wanyu/IteraTeR_v2 huggingface dataset:
Original:  We introduce the stable division property for modules (and ideals): a normed module M over the ring of polynomials in d variables has the stable division property if it has a generating set  f_1, ..., f_k  such that every h \in M can be written as h = \sum_i a_i f_i for some polynomials a_i such that \sum \|a_i f_i\| \leq C\|h\|.
Revised:  We introduce the stable division property for modules (and ideals): a normed module M over the ring of polynomials in d variables has the stable division property if it has a generating set \{ f_1, ..., f_k such that every h \in M can be written as h = \sum_i a_i f_i for some polynomials a_i such that \sum \|a_i f_i\| \leq C\|h\|.
Edit intent: fluency
Original:  We introduce the stable division property for modules (and ideals): a normed module M over the ring of polynomials in d variables has the stable division property if it has a generating set  f_1, ..., f_k  such that every h \in M can be written as h = \sum_i a_i f_i for some polynomials a_i such that \sum \|a_i f_i\| \leq C\|h\|.
Revised:  We introduce the stable division property for modules (and ideals): a normed module M over the ring of polynomials in d variables has the stable division property if it has a generating set f_1, ..., f_k \ such that every h \in M can be written as h = \sum_i a_i f_i for some polynomials a_i such that \sum \|a_i f_i\| \leq C\|h\|.
Edit intent: fluency
Original:  We then show that a module M that has the stable division property (with respect to the appropriate norm) is p-essentially normal for all p > d . This result is used to give a new, unified proof that certain classes of graded submodules are essentially normal.
Revised:  We then show that a module M that has the stable division property (with respect to the appropriate norm) is p-essentially normal for p > d . This result is used to give a new, unified proof that certain classes of graded submodules are essentially normal.
Edit intent: clarity
Original:  FDR is the expected False Discovery Proportion (FDP), that is, the expected proportion of false rejections among all rejected hypotheses.
Revised:  FDR is the expected False Discovery Proportion (FDP), that is, the expected fraction of false positives among rejected hypotheses.
Edit intent: clarity
Original:  We present these ideas in several model examples.
Revised:  We illustrate these ideas in several model examples.
Edit intent: clarity
Original: The weight space of the Ising perceptron is explored by a random walk process where single weight flips are performed until the new presented pattern is learned .
Revised: Several variants of a stochastic local search process for constructing the synaptic weights of an Ising perceptron are studied .
Edit intent: clarity
Original:  This strategy, namely%DIFDELCMD < {\tt %%% WalkLearning , is simple to implement in practice and most efficient for solving the learning problem of moderate system size .
Revised:  This strategy, namely%DIFDELCMD < {\tt %%% , is simple to implement in practice and most efficient for solving the learning problem of moderate system size .
Edit intent: clarity
Original: Given a spatio-temporal network (ST network) whose edge properties vary with time, a time-sub-interval minimum spanning tree (TSMST) is a collection of distinct minimum spanning trees of the ST network .
Revised: Given a spatio-temporal network (ST network) where edge properties vary with time, a time-sub-interval minimum spanning tree (TSMST) is a collection of distinct minimum spanning trees of the ST network .
Edit intent: fluency
Original: Given a spatio-temporal network (ST network) whose edge properties vary with time, a time-sub-interval minimum spanning tree (TSMST) is a collection of distinct minimum spanning trees of the ST network .
Revised: Given a spatio-temporal network (ST network) whose edge properties vary with time, a time-sub-interval minimum spanning tree (TSMST) is a collection of minimum spanning trees of the ST network .
Edit intent: clarity
Original:  In contrast, this paper proposes novel algorithms to find TSMST for large ST networks by accounting for both separable and non-separable piecewise linear edge weight functions.
Revised:  In contrast, we propose novel algorithms to find TSMST for large ST networks by accounting for both separable and non-separable piecewise linear edge weight functions.
