wanyu/IteraTeR_v2 huggingface dataset:
Original: We present a new framework for studying C \infty\C -words, that are arbitrarily many times differentiable words over the alphabet \Sigma%DIFDELCMD < } %%% = 1,2%DIFDELCMD < }%%% . After introducing an equivalence relation on C \infty\C -words , whose classes are called minimal classes and represent all the C \infty%DIFDELCMD < }%%% -words, we define a vertical coding of these words based on a three-letter alphabet, and a set of functions operating over this representation.
Revised: We present a new framework for studying C \infty\C -words, that are arbitrarily many times differentiable words over the alphabet %DIFDELCMD < } %%% = 1,2%DIFDELCMD < }%%% . After introducing an equivalence relation on C \infty\C -words , whose classes are called minimal classes and represent all the C \infty%DIFDELCMD < }%%% -words, we define a vertical coding of these words based on a three-letter alphabet, and a set of functions operating over this representation.
Edit intent: coherence
Original: We present a new framework for studying C \infty\C -words, that are arbitrarily many times differentiable words over the alphabet \Sigma%DIFDELCMD < } %%% = 1,2%DIFDELCMD < }%%% . After introducing an equivalence relation on