wanyu/IteraTeR_v2 huggingface dataset:
Original:  the direct factors are the right regular representation of S_n and the image of the left regular action of%DIFDELCMD < \Aut%%% (T(S)) on S_n . This strengthens a previous result of the author, where the automorphism group was factored as a semidirect product.
Revised:  the direct factors are the right regular representation of S_n and the image of the left regular action of%DIFDELCMD < \Aut%%% . This strengthens a previous result of the author, where the automorphism group was factored as a semidirect product.
Edit intent: clarity
Original:  the direct factors are the right regular representation of S_n and the image of the left regular action of%DIFDELCMD < \Aut%%% (T(S)) on S_n . This strengthens a previous result of the author, where the automorphism group was factored as a semidirect product.
Revised:  the direct factors are the right regular representation of S_n and the image of the left regular action of%DIFDELCMD < \Aut%%% (T(S)) on S_n . This strengthens some of the results in the literature where the automorphism group was factored as a semidirect product.
Edit intent: clarity
Original:  the direct factors are the right regular representation of S_n and the image of the left regular action of%DIFDELCMD < \Aut%%% (T(S)) on S_n . This strengthens a previous result of the author, where the automorphism group was factored as a semidirect product.
Revised:  the direct factors are the right regular representation of S_n and the image of the left regular action of%DIFDELCMD < \Aut%%% (T(S)) on S_n . This strengthens a previous result of the author, where the automorphism group was expressed as a semidirect product.
Edit intent: clarity
Original: The automorphism group of the modified bubble-sort graph was shown by Feng to be the semidirect product S_n%DIFDELCMD < \rtimes %%% D_{2n.
Revised: The modified bubble-sort graph was shown by Feng to be the semidirect product S_n%DIFDELCMD < \rtimes %%% D_{2n.
Edit intent: clarity
Original: The automorphism group of the modified bubble-sort graph was shown by Feng to be the semidirect product S_n%DIFDELCMD < \rtimes %%% D_{2n.
Revised: The automorphism group of the modified bubble-sort graph %DIFDELCMD < \rtimes %%% D_{2n.
Edit intent: coherence
Original:  Let S be a set of transpositions generating the symmetric group S_n, where n }%DIFDELCMD < \ge %%% 3. It is shown that if the girth of the transposition graph of S is at least 5, then the , for all n }\ge  automorphism group of the Cayley graph%DIFDELCMD < \Cay%%% (S_n,S) is the direct product S_n \times%DIFDELCMD < \Aut%%% (T(S)), where T(S) is the transposition graph of S.
Revised:  Let S be a set of transpositions generating the symmetric group S_n, where n }%DIFDELCMD < \ge %%% 3. It is shown that if the girth of the transposition graph of S is at least 5, then the , for all n }\ge automorphism group of the %DIFDELCMD < \Cay%%% (S_n,S) is the direct product S_n \times%DIFDELCMD < \Aut%%% (T(S)), where T(S) is the transposition graph of S.
Edit intent: clarity
Original: Random s-intersection graphs have recently received much interest 1-9 . In a random s-intersection graph , each vertex is assigned to a set of items in some  manner, and two vertices have an edge in between if and only if they share at least s items .
Revised: Random s-intersection graphs have recently received much interest . In a random s-intersection graph , each vertex is assigned to a set of items in some manner, and two vertices have an edge in between if and only if they share at least s items .
Edit intent: coherence
Original: Random s-intersection graphs have recently received much interest 1-9 . In a random s-intersection graph , each vertex is assigned to a set of items in some  manner, and two vertices have an edge in between if and only if they share at least s items .
Revised: Random s-intersection graphs have recently received much interest 1-9 . In a random s-intersection graph is constructed by first assigning each vertex a set of items in some manner, and two vertices have an edge in between if and only if they share at least s items .
Edit intent: coherence
Original: Random s-intersection graphs have recently received much interest 1-9 . In a random s-intersection graph , each vertex is assigned to a set of items in some  manner, and two vertices have an edge in between if and only if they share at least s items .
Revised: Random s-intersection graphs have recently received much interest 1-9 . In a random s-intersection graph , each vertex is assigned to a set of items in some manner, and then putting an undirected edge between all pairs of vertices that share at least s items .
Edit intent: clarity
Original:  In particular , in a uniform random s-intersection graph,  each vertex independently selects the same number of items uniformly at random from a common item pool , while in a binomial random s-intersection graph, each item in some item pool is independently attached to each vertex with the same probability.
Revised:  In particular , in a uniform random s-intersection graph, where each vertex independently selects the same number of items uniformly at random from a common item pool , while in a binomial random s-intersection graph, each item in some item pool is independently attached to each vertex with the same probability.
Edit intent: clarity
Original:  In particular , in a uniform random s-intersection graph,  each vertex independently selects the same number of items uniformly at random from a common item pool , while in a binomial random s-intersection graph, each item in some item pool is independently attached to each vertex with the same probability.
Revised:  In particular , in a uniform random s-intersection graph, each vertex independently selects the same number of items uniformly at random from a common item pool , while in a binomial random s-intersection graph, where each item from a pool is independently attached to each vertex with the same probability.
