wanyu/IteraTeR_v2 huggingface dataset:
Original:  We prove a unifying nodal domain theorem for the graph p-Laplacian for any p\geq 1.
Revised:  We prove a nodal domain theorem for the graph p-Laplacian for any p\geq 1.
Edit intent: clarity
Original:  We show that the bounds are tight for p\geq 1 by studying the eigenvectors of the graph p-Laplacian for two example graphswhere the bounds on the nodal domains are attained.
Revised:  We show that the bounds are tight for p\geq 1 as the bounds are attained by the eigenfunctions of the graph p-Laplacian for two example graphswhere the bounds on the nodal domains are attained.
Edit intent: clarity
Original:  We show that the bounds are tight for p\geq 1 by studying the eigenvectors of the graph p-Laplacian for two example graphswhere the bounds on the nodal domains are attained.
Revised:  We show that the bounds are tight for p\geq 1 by studying the eigenvectors of the graph p-Laplacian on two graphs. Finally, using the properties of the nodal domains are attained.
Edit intent: clarity
Original: In this paper,  we prove that the decision version of the Frobenius problem is Sigma_2^P-complete as a main theorem .
Revised: In this paper, as a main theorem, we prove that the decision version of the Frobenius problem is Sigma_2^P-complete as a main theorem .
Edit intent: coherence
Original: Given a set A = a_1, ..., a_n%DIFDELCMD < } %%% of coprime integers such that 2 <= a_1 < ...
Revised: Given a a_1, ..., a_n%DIFDELCMD < } %%% of coprime integers such that 2 <= a_1 < ...
Edit intent: clarity
Original:  < a_n , we call the greatest integer that has no nonnegative integer combination of A the Frobenius number, and denote it by g(A).
Revised:  < a_n , we call the greatest integer that cannot be represented as a nonnegative integer combination of A the Frobenius number, and denote it by g(A).
Edit intent: clarity
Original:  < a_n , we call the greatest integer that has no nonnegative integer combination of A the Frobenius number, and denote it by g(A).
Revised:  < a_n , we call the greatest integer that has no nonnegative integer combination of A the Frobenius number, and we denote it as g(A).
Edit intent: style
Original:  We call the decision problem determining whether g(A) >= k for a given pair (A, k) the  Frobenius problem, and denote it by Frobenius Problem, where A = a_1, ..., a_n%DIFDELCMD < } %%% is a set of coprime integers such that 2 <= a_1 < ...
Revised:  We call a problem of finding g(A) for a given A the Frobenius problem; moreover, we call a problem of determining whether g(A) >= k for a given pair (A, k) the Frobenius problem, and denote it by Frobenius Problem, where A = a_1, ..., a_n%DIFDELCMD < } %%% is a set of coprime integers such that 2 <= a_1 < ...
Edit intent: clarity
Original:  We call the decision problem determining whether g(A) >= k for a given pair (A, k) the  Frobenius problem, and denote it by Frobenius Problem, where A = a_1, ..., a_n%DIFDELCMD < } %%% is a set of coprime integers such that 2 <= a_1 < ...
Revised:  We call the decision problem determining whether g(A) >= k for a given pair (A, k) the decision version of the Frobenius problem, and denote it by Frobenius Problem, where A = a_1, ..., a_n%DIFDELCMD < } %%% is a set of coprime integers such that 2 <= a_1 < ...
Edit intent: clarity
Original:  We call the decision problem determining whether g(A) >= k for a given pair (A, k) the  Frobenius problem, and denote it by Frobenius Problem, where A = a_1, ..., a_n%DIFDELCMD < } %%% is a set of coprime integers such that 2 <= a_1 < ...
Revised:  We call the decision problem determining whether g(A) >= k for a given pair (A, k) the Frobenius problem, a_1, ..., a_n%DIFDELCMD < } %%% is a set of coprime integers such that 2 <= a_1 < ...
Edit intent: coherence
Original:  We call the decision problem determining whether g(A) >= k for a given pair (A, k) the  Frobenius problem, and denote it by Frobenius Problem, where A = a_1, ..., a_n%DIFDELCMD < } %%% is a set of coprime integers such that 2 <= a_1 < ...
Revised:  We call the decision problem determining whether g(A) >= k for a given pair (A, k) the Frobenius problem, and denote it by Frobenius Problem, where A = a_1, ..., a_n%DIFDELCMD < } %%% where A is a finite set of coprime integers such that 2 <= a_1 < ...
Edit intent: clarity
Original: We propose a probabilistic Hoare logic aHL based on the union bound, a basic fact from probability theory.
Revised: We propose a probabilistic Hoare logic aHL based on the union bound, a tool from basic probability theory.
Edit intent: clarity
Original:  While this principle is simple, it is an extremely common tool in the analysis of randomized algorithms.