Edit intent: coherence
Original:  Because specific recognition may provide a solution , we thoroughly analyze an idealized quasispecies model for enzymatic replication, with replication rates that are either a decreasing (self-specific) or increasing (cross-specific) function of the Hamming distance between the recognition or "tag" sequences of enzyme and substrate.
Revised:  Because specific recognition can enforce these propensities , we thoroughly analyze an idealized quasispecies model for enzymatic replication, with replication rates that are either a decreasing (self-specific) or increasing (cross-specific) function of the Hamming distance between the recognition or "tag" sequences of enzyme and substrate.
Edit intent: clarity
Original:  Since all-pairs min cut and the minimum cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum cycle basis in O(n\log ^5 n) time and O(n\log n) space and an explicit representation with additional O(C) time and space where C is the size of the basis.
Revised:  Since all-pairs min cut and the minimum cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum cycle basis in O(n\log ^5 n) time and O(n\log n) space . Additionally, an explicit representation with additional O(C) time and space where C is the size of the basis.
Edit intent: coherence
Original:  Since all-pairs min cut and the minimum cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum cycle basis in O(n\log ^5 n) time and O(n\log n) space and an explicit representation with additional O(C) time and space where C is the size of the basis.
Revised:  Since all-pairs min cut and the minimum cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum cycle basis in O(n\log ^5 n) time and O(n\log n) space and an explicit representation can be obtained in O(C) time and space where C is the size of the basis.
Edit intent: clarity
Original:  We deterministically remove this assumption with an additional \log^2 n factor in the running time .
Revised:  We deterministically remove this assumption with an additional \log^2 n factor in the preprocessing times .
Edit intent: clarity
Original: The possibility of microscopic swim by extraction of energy from an external flow is discussed, focusing on the migration of a simple trimer across a linear shear flow.
Revised: The possibility of microscopic swimming by extraction of energy from an external flow is discussed, focusing on the migration of a simple trimer across a linear shear flow.
Edit intent: fluency
Original:  The geometric properties of swim , together with the possible generalization to the case of a vesicle, are analyzed.
Revised:  The geometric properties of swimming , together with the possible generalization to the case of a vesicle, are analyzed.
Edit intent: fluency
Original:  The swimmer takes advantage of the external flow by both extracting energy for swim and "sailing" through it.
Revised:  The swimmer takes advantage of the external flow by both extracting energy for swimming and "sailing" through it.
Edit intent: fluency
Original:  This effect turns out to be connected with violation of the scallop theorem in the presence of an external flow field .
Revised:  This effect turns out to be connected with the non-applicability of the scallop theorem in the presence of an external flow field .
Edit intent: clarity
Original:  This effect turns out to be connected with violation of the scallop theorem in the presence of an external flow field .
Revised:  This effect turns out to be connected with violation of the scallop theorem in the presence of external flow fields .
Edit intent: fluency
Original: Unlike static documents, versioned documents are continuously edited by one or more authors.
Revised: Unlike static documents, version controlled documents are continuously edited by one or more authors.
Edit intent: clarity
Original:   Blin, Fertin, Sikora, and Vialette recently proved that deciding whether the exemplar distance between two genomes with duplicate genes is zero is  NP-hard even if each gene appears at most two times in each genome .
Revised: Given two genomes with duplicate genes, Blin, Fertin, Sikora, and Vialette recently proved that deciding whether the exemplar distance between two genomes with duplicate genes is zero is NP-hard even if each gene appears at most two times in each genome .
Edit intent: clarity
Original:   Blin, Fertin, Sikora, and Vialette recently proved that deciding whether the exemplar distance between two genomes with duplicate genes is zero is  NP-hard even if each gene appears at most two times in each genome .
Revised: Zero Exemplar Distance Blin, Fertin, Sikora, and Vialette recently proved that deciding whether the exemplar distance between two genomes with duplicate genes is zero is NP-hard even if each gene appears at most two times in each genome .