Edit intent: clarity
Original:  In particular , in a uniform random s-intersection graph,  each vertex independently selects the same number of items uniformly at random from a common item pool , while in a binomial random s-intersection graph, each item in some item pool is independently attached to each vertex with the same probability.
Revised:  In particular , in a uniform random s-intersection graph, each vertex independently selects the same number of items uniformly at random from a common item pool , while in a binomial random s-intersection graph, each item in some item pool is independently assigned to each vertex with the same probability.
Edit intent: clarity
Original:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and  online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Revised:  e.g., using uniform random s-intersection graph for cryptanalysis %DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Edit intent: coherence
Original:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and  online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Revised:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% 8-10 and online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Edit intent: coherence
Original:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and  online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Revised:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and online social networks , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Edit intent: coherence
Original:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and  online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Revised:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and online social networks 11, 12 , and using a binomial random s-intersection 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Edit intent: coherence
Original:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and  online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Revised:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis %DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Edit intent: coherence
Original:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and  online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Revised:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Edit intent: coherence
Original:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and  online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Revised:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Edit intent: coherence
Original:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and  online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 . For binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Revised:  e.g., using uniform random s-intersection graph for cryptanalysis 14, 15%DIFDELCMD < ]%%% , and to model secure wireless sensor networks 8-10 and online social networks 11, 12 , and using a binomial random s-intersection graph for clustering analysis 17%DIFDELCMD < ]%%% , classification 18 and the design of integrated circuits 34 (binomial random s-intersection graphs). In this paper, for binomial/uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree.
Edit intent: coherence
Original:  Specifically, we derive the asymptotically exact probabilities and asymptotic zero-one laws for the following three properties:
Revised:  Specifically, we derive the asymptotically exact probabilities and zero-one laws for the following three properties:
Edit intent: clarity
Original:  (i) k-vertex-connectivity, (ii) k-edge-connectivity and (iii) the property of minimum vertex degree being at least k . Recently, similar results have been obtained by Bloznelis and Rybarczyk 5%DIFDELCMD < ]%%% , but their findings are only for uniform random s-intersection graphs, not for binomial random s-intersection graphs, and require parameter conditions which are disjoint from ours -
Revised:  (i) k-vertex-connectivity, (ii) k-edge-connectivity and (iii) the property of minimum vertex degree being at least k 5%DIFDELCMD < ]%%% , but their findings are only for uniform random s-intersection graphs, not for binomial random s-intersection graphs, and require parameter conditions which are disjoint from ours -
Edit intent: coherence
Original:  (i) k-vertex-connectivity, (ii) k-edge-connectivity and (iii) the property of minimum vertex degree being at least k . Recently, similar results have been obtained by Bloznelis and Rybarczyk 5%DIFDELCMD < ]%%% , but their findings are only for uniform random s-intersection graphs, not for binomial random s-intersection graphs, and require parameter conditions which are disjoint from ours -
Revised:  (i) k-vertex-connectivity, (ii) k-edge-connectivity and (iii) the property of minimum vertex degree being at least k . Recently, similar results have been obtained by Bloznelis and Rybarczyk %DIFDELCMD < ]%%% , but their findings are only for uniform random s-intersection graphs, not for binomial random s-intersection graphs, and require parameter conditions which are disjoint from ours -
Edit intent: coherence
Original: Random key graphs have been used in secure wireless sensor networks (WSNs) and various other applications.
Revised: The Eschenauer--Gligor (EG) random key predistribution scheme has been widely recognized as a typical approach to secure communications in wireless sensor networks (WSNs) and various other applications.
Edit intent: clarity
Original:  With parameters K,P, and p scaled with n, we derive asymptotically exact probabilities for three related graph properties in G_{on remains connected despite the deletion of any (k-1) vertices (resp. edges).
Revised:  With parameters K,P, and p scaled with n, we derive asymptotically exact probabilities for three related graph properties in G_{on remains connected despite the failure of any (k-1) vertices (resp. edges).
Edit intent: clarity
Original:  We propose here some new characterizations linking indexed languages to context-free languages: the class of indexed languages is the image of the Dyck language by a class of context-free transducers, it is also the class of images, by a projection defined by an FO-formula of nested words labelled by a Dyck word .
Revised:  Here we propose a characterizations of indexed languages by context-free languages: the class of indexed languages is the image of the Dyck language by a class of context-free transducers, it is also the class of images, by a projection defined by an FO-formula of nested words labelled by a Dyck word .
Edit intent: clarity
Original:  By shaping a pulse we mean computing its magnitude and length, which results in reaching the desired control objective.
Revised:  By shaping we mean computing its magnitude and length, which results in reaching the desired control objective.
Edit intent: clarity
Original:  By shaping a pulse we mean computing its magnitude and length, which results in reaching the desired control objective.
Revised:  By shaping a pulse we mean computing the magnitude and the length of a pulse, application of which results in reaching the desired control objective.
