exhaustive research of recent scientific publications and thematically cover the largest part of the industrial automation pyramid layers.
Revised:  The concepts presented in this article thematically cover the largest part of the industrial automation pyramid layers.
Edit intent: clarity
Original: We develop two deterministic approximation algorithms for the maximization of non-monotone submodular functions under cardinality constraint : both are based upon the novel idea of interlacing two greedy procedures.
Revised: A deterministic approximation algorithm is presented for the maximization of non-monotone submodular functions under cardinality constraint : both are based upon the novel idea of interlacing two greedy procedures.
Edit intent: clarity
Original: We develop two deterministic approximation algorithms for the maximization of non-monotone submodular functions under cardinality constraint : both are based upon the novel idea of interlacing two greedy procedures.
Revised: We develop two deterministic approximation algorithms for the maximization of non-monotone submodular functions under cardinality constraint : both are based upon the idea of interlacing two greedy procedures.
Edit intent: clarity
Original:  Our algorithm FastInterlaceGreedy uses interlaced, thresholded greedy procedures to obtain  ratio 1/4 - \varepsilon in O \left( \frac{n{\varepsilon} } {\epsilon} } \log \left( \frac{n{\varepsilon} } {\epsilon} } \right) \right) queries of the objective  , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Revised:  The algorithm uses interlaced, thresholded greedy procedures to obtain ratio 1/4 - \varepsilon in O \left( \frac{n{\varepsilon} } {\epsilon} } \log \left( \frac{n{\varepsilon} } {\epsilon} } \right) \right) queries of the objective , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Edit intent: clarity
Original:  Our algorithm FastInterlaceGreedy uses interlaced, thresholded greedy procedures to obtain  ratio 1/4 - \varepsilon in O \left( \frac{n{\varepsilon} } {\epsilon} } \log \left( \frac{n{\varepsilon} } {\epsilon} } \right) \right) queries of the objective  , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Revised:  Our algorithm FastInterlaceGreedy uses interlaced, thresholded greedy procedures to obtain ratio 1/4 - \epsilon in O \left( \frac{n{\varepsilon} } {\epsilon} } \log \left( \frac{n{\varepsilon} } {\epsilon} } \right) \right) queries of the objective , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Edit intent: fluency
Original:  Our algorithm FastInterlaceGreedy uses interlaced, thresholded greedy procedures to obtain  ratio 1/4 - \varepsilon in O \left( \frac{n{\varepsilon} } {\epsilon} } \log \left( \frac{n{\varepsilon} } {\epsilon} } \right) \right) queries of the objective  , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Revised:  Our algorithm FastInterlaceGreedy uses interlaced, thresholded greedy procedures to obtain ratio 1/4 - \varepsilon in O \left( {\varepsilon} } {\epsilon} } \log \left( \frac{n{\varepsilon} } {\epsilon} } \right) \right) queries of the objective , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Edit intent: fluency
Original:  Our algorithm FastInterlaceGreedy uses interlaced, thresholded greedy procedures to obtain  ratio 1/4 - \varepsilon in O \left( \frac{n{\varepsilon} } {\epsilon} } \log \left( \frac{n{\varepsilon} } {\epsilon} } \right) \right) queries of the objective  , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Revised:  Our algorithm FastInterlaceGreedy uses interlaced, thresholded greedy procedures to obtain ratio 1/4 - \varepsilon in O \left( \frac{n{\varepsilon} } \frac{n{\epsilon} } \log \left( \frac{n{\varepsilon} } {\epsilon} } \right) \right) queries of the objective , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Edit intent: fluency
Original:  Our algorithm FastInterlaceGreedy uses interlaced, thresholded greedy procedures to obtain  ratio 1/4 - \varepsilon in O \left( \frac{n{\varepsilon} } {\epsilon} } \log \left( \frac{n{\varepsilon} } {\epsilon} } \right) \right) queries of the objective  , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Revised:  Our algorithm FastInterlaceGreedy uses interlaced, thresholded greedy procedures to obtain ratio 1/4 - \varepsilon in O \left( \frac{n{\varepsilon} } {\epsilon} } \log \left( {\varepsilon} } {\epsilon} } \right) \right) queries of the objective , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Edit intent: fluency
Original:  Our algorithm FastInterlaceGreedy uses interlaced, thresholded greedy procedures to obtain  ratio 1/4 - \varepsilon in O \left( \frac{n{\varepsilon} } {\epsilon} } \log \left( \frac{n{\varepsilon} } {\epsilon} } \right) \right) queries of the objective  , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Revised:  Our algorithm FastInterlaceGreedy uses interlaced, thresholded greedy procedures to obtain ratio 1/4 - \varepsilon in O \left( \frac{n{\varepsilon} } {\epsilon} } \log \left( \frac{n{\varepsilon} } {\epsilon} } \right) \right) queries of the objective function , which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem.
Edit intent: clarity
Original:  We validate our algorithms in the context of two applications of non-monotone submodular maximization, on which FastInterlaceGreedy outperforms the fastest deterministic and randomized algorithms in prior literature.
Revised:  The algorithm is validated in the context of two applications of non-monotone submodular maximization, on which FastInterlaceGreedy outperforms the fastest deterministic and randomized algorithms in prior literature.
Edit intent: clarity
Original:  We validate our algorithms in the context of two applications of non-monotone submodular maximization, on which FastInterlaceGreedy outperforms the fastest deterministic and randomized algorithms in prior literature.
Revised:  We validate our algorithms in the context of two applications of non-monotone submodular maximization, on which it outperforms the fastest deterministic and randomized algorithms in prior literature.
Edit intent: clarity
Original: Kaniadakis and Quarati (1994) proposed a Fokker-Planck equation with quadratic drift as a PDE model for the dynamics of Bose-Einstein condensation in a homogeneous Bose gas .
Revised: Kaniadakis and Quarati (1994) proposed a Fokker--Planck equation with quadratic drift as a PDE model for the dynamics of Bose-Einstein condensation in a homogeneous Bose gas .
Edit intent: fluency
Original: Kaniadakis and Quarati (1994) proposed a Fokker-Planck equation with quadratic drift as a PDE model for the dynamics of Bose-Einstein condensation in a homogeneous Bose gas .
Revised: Kaniadakis and Quarati (1994) proposed a Fokker-Planck equation with quadratic drift as a PDE model for the dynamics of bosons in the spatially homogeneous setting .
Edit intent: clarity
Original:  We present a thoroughly validated time-implicit numerical scheme capable of simulating solutions for arbitrarily large time, and thus enabling a numerical study of the condensation process in the Kaniadakis-Quarati model.
Revised:  We present a thoroughly validated time-implicit numerical scheme capable of simulating solutions for arbitrarily large time, and thus enabling a numerical study of the condensation process in the Kaniadakis--Quarati model.
Edit intent: fluency
Original:  We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the Kaniadakis-Quarati model in 3D form a condensate in finite time and converge in entropy to the unique minimiser of the natural entropy functional at an exponential rate.
Revised:  We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the Kaniadakis--Quarati model in 3D form a condensate in finite time and converge in entropy to the unique minimiser of the natural entropy functional at an exponential rate.
Edit intent: fluency