Edit intent: clarity
Original:   Blin, Fertin, Sikora, and Vialette recently proved that deciding whether the exemplar distance between two genomes with duplicate genes is zero is  NP-hard even if each gene appears at most two times in each genome .
Revised:  Blin, Fertin, Sikora, and Vialette recently proved that Zero Exemplar Distance NP-hard even if each gene appears at most two times in each genome .
Edit intent: clarity
Original:  from the other direction, we show that this problem admits a  polynomial-time algorithm if only one of the two genomes has duplicate genes, and is fixed-parameter tractable if the parameter is the maximum number of chromosomes in each genome.
Revised:  from the other direction, we show that this problem admits a admit polynomial-time algorithm if only one of the two genomes has duplicate genes, and is fixed-parameter tractable if the parameter is the maximum number of chromosomes in each genome.
Edit intent: fluency
Original: We introduce a general mathematical model of symbiosis between different entities by taking into account the influence of each species on the carrying capacities of the others.
Revised: We introduce a mathematical model of symbiosis between different entities by taking into account the influence of each species on the carrying capacities of the others.
Edit intent: clarity
Original: We introduce a general mathematical model of symbiosis between different entities by taking into account the influence of each species on the carrying capacities of the others.
Revised: We introduce a general mathematical model of symbiosis between different species by taking into account the influence of each species on the carrying capacities of the others.
Edit intent: clarity
Original:  As a consequence of our findings, we are planning to move towards a new crown indicator, which relies on the alternative normalization mechanism.
Revised:  As a consequence of our findings, we are currently moving towards a new crown indicator, which relies on the alternative normalization mechanism.
Edit intent: clarity
Original: The problem of evaluating a large number of factors in terms of their relevance to an outcome of interest arises in many research areas such as genetics, image processing, astrophysics, and neuroscience.
Revised: The problem of evaluating a large number of factors in terms of their relevance to an outcome of interest . Such studies are abundant in genetics, image processing, astrophysics, and neuroscience.
Edit intent: clarity
Original:  In this paper, we argue that treating such problems as large-scale hypothesis testing does not reflect the usual motivation behind these studies, which is to select a subset of promising factors for further investigation. and leads investigators to rely on arbitrary selection mechanisms (e.g., setting the false discovery rate at 0.05) or unrealistic loss functions.
Revised:  In this paper, we argue that treating these problems as large-scale hypothesis testing does not reflect the usual motivation behind these studies, which is to select a subset of promising factors for further investigation. and leads investigators to rely on arbitrary selection mechanisms (e.g., setting the false discovery rate at 0.05) or unrealistic loss functions.
Edit intent: fluency
Original:  In this paper, we argue that treating such problems as large-scale hypothesis testing does not reflect the usual motivation behind these studies, which is to select a subset of promising factors for further investigation. and leads investigators to rely on arbitrary selection mechanisms (e.g., setting the false discovery rate at 0.05) or unrealistic loss functions.
Revised:  In this paper, we argue that treating such problems as large-scale hypothesis tests does not reflect the usual motivation behind these studies, which is to select a subset of promising factors for further investigation. and leads investigators to rely on arbitrary selection mechanisms (e.g., setting the false discovery rate at 0.05) or unrealistic loss functions.
Edit intent: fluency
Original:  Using simulated data, we show that our model performs substantially better than alternative methods such as those based on the false discovery rate.
Revised:  Using simulated data, we demonstrate that our model performs substantially better than alternative methods such as those based on the false discovery rate.
Edit intent: fluency
Original:  Using simulated data, we show that our model performs substantially better than alternative methods such as those based on the false discovery rate.
Revised:  Using simulated data, we show that our model performs substantially better than alternative methods based on the false discovery rate.