Revised:  While this principle is simple, it is an extremely common tool for analyzing randomized algorithms.
Edit intent: clarity
Original:  From a formal verification perspective , the union bound allows  compositional reasoning over possible ways an algorithm may go wrong.
Revised:  In formal verification terms , the union bound allows compositional reasoning over possible ways an algorithm may go wrong.
Edit intent: clarity
Original:  The union bound is also a flexible abstraction for expressing complex facts about probabilities---even though our target programs and properties are probabilistic , assertions in our logic are standard first-order formulas and do not involve probabilities or distributions .
Revised:  It also enables a clean separation between reasoning about probabilities and reasoning about events, which are expressed as standard first-order formulas in our logic. Notably , assertions in our logic are standard first-order formulas and do not involve probabilities or distributions .
Edit intent: clarity
Original:  We implement our logic and show how to prove accuracy for several examples from the differential privacy literature, both interactive and non-interactive.
Revised:  We show how to prove accuracy for several examples from the differential privacy literature, both interactive and non-interactive.
Edit intent: clarity
Original: In Chung-Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution (for which we fix an exponent 2 < \beta < 3) , and two vertices form an edge independently with probability proportional to the product of their weights.
Revised: In Chung-Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution , and two vertices form an edge independently with probability proportional to the product of their weights.
Edit intent: coherence
Original:   Modern, more realistic variants of this model also equip each vertex with a random position in a specific underlying geometry , which is typically Euclidean, such as the unit square, circle, or torus .
Revised:  Modern, more realistic variants of this model also equip each vertex with a random position in a specific underlying geometry .
Edit intent: coherence
Original:  We show that specific choices, such as the underlying geometry being Euclidean or the dependence on the distance being inversely polynomial, do not significantly influence the average distance, by studying a generic augmented version of Chung-Lu random graphs.
Revised:  In this paper we study a generic augmented version of Chung-Lu random graphs.
Edit intent: clarity
Original:  Specifically, we analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in Chung-Lu random graphs , i. e., proportional to the product of their weights.
Revised:  We analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in Chung-Lu random graphs , i. e., proportional to the product of their weights.
Edit intent: coherence
Original:  Specifically, we analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in Chung-Lu random graphs , i. e., proportional to the product of their weights.
Revised:  Specifically, we analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in Chung-Lu random graphs .
Edit intent: coherence
Original:  We study  optimal investment under an expected utility criterion and prove the existence of optimal strategies.
Revised:  We study the problem of optimal investment under an expected utility criterion and prove the existence of optimal strategies.
Edit intent: clarity
Original:  We study  optimal investment under an expected utility criterion and prove the existence of optimal strategies.
Revised:  We study optimal investment under an expected utility criterion and look for conditions ensuring the existence of optimal strategies.
Edit intent: clarity
Original:  Previous results require a certain restrictive hypothesis on the non-triviality of the tails of asset return distributions.
Revised:  Previous results required a certain restrictive hypothesis on the non-triviality of the tails of asset return distributions.
Edit intent: fluency
Original:  Previous results require a certain restrictive hypothesis on the non-triviality of the tails of asset return distributions.
Revised:  Previous results require a certain restrictive hypothesis on the tails of asset return distributions.
Edit intent: clarity
Original:  Using a different method, we manage to remove this hypothesis in the present article , at the price of stronger assumptions on the moments of asset returns .
Revised:  Using a different method, we manage to remove this hypothesis , at the price of stronger assumptions on the moments of asset returns .
Edit intent: coherence
Original:  Our main contribution is a generic variety theorem that covers e.g.
Revised:  Our main contribution is a variety theorem that covers e.g.
Edit intent: clarity
Original:  In addition it gives a number of new results, such as an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words.
Revised:  In addition we derive a number of new results, such as an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words.
Edit intent: clarity
Original:  In addition it gives a number of new results, such as an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words.
Revised:  In addition it gives a number of new results, including an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words.
Edit intent: fluency
Original:  In a model financial network, we show that the origin of instabilityresides in the presence of specific types of cyclical structures , regardless of many of the details of the distress propagation mechanism.
Revised:  In a model financial network, we show that the details of the distress propagation mechanism.
Edit intent: clarity
Original:  On the positive side, we consider a simple randomized greedy matching algorithm , which we call RGMA, and we study its performance on the special class of bush graphs, i.e. on edge-weighted graphs where all edges with equal weight form a star (bush ).
Revised:  On the positive side, we present a randomized approximation algorithm (RGMA) for GreedyMatching on a special class of bush graphs, i.e. on edge-weighted graphs where all edges with equal weight form a star (bush ).
Edit intent: clarity
Original:  The importance of the performance of RGMA on general bush graphs is highlighted by the fact that, as our results show, a \rho-approximation }  ratio of RGMA for GreedyMatching on general bush graphs implies that, for every \epsilon <1, the randomized algorithm MRG (first defined in \mbox{ ADFS95 ]  a (\rho-\epsilon)-approximation of the maximum cardinality matching in unweighted graphs.
