wanyu/IteraTeR_v2 huggingface dataset:
Original:  At the time, we could only state that a generic divisor is typical, and hence unlikely to be encountered if one implements these algorithms over a very large finite field.
Revised:  At the time, we could only state that a generic divisor is typical, and hence unlikely to be encountered if one implemented these algorithms over a very large finite field.
Edit intent: fluency
Original:  This article pins down an explicit sufficient characterization of these typical divisors, and proves that if the algorithms in our previous article can be carried out, then this provides a guarantee that the resulting output is correct and that the resulting divisor is also typical.
Revised:  This article pins down an explicit characterization of these typical divisors, and proves that if the algorithms in our previous article can be carried out, then this provides a guarantee that the resulting output is correct and that the resulting divisor is also typical.
Edit intent: clarity
Original:  As a byproduct of our work , we obtain a further speedup of approximately 15\% on our earlier algorithms for C_{3,4} curves.
Revised:  As a byproduct , we obtain a further speedup of approximately 15\% on our earlier algorithms for C_{3,4} curves.
Edit intent: clarity
Original:  As a byproduct of our work , we obtain a further speedup of approximately 15\% on our earlier algorithms for C_{3,4} curves.
Revised:  As a byproduct of our work , we obtain a further speedup of approximately 15\% on our previous algorithms for C_{3,4} curves.
Edit intent: fluency
Original:  We give general algorithms for Jacobian group arithmetic with these typical divisors, and prove that if our algorithms can be carried out, then this provides a guarantee that the resulting output is correct and that the resulting divisor is also typical.
Revised:  We give general algorithms for Jacobian group arithmetic with these typical divisors, and prove that if our algorithms can be carried out, then this provides a guarantee that the resulting output is correct and that the resulting input and/or output divisors are also typical.
Edit intent: clarity
Original:  It also gives a tighter paucity bound , and it shows that the set of lengths n at which there is a paucity of trivial strings is not only infinite, but has positive Schnirelmann density .
Revised:  It also gives a tighter paucity bound , and it gives a tighter paucity bound .
Edit intent: coherence
Original:  A property that can come to rescue is that deep sequencing of every cell should not be necessary to capture all distinct genomes, as the majority of cells are biological replicates and therefore samples are often sparse in that sense.
Revised:  A property that can come to rescue is that deep sequencing of every cell should not be necessary to capture all distinct genomes, as the majority of cells are biological replicates . Biologically important samples are often sparse in that sense.
Edit intent: clarity
Original:  In this paper, we give a new algorithm based on the depth first search (DFS) strategy.
Revised:  In this paper, we give a new algorithm based on the depth first search strategy.
Edit intent: clarity
Original:  here we generalise several of these results by proving hardness for some more general classes of such problems.
Revised:  here we substantially generalise some of these existing results by proving hardness for some more general classes of such problems.
Edit intent: clarity
Original:  In the special case that the property in question depends only on the number of edges in the subgraph, we give a slight strengthening of this result  .
Revised:  In the special case that the property in question depends only on the number of edges in the subgraph, we give a strengthening of this result .
Edit intent: coherence
Original: This paper studies complete rewriting systems and biautomaticity for three interesting classes of finite-rank homogeneous monoids: Chinese monoids, hypoplactic monoids, and Sylvester monoids.
Revised: This paper studies complete rewriting systems and biautomaticity for three interesting classes of finite-rank homogeneous monoids: Chinese monoids, hypoplactic monoids, and sylvester monoids.
Edit intent: fluency
Original:  For Sylvester monoids, which are not finitely presented, we prove that the standard presentation is an infinite complete rewriting system, and construct biautomatic structures.
Revised:  For sylvester monoids, which are not finitely presented, we prove that the standard presentation is an infinite complete rewriting system, and construct biautomatic structures.
Edit intent: fluency
Original:  Here we show that any single beamsplitter that nontrivially mixes two modes, also densely generates the set of m by m unitary transformations (or orthogonal transformations, in the real case) on  m>=3 modes.