Edit intent: clarity
Original:  An advantage of using pulses for biological control problems is that pulses can easily be implemented in a laboratory setting by scientists.
Revised:  One of the advantages of applying this framework to biological systems is that pulses can easily be implemented in a laboratory setting by scientists.
Edit intent: clarity
Original:  We show that it is possible to estimate the  pulses fulfilling the objective, if the trajectories of the controlled system can be bounded from above and below by the trajectories of monotone systems.
Revised:  We show that it is possible to estimate the set of the pulses fulfilling the objective, if the trajectories of the controlled system can be bounded from above and below by the trajectories of monotone systems.
Edit intent: clarity
Original:  This result can be generalised to systems with uncertain parameter values under some mild assumptions on the set of admissible parameters.
Revised:  This result can be generalised to systems with parametric uncertainty under some mild assumptions on the set of admissible parameters.
Edit intent: clarity
Original:  Secondly, we consider a problem of inducing  oscillatory behaviour in an eight species generalised repressilator, which is a monotone system.
Revised:  Secondly, we consider a problem of inducing an oscillatory behaviour in an eight species generalised repressilator, which is a monotone system.
Edit intent: fluency
Original:  We illustrate the results on some example genetic circuits .
Revised:  We illustrate the results with a genetic bistable system commonly used in synthetic biology .
Edit intent: clarity
Original:  The latter are used first as fitness to estimate the unknown node degrees through a standard configuration model ;
Revised:  The latter are used as fitness to estimate the unknown node degrees through a standard configuration model ;
Edit intent: coherence
Original:  In this paper we prove  tropical Nullstellensatz and moreover we show  effective formulation of this theorem.
Revised:  In this paper we prove a tropical Nullstellensatz and moreover we show effective formulation of this theorem.
Edit intent: fluency
Original:  In this paper we prove  tropical Nullstellensatz and moreover we show  effective formulation of this theorem.
Revised:  In this paper we prove tropical Nullstellensatz and moreover we show an effective formulation of this theorem.
Edit intent: fluency
Original:  Nullstellensatz is a next natural step in building algebraic theory of tropical polynomials and  effective version is relevant for computational aspects of this field.
Revised:  Nullstellensatz is a natural step in building algebraic theory of tropical polynomials and effective version is relevant for computational aspects of this field.
Edit intent: clarity
Original:  Nullstellensatz is a next natural step in building algebraic theory of tropical polynomials and  effective version is relevant for computational aspects of this field.
Revised:  Nullstellensatz is a next natural step in building algebraic theory of tropical polynomials and its effective version is relevant for computational aspects of this field.
Edit intent: clarity
Original: The proof of efficiency of our algorithm is similar to Yoshinaka\mbox{ Yoshinaka:20080pt . Notice that the judgment of "x\sim y?" ends in linear order time.
Revised: 0pt . Notice that the judgment of "x\sim y?" ends in linear order time.
Edit intent: clarity
Original: The paper highlights the role of a particular feedback mechanism at the cellular level in the robustness and modulation properties of oscillations at the circuit level.
Revised: The paper highlights the role of a positive feedback gating mechanism at the cellular level in the robustness and modulation properties of oscillations at the circuit level.
Edit intent: clarity
Original: The paper highlights the role of a particular feedback mechanism at the cellular level in the robustness and modulation properties of oscillations at the circuit level.
Revised: The paper highlights the role of a particular feedback mechanism at the cellular level in the robust- ness and modulation properties of oscillations at the circuit level.
Edit intent: fluency
Original: The paper highlights the role of a particular feedback mechanism at the cellular level in the robustness and modulation properties of oscillations at the circuit level.
Revised: The paper highlights the role of a particular feedback mechanism at the cellular level in the robustness and modulation properties of rhythmic activities at the circuit level.
Edit intent: clarity
Original:  The results are presented in the context of half-center oscillators, which are simple rhythmic circuits composed of two  inhibitory neuronal populations reciprocally connected.
Revised:  The results are presented in the context of half-center oscillators, which are simple rhythmic circuits composed of two reciprocally connected inhibitory neuronal populations reciprocally connected.
Edit intent: clarity
Original:  It is well known that such rhythms rely on a particular excitability property, the post-inhibitory rebound, to which two distinct ionic currents contribute, I_{h slow-positive feedback nature at the cellular level is fundamental for the robustness and modulation at the circuit level.
Revised:  It is well known that such rhythms rely on a particu- lar excitability property, the post-inhibitory rebound, to which two distinct ionic currents contribute, I_{h slow-positive feedback nature at the cellular level is fundamental for the robustness and modulation at the circuit level.
Edit intent: fluency
Original:  It is well known that such rhythms rely on a particular excitability property, the post-inhibitory rebound, to which two distinct ionic currents contribute, I_{h slow-positive feedback nature at the cellular level is fundamental for the robustness and modulation at the circuit level.
Revised:  It is well known that such rhythms rely on a particular excitability property, the post-inhibitory rebound, to which two distinct ionic currents contribute, I_{h slow-positive feedback at the cellular level is fundamental for the robustness and modulation at the circuit level.
Edit intent: clarity