Revised:  The importance of the performance of RGMA on general bush graphs is highlighted by the fact that, as our results show, a \rho-approximation } ratio of RGMA for GreedyMatching on general bush graphs implies that, for every \epsilon <1, the randomized algorithm MRG (first defined in \mbox{ ADFS95 ] gives a (\rho-\epsilon)-approximation of the maximum cardinality matching in unweighted graphs.
Edit intent: clarity
Original:  The importance of the performance of RGMA on general bush graphs is highlighted by the fact that, as our results show, a \rho-approximation }  ratio of RGMA for GreedyMatching on general bush graphs implies that, for every \epsilon <1, the randomized algorithm MRG (first defined in \mbox{ ADFS95 ]  a (\rho-\epsilon)-approximation of the maximum cardinality matching in unweighted graphs.
Revised:  The importance of the performance of RGMA on general bush graphs is highlighted by the fact that, as our results show, a \rho-approximation } ratio of RGMA for GreedyMatching on general bush graphs implies that, for every \epsilon <1, the randomized algorithm MRG (first defined in \mbox{ ADFS95 ] a (\rho-\epsilon)-approximation for the maximum cardinality matching in unweighted graphs.
Edit intent: fluency
Original:  We consider here OCAs with counter visibility : Whenever the automaton produces a letter, it outputs the current counter value along with it.
Revised:  In this paper, we consider OCAs with counter visibility : Whenever the automaton produces a letter, it outputs the current counter value along with it.
Edit intent: clarity
Original:  We consider here OCAs with counter visibility : Whenever the automaton produces a letter, it outputs the current counter value along with it.
Revised:  We consider here OCAs with counter observability : Whenever the automaton produces a letter, it outputs the current counter value along with it.
Edit intent: clarity
Original:  We show that universality and inclusion for that model are in PSPACE , thus no harder than the corresponding problems for finite automata , which can actually be considered as a special case .
Revised:  We show that universality and inclusion for that model are PSPACE-complete , thus no harder than the corresponding problems for finite automata , which can actually be considered as a special case .
Edit intent: clarity
Original:  We show that universality and inclusion for that model are in PSPACE , thus no harder than the corresponding problems for finite automata , which can actually be considered as a special case .
Revised:  We show that universality and inclusion for that model are in PSPACE , thus no harder than the corresponding problems for finite automata .
Edit intent: coherence
Original:  In fact,  we show that OCAs with counter visibility are effectively determinizable and closed under all boolean operations .
Revised:  In fact, we show that OCAs with counter observability are effectively determinizable and closed under all boolean operations .
Edit intent: clarity
Original:  In this work, we present a  truncated LU factorization called %DIFDELCMD < {\bf %%% Spectrum-Revealing LU (SRLU) for effective low-rank matrix approximation, and develop the first algorithm to compute an SRLU factorization , which is both efficient and reliable.
Revised:  In this work, we present a truncated LU factorization called %DIFDELCMD < {\bf %%% Spectrum-Revealing LU (SRLU) for effective low-rank matrix approximation, and develop a fast algorithm to compute an SRLU factorization , which is both efficient and reliable.
Edit intent: clarity
Original:  Numeric experiments illustrate that SRLU preserves sparsity, highlights important data features and variables, can be efficiently updated, and calculates data approximations nearly as accurately as the SVD . To the best of our knowledge this is the first practical variant of the LU decomposition for efficient and effective low-rank matrix approximation.
Revised:  Numeric experiments illustrate that SRLU preserves sparsity, highlights important data features and variables, can be efficiently updated, and calculates data approximations nearly as accurately as possible . To the best of our knowledge this is the first practical variant of the LU decomposition for efficient and effective low-rank matrix approximation.
Edit intent: clarity
Original:  The proposed deringing filter is shown to improve the quality of both Daala and the Alliance for Open Media (AOM) AV1 video codec .
Revised:  The proposed deringing filter is shown to improve the quality of the Alliance for Open Media (AOM) AV1 video codec .
Edit intent: clarity
Original:  In turn, this allows for a detailed valuation analysis, stress testing and risk analysis via sensitivities .
Revised:  This allows for a detailed valuation analysis, stress testing and risk analysis via sensitivities .
Edit intent: coherence
Original:  We refer to the full paper 5%DIFDELCMD < ] %%% for a more complete mathematical treatment .
Revised:  We refer to the full paper %DIFDELCMD < ] %%% for a more complete mathematical treatment .
Edit intent: fluency
Original:  We refer to the full paper 5%DIFDELCMD < ] %%% for a more complete mathematical treatment .
Revised:  We refer to the full paper 5%DIFDELCMD < ] %%% .