Revised:  Here we show that any single beamsplitter that nontrivially mixes two modes, also densely generates the set of unitary transformations (or orthogonal transformations, in the real case) on m>=3 modes.
Edit intent: clarity
Original:  (We prove the same result for any 2-mode real optical gate, and for any 2-mode optical gate combined with a generic phaseshifter.) Experimentally, this means that one does not need tunable beamsplitters or phaseshifters for universality: any nontrivial beamsplitter is universal for linear optics.
Revised:  (We prove the same result for any two-mode real optical gate, and for any 2-mode optical gate combined with a generic phaseshifter.) Experimentally, this means that one does not need tunable beamsplitters or phaseshifters for universality: any nontrivial beamsplitter is universal for linear optics.
Edit intent: fluency
Original:  (We prove the same result for any 2-mode real optical gate, and for any 2-mode optical gate combined with a generic phaseshifter.) Experimentally, this means that one does not need tunable beamsplitters or phaseshifters for universality: any nontrivial beamsplitter is universal for linear optics.
Revised:  (We prove the same result for any 2-mode real optical gate, and for any two-mode optical gate combined with a generic phaseshifter.) Experimentally, this means that one does not need tunable beamsplitters or phaseshifters for universality: any nontrivial beamsplitter is universal for linear optics.
Edit intent: fluency
Original:  We leave open the problem of classifying optical gates that act on 3 or more modes.
Revised:  We leave open the problem of classifying optical gates that act on three or more modes.
Edit intent: fluency
Original: We consider the enumeration of dense substructures in the form of maximal cliques from an uncertain graph, which is a probability distribution on a set of deterministic graphs.
Revised: We consider mining dense substructures (maximal cliques) from an uncertain graph, which is a probability distribution on a set of deterministic graphs.
Edit intent: clarity
Original:  We }  present a precise definition of an \alpha -maximal clique in an uncertain graph , for parameter 0 < \alpha%DIFDELCMD < \le %%% 1. We present matching upper and lower bounds on the number of \alpha -maximal cliques possible within an uncertain graph.
Revised:  We } present a precise definition of an \alpha -maximal clique in an uncertain graph , for parameter 0 < \alpha%DIFDELCMD < \le %%% 1. We present matching upper and lower bounds on the number of \alpha -maximal cliques possible within an uncertain graph.
Edit intent: fluency
Original:  We }  present a precise definition of an \alpha -maximal clique in an uncertain graph , for parameter 0 < \alpha%DIFDELCMD < \le %%% 1. We present matching upper and lower bounds on the number of \alpha -maximal cliques possible within an uncertain graph.
Revised:  We } present a precise definition of an \alpha -maximal clique in an uncertain graph %DIFDELCMD < \le %%% 1. We present matching upper and lower bounds on the number of \alpha -maximal cliques possible within an uncertain graph.
Edit intent: clarity
Original:  We }  present a precise definition of an \alpha -maximal clique in an uncertain graph , for parameter 0 < \alpha%DIFDELCMD < \le %%% 1. We present matching upper and lower bounds on the number of \alpha -maximal cliques possible within an uncertain graph.
Revised:  We } present a precise definition of an \alpha -maximal clique in an uncertain graph , for parameter 0 < \alpha%DIFDELCMD < \le %%% . We present matching upper and lower bounds on the number of \alpha -maximal cliques possible within an uncertain graph.
Edit intent: coherence
Original:  We also present an algorithm to enumerate \alpha -maximal cliques in an uncertain graph  , and an experimental evaluation that suggests a runtime that is proportional to the size of the output .
Revised:  We present an algorithm to enumerate \alpha -maximal cliques in an uncertain graph , and an experimental evaluation that suggests a runtime that is proportional to the size of the output .
Edit intent: clarity
Original:  We also present an algorithm to enumerate \alpha -maximal cliques in an uncertain graph  , and an experimental evaluation that suggests a runtime that is proportional to the size of the output .
Revised:  We also present an algorithm to enumerate \alpha -maximal cliques in an uncertain graph , and an experimental evaluation that suggests a runtime that is proportional to the size of the output .