Edit intent: clarity
Original: We study the fundamental network capacity of a single-hop wireless downlink under two assumptions: (1) channels are never explicitly probed and thus instantaneous states are never known;
Revised: We study the fundamental network capacity of a single-hop wireless downlink under two assumptions: (1) Channels are not explicitly measured and thus instantaneous states are never known;
Edit intent: clarity
Original: We study the fundamental network capacity of a single-hop wireless downlink under two assumptions: (1) channels are never explicitly probed and thus instantaneous states are never known;
Revised: We study the fundamental network capacity of a single-hop wireless downlink under two assumptions: (1) channels are never explicitly probed and thus instantaneous states are unknown,
Edit intent: fluency
Original:  (2) channels are modeled as Markov ON/OFF  chains.
Revised:  (2) Channels are modeled as Markov ON/OFF chains.
Edit intent: fluency
Original:  This is an important network model to explore because practically channel probing may be infeasible, and in this case we shall use channel memory  to improve network throughput.
Revised:  This is an important network model to explore because channel probing may be infeasible, and in this case we shall use channel memory to improve network throughput.
Edit intent: coherence
Original:  Computing in closed-form the capacity region of this network is difficult because it involves solving partially observable Markov decision processes .
Revised:  Computing in closed form the capacity region of this network is difficult because it involves solving partially observable Markov decision processes .
Edit intent: fluency
Original:  Computing in closed-form the capacity region of this network is difficult because it involves solving partially observable Markov decision processes .
Revised:  Computing in closed-form the capacity region of this network is difficult because it involves solving a high dimension partially observed Markov decision problem .
Edit intent: clarity
Original:  Instead, in this paper we construct an inner and an outer bound on the capacity region .
Revised:  Instead, in this paper we construct an inner and outer bound on the capacity region .
Edit intent: fluency
Original:  We show that the two bounds are tight when the network serves a large number of users with balanced traffic .
Revised:  We show that the two bounds are tight when the number of users with balanced traffic .
Edit intent: clarity
Original:  This paper uses analytical tools such as a novel frame-based Lyapunov analysis to prove network stability.
Revised:  The stability analysis uses a novel frame-based Lyapunov analysis to prove network stability.
Edit intent: clarity
Original:  In constructing the outer capacity bound, using stochastic coupling and state aggregation we provide a methodology to upper bound the optimal performance of a restless bandit problem by creating a related multi-armed bandit system.
Revised:  In constructing the outer capacity bound, using stochastic coupling and state aggregation to bound the performance of a restless bandit problem by creating a related multi-armed bandit system.
Edit intent: clarity
Original:  In constructing the outer capacity bound, using stochastic coupling and state aggregation we provide a methodology to upper bound the optimal performance of a restless bandit problem by creating a related multi-armed bandit system.
Revised:  In constructing the outer capacity bound, using stochastic coupling and state aggregation we provide a methodology to upper bound the optimal performance of a restless bandit problem using a related multi-armed bandit system.
Edit intent: clarity
Original:  Our framework and methodology may be useful in problems in cognitive radio networks, opportunistic scheduling with delayed/uncertain channel state information, and restless bandit problems.
Revised:  Our results are useful in cognitive radio networks, opportunistic scheduling with delayed/uncertain channel state information, and restless bandit problems.
Edit intent: clarity
Original:  We show that any particular type of adaptive response is neither sufficient nor necessary for an effective adaptive enlargement of dynamic range.
Revised:  We show that any particular type of adaptive response is neither sufficient nor necessary for adaptive enlargement of dynamic range.
Edit intent: clarity
Original:  We propose that a general mechanism for adaptive dynamic range enlargement comes about from the activity-dependent modulation of protein responsiveness by multiple biochemical modification  .
Revised:  We propose that a general mechanism for input dynamic range enlargement comes about from the activity-dependent modulation of protein responsiveness by multiple biochemical modification .
Edit intent: clarity
Original:  Therefore hierarchical biochemical processes such as methylation and phosphorylation are natural candidates to induce such adaptive dynamic range enlargement .