Edit intent: clarity
Original:  The adequacy of the model for these tasks is validated on a  dataset of the most Tweeted and shared videos over 6 months  .
Revised:  The adequacy of the model for these tasks is validated on a large dataset of the most Tweeted and shared videos over 6 months .
Edit intent: clarity
Original: Explaining and predicting popularity of an online item is an important problem in both the science of collective behavior, and the practice of managing information dissemination and consumption.
Revised: Explaining and predicting the popularity of online multimedia content is an important problem in both the science of collective behavior, and the practice of managing information dissemination and consumption.
Edit intent: clarity
Original: Explaining and predicting popularity of an online item is an important problem in both the science of collective behavior, and the practice of managing information dissemination and consumption.
Revised: Explaining and predicting popularity of an online item is an important problem for the practice of information dissemination and consumption.
Edit intent: clarity
Original:  Many recent work advance our understanding of popularity, but one notable gap remains: to precisely quantify the relationship between external promotions and subsequent popularity dynamics .
Revised:  Recent work advances our understanding of popularity, but one notable gap remains: to precisely quantify the relationship between external promotions and subsequent popularity dynamics .
Edit intent: coherence
Original:  Many recent work advance our understanding of popularity, but one notable gap remains: to precisely quantify the relationship between external promotions and subsequent popularity dynamics .
Revised:  Many recent work advance our understanding of popularity, but one important gap remains: to precisely quantify the relationship between external promotions and subsequent popularity dynamics .
Edit intent: clarity
Original:  Many recent work advance our understanding of popularity, but one notable gap remains: to precisely quantify the relationship between external promotions and subsequent popularity dynamics .
Revised:  Many recent work advance our understanding of popularity, but one notable gap remains: to precisely quantify the relationship between the popularity of an online item and the external promotions it receives .
Edit intent: clarity
Original:  This is done via a novel mathematical model, called the Hawkes intensity process, that explains the complex popularity history of each video according to its content, network, and sensitivity to promotion.
Revised:  This is done via a novel mathematical model, the Hawkes intensity process, that explains the complex popularity history of each video according to its content, network, and sensitivity to promotion.
Edit intent: fluency
Original:  This is done via a novel mathematical model, called the Hawkes intensity process, that explains the complex popularity history of each video according to its content, network, and sensitivity to promotion.
Revised:  This is done via a novel mathematical model, called the Hawkes intensity process, which is able to explain the complex popularity history of each video according to its content, network, and sensitivity to promotion.
Edit intent: clarity
Original:  This model also allows us to forecast the effects of future promotions more accurately than approaches based on popularity history alone, and to identify videos that has a high potential to become viral, or those for which promotions will have hardly any effect.
Revised:  This model also allows us to forecast the effects of future promotions more accurately than approaches based on popularity history alone, and to identify videos that have a high potential to become viral, or those for which promotions will have hardly any effect.
Edit intent: fluency
Original:  This model supplies a prototypical description of videos, called an endo-exo map, which allows us to explain the popularity as the joint effects of two intrinsic measures and the amount of discussions from the outside world.
Revised:  This model supplies a prototypical description of videos, called an endo-exo map, which allows us to explain popularity as the joint effects of two intrinsic measures and the amount of discussions from the outside world.
Edit intent: fluency
Original:  This model supplies a prototypical description of videos, called an endo-exo map, which allows us to explain the popularity as the joint effects of two intrinsic measures and the amount of discussions from the outside world.
Revised:  This model supplies a prototypical description of videos, called an endo-exo map, which allows us to explain the popularity as the joint effect of two intrinsic measures and the amount of discussions from the outside world.
Edit intent: fluency
Original:  This model supplies a prototypical description of videos, called an endo-exo map, which allows us to explain the popularity as the joint effects of two intrinsic measures and the amount of discussions from the outside world.
Revised:  This model supplies a prototypical description of videos, called an endo-exo map, which allows us to explain the popularity as the joint effects of two intrinsic measures and the amount of discussion from the outside world.
Edit intent: fluency
Original: Explaining and predicting the popularity of online multimedia content is an important problem for the practice of information dissemination  and consumption.
Revised: Modeling and predicting the popularity of online multimedia content is an important problem for the practice of information dissemination and consumption.
Edit intent: clarity
Original: Explaining and predicting the popularity of online multimedia content is an important problem for the practice of information dissemination  and consumption.
Revised: Explaining and predicting the popularity of online content is a significant problem for the practice of information dissemination and consumption.
Edit intent: clarity
Original:  Recent work  advances our understanding of popularity, but one important gap remains: to precisely quantify the relationship between the popularity of an online item and the external promotions it receives.
Revised:  Recent work advances our understanding of popularity, but one major gap remains: to precisely quantify the relationship between the popularity of an online item and the external promotions it receives.
Edit intent: clarity