Edit intent: fluency
Original:  We also present an algorithm to enumerate \alpha -maximal cliques in an uncertain graph  , and an experimental evaluation that suggests a runtime that is proportional to the size of the output .
Revised:  We also present an algorithm to enumerate \alpha -maximal cliques in an uncertain graph , and an experimental evaluation showing the practical utility of the algorithm .
Edit intent: clarity
Original: We model and analyze multiple-input multiple-output heterogeneous cellular networks (MIMO HetNets) with K classes or tiers of base stations (BSs), which may differ in terms of transmit power, deployment density, number of transmit antennas, number of users served, transmission scheme, and path loss exponent.
Revised: We model and analyze heterogeneous cellular networks (MIMO HetNets) with K classes or tiers of base stations (BSs), which may differ in terms of transmit power, deployment density, number of transmit antennas, number of users served, transmission scheme, and path loss exponent.
Edit intent: clarity
Original:  We show that the cell selection rules in MIMO HetNets may differ significantly from the single-antenna HetNets due to the possible differences in multi-antenna transmission schemes across tiers.
Revised:  We show that the cell selection rules in multi-antenna HetNets may differ significantly from the single-antenna HetNets due to the possible differences in multi-antenna transmission schemes across tiers.
Edit intent: clarity
Original:  While it is challenging to derive exact cell selection rules even for maximizing signal-to-interference-plus-noise-ratio (SINR) at the receiver, our analysis concretely demonstrates that adding a fixed tier-dependent  bias in the received power while doing cell selection leads to a near optimal performance both for downlink coverage and rate .
Revised:  While it is challenging to derive exact cell selection rules even for maximizing signal-to-interferenceplus-noise-ratio (SINR) at the receiver, our analysis concretely demonstrates that adding a fixed tier-dependent bias in the received power while doing cell selection leads to a near optimal performance both for downlink coverage and rate .
Edit intent: fluency
Original:  While it is challenging to derive exact cell selection rules even for maximizing signal-to-interference-plus-noise-ratio (SINR) at the receiver, our analysis concretely demonstrates that adding a fixed tier-dependent  bias in the received power while doing cell selection leads to a near optimal performance both for downlink coverage and rate .
Revised:  While it is challenging to derive exact cell selection rules even for maximizing signal-to-interference-plus-noise-ratio (SINR) at the receiver, we show that adding an appropriately chosen tier-dependent bias in the received power while doing cell selection leads to a near optimal performance both for downlink coverage and rate .
Edit intent: clarity
Original:  While it is challenging to derive exact cell selection rules even for maximizing signal-to-interference-plus-noise-ratio (SINR) at the receiver, our analysis concretely demonstrates that adding a fixed tier-dependent  bias in the received power while doing cell selection leads to a near optimal performance both for downlink coverage and rate .
Revised:  While it is challenging to derive exact cell selection rules even for maximizing signal-to-interference-plus-noise-ratio (SINR) at the receiver, our analysis concretely demonstrates that adding a fixed tier-dependent bias in the received power yields a close approximation .
Edit intent: clarity
Original:  Due to this connection with biasing, MIMO HetNets may balance load more naturally  in certain regimes compared to single-antenna HetNets, where an artificial selection bias is often needed  .
Revised:  Due to this connection with biasing, multi-antenna HetNets may balance load more naturally in certain regimes compared to single-antenna HetNets, where an artificial selection bias is often needed .
Edit intent: clarity
Original:  Details of numerical experiments are given that verify that these asymptotic formulas yield the expected quantitative agreement with Monte Carlo results for large finite networks  .
Revised:  Results of numerical experiments are given that verify that these asymptotic formulas yield the expected quantitative agreement with Monte Carlo results for large finite networks .
Edit intent: fluency
Original:  Details of numerical experiments are given that verify that these asymptotic formulas yield the expected quantitative agreement with Monte Carlo results for large finite networks  .
Revised:  Details of numerical experiments then demonstrate that these asymptotic formulas yield the expected quantitative agreement with Monte Carlo results for large finite networks .
Edit intent: clarity
Original:  Details of numerical experiments are given that verify that these asymptotic formulas yield the expected quantitative agreement with Monte Carlo results for large finite networks  .
Revised:  Details of numerical experiments are given that verify that these asymptotic formulas agree qualitatively with Monte Carlo results for large finite networks .
Edit intent: clarity
Original:  On the other hand, in a stress cascade, illiquidity that hits a given bank will create a shock to the liability side of the balance sheet of each of its debtor banks.
Revised:  On the other hand, funding illiquidity that hits a given bank will create a shock to the liability side of the balance sheet of each of its debtor banks.
Edit intent: clarity
Original:  Such "upstream" shocks can cause further illiquidity stresses that may build up to create a global illiquidity cascade.
Revised:  Under some circumstances, such "upstream" shocks can cause further illiquidity stresses that may build up to create a global illiquidity cascade.
Edit intent: clarity
Original:  Such "upstream" shocks can cause further illiquidity stresses that may build up to create a global illiquidity cascade.
Revised:  Such "upstream" shocks can cause illiquidity in further banks that may build up to create a global illiquidity cascade.
Edit intent: clarity
Original:  Such "upstream" shocks can cause further illiquidity stresses that may build up to create a global illiquidity cascade.
Revised:  Such "upstream" shocks can cause further illiquidity stresses that may build up to create an illiquidity cascade.
Edit intent: clarity
Original:  These experiments illustrate clearly our main conclusion that in financial networks , the average default probability is inversely related to strength of banks'  stress response and therefore to the overall level of stress in the network.
Revised:  The experiments clearly support the main conclusion that in financial networks , the average default probability is inversely related to strength of banks' stress response and therefore to the overall level of stress in the network.
Edit intent: clarity
Original:  These experiments illustrate clearly our main conclusion that in financial networks , the average default probability is inversely related to strength of banks'  stress response and therefore to the overall level of stress in the network.
Revised:  These experiments illustrate clearly our main conclusion that in financial networks , the average default probability is inversely related to strength of banks' stress response and the eventual level of stress in the network.
Edit intent: clarity
Original:  This paper introduces a deliberately simplified financial network model that combines the default and liquidity stress mechanisms into a "double cascade mapping".
Revised:  This paper introduces a financial network model that combines the default and liquidity stress mechanisms into a "double cascade mapping".
Edit intent: clarity
Original:  Unlike simpler models, this model can therefore quantify how illiquidity or default of one bank influences the eventual overall level of liquidity stress and default in the system.
Revised:  Unlike simpler models, this model can therefore quantify how illiquidity or default of one bank influences the overall level of liquidity stress and default in the system.
Edit intent: clarity
Original:  Given a finite number of elements and a family of rankings of those elements, we consider that two elements compete {\em  when they change their relative positions in at least two of those rankings.
Revised:  Given a finite number of elements and a family of rankings of those elements, we say that two elements compete {\em when they change their relative positions in at least two of those rankings.
Edit intent: clarity
Original:  Given a finite number of elements and a family of rankings of those elements, we consider that two elements compete {\em  when they change their relative positions in at least two of those rankings.
Revised:  Given a finite number of elements and a family of rankings of those elements, we consider that two elements {\em when they change their relative positions in at least two of those rankings.
Edit intent: clarity
Original:  Given a finite number of elements and a family of rankings of those elements, we consider that two elements compete {\em  when they change their relative positions in at least two of those rankings.
Revised:  Given a finite number of elements and a family of rankings of those elements, we consider that two elements compete {\em when they change their relative positions in at least two rankings.
Edit intent: fluency
Original:  This allows to define an undirected graph , the so-called competitivity graph, by linking elements that compete.
Revised:  This allows to define an undirected graph by connecting elements that compete.
Edit intent: clarity
Original:  Moreover, we relate{\em   competitivity graphs with other well-known families of graphs, such as permutation graphs, comparability graphs and chordal graphs {\em  {\em  {\em   .