Revised:  Therefore hierarchical biochemical processes such as methylation and phosphorylation are natural candidates to induce this property in signalling systems .
Edit intent: clarity
Original:  In contrast to the commonly held view, we show that any particular type of adaptive response is neither sufficient nor necessary for adaptive enlargement of dynamic range.
Revised:  We show that any particular type of adaptive response is neither sufficient nor necessary for adaptive enlargement of dynamic range.
Edit intent: clarity
Original:  A general mechanism for input dynamic range enlargement comes about from the activity-dependent modulation of protein responsiveness by multiple biochemical modification, regardless of the type of adaptive response it induces.
Revised:  A general mechanism for input dynamic range enlargement can come about from the activity-dependent modulation of protein responsiveness by multiple biochemical modification, regardless of the type of adaptive response it induces.
Edit intent: clarity
Original:  Therefore hierarchical biochemical processes such as methylation and phosphorylation are natural candidates to induce this property in signalling systems.
Revised:  Therefore hierarchical biochemical processes such as methylation and phosphorylation are natural candidates to induce this property in signaling systems.
Edit intent: fluency
Original:  However, proteins are often involved in multiple biological pathways which complicates inference about interactions between proteins.
Revised:  Proteins are often involved in multiple biological pathways which complicates inference about interactions between proteins.
Edit intent: coherence
Original:  However, proteins are often involved in multiple biological pathways which complicates inference about interactions between proteins.
Revised:  However, proteins are often involved in multiple biological pathways which complicates interpretation of interactions between proteins.
Edit intent: clarity
Original:  Gene expression data is informative here since genes within a particular pathway tend to have more correlated expression patterns than genes from distinct pathways.
Revised:  Gene expression data can assist our inference since genes within a particular pathway tend to have more correlated expression patterns than genes from distinct pathways.
Edit intent: clarity
Original:  We investigate UNO from the viewpoint of combinatorial algorithmic game theory by giving some simple and concise mathematical models for it.
Revised:  We investigate UNO from the viewpoint of algorithmic combinatorial game theory. We define simple and concise mathematical models for it.
Edit intent: clarity
Original:  As a result of analyzing their computational complexities ,  we prove that even a single-player version of UNO is NP-complete, while it becomes in P in some restricted cases .
Revised:  As a result of analyzing their computational complexities , we prove that even a single-player version of UNO is NP-complete, although some restricted cases .
Edit intent: clarity
Original:  We also show that uncooperative two-player 's version is PSPACE-complete .
Revised:  We also show that uncooperative two-player version is also in P .
Edit intent: clarity
Original:  We evaluated these three kinds of features and their combinations on several real EEG signals . Results obtained with various EEG datasets show that our novel features can lead to BCI designs with improved classification performance, notably when using and combining the three kinds of feature ( band power , multifractal cumulants, predictive complexity) together.
Revised:  We evaluated these three kinds of features and their combinations on several real EEG signals . Results obtained show that our novel features can lead to BCI designs with improved classification performance, notably when using and combining the three kinds of feature ( band power , multifractal cumulants, predictive complexity) together.
Edit intent: clarity
Original:  We evaluated these three kinds of features and their combinations on several real EEG signals . Results obtained with various EEG datasets show that our novel features can lead to BCI designs with improved classification performance, notably when using and combining the three kinds of feature ( band power , multifractal cumulants, predictive complexity) together.
Revised:  We evaluated these three kinds of features and their combinations on several real EEG signals . Results obtained with various EEG datasets show that our novel features can lead to BCI designs with improved classification performance, notably when using and combining the three kinds of feature ( band-power , multifractal cumulants, predictive complexity) together.
Edit intent: fluency
Original:  Pebbling has proven to be a useful tool for studying resolution-based proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing size-space trade-offs.
Revised:  Pebbling has proven a useful tool for studying resolution-based proof systems when comparing the strength of different subsystems, showing bounds on proof space, and establishing size-space trade-offs.
Edit intent: fluency