Revised:  Moreover, we relate{\em . We study the relation of competitivity graphs with other well-known families of graphs, such as permutation graphs, comparability graphs and chordal graphs {\em {\em {\em .
Edit intent: clarity
Original:  Moreover, we relate{\em   competitivity graphs with other well-known families of graphs, such as permutation graphs, comparability graphs and chordal graphs {\em  {\em  {\em   .
Revised:  Moreover, we relate{\em competitivity graphs with other well-known families of graphs, such as permutation graphs, comparability graphs and chordal graphs {\em competitivity graph {\em {\em .
Edit intent: fluency
Original: In this paper we show a new technique to analyze families of rankings  . Given a finite number of elements and a family of rankings of those elements, we say that two elements %DIFDELCMD < {\em %%% compete when they change their relative positions in at least two rankings.
Revised: In this paper we introduce a new technique to analyze families of rankings . Given a finite number of elements and a family of rankings of those elements, we say that two elements %DIFDELCMD < {\em %%% compete when they change their relative positions in at least two rankings.
Edit intent: clarity
Original: In this paper we show a new technique to analyze families of rankings  . Given a finite number of elements and a family of rankings of those elements, we say that two elements %DIFDELCMD < {\em %%% compete when they change their relative positions in at least two rankings.
Revised: In this paper we show a new technique to analyze families of rankings . Given a finite number of elements and a family of rankings of those elements, we say that two elements %DIFDELCMD < {\em %%% when they change their relative positions in at least two rankings.
Edit intent: clarity
Original: In this paper we show a new technique to analyze families of rankings  . Given a finite number of elements and a family of rankings of those elements, we say that two elements %DIFDELCMD < {\em %%% compete when they change their relative positions in at least two rankings.
Revised: In this paper we show a new technique to analyze families of rankings . Given a finite number of elements and a family of rankings of those elements, we say that two elements %DIFDELCMD < {\em %%% compete compete when they exchange their relative positions in at least two rankings.
Edit intent: clarity
Original:  This allows  to define an undirected graph by connecting elements that compete.
Revised:  This allows us to define an undirected graph by connecting elements that compete.
Edit intent: coherence
Original:  We call this graph a %DIFDELCMD < {\em %%% competitivity graph .
Revised:  We call this graph a %DIFDELCMD < {\em %%% competitivity graph .
Edit intent: fluency
Original:  We study the relation of competitivity graphs with other well-known families of graphs, such as permutation graphs, comparability graphs and chordal graphs.
Revised:  We study the relationship of competitivity graphs with other well-known families of graphs, such as permutation graphs, comparability graphs and chordal graphs.
Edit intent: fluency
Original:  In addition to this, we also introduce certain important sets of nodes in a %DIFDELCMD < {\em %DIFDELCMD < %%% competitivity graph . For example, nodes that compete among them form a %DIFDELCMD < {\em %DIFDELCMD < %%% competitivity set and nodes connected by chains of competitors form a set of %DIFDELCMD < {\em %%% eventual competitors .
Revised:  In addition to this, we also introduce certain important sets of nodes in a %DIFDELCMD < {\em %DIFDELCMD < %%% competitivity graph . For example, nodes that compete among them form a %DIFDELCMD < {\em %DIFDELCMD < %%% competitivity set and nodes connected by chains of competitors form a set of %DIFDELCMD < {\em %%% eventual competitors .
Edit intent: fluency
Original:  We analyze these sets and we show a technique to obtain them easily from a family of rankings.
Revised:  We analyze hese sets and we show a technique to obtain them easily from a family of rankings.
Edit intent: fluency
Original:  We analyze these sets and we show a technique to obtain them easily from a family of rankings.
Revised:  We analyze these sets and we show a method to obtain sets of eventual competitors directly from a family of rankings.
Edit intent: clarity
Original: We consider the variant of monotone simultaneous embeddings of directed graphs where all graphs are directed paths and directions of monotonicity are different .
Revised: We study monotone simultaneous embeddings of directed graphs where all graphs are directed paths and directions of monotonicity are different .
Edit intent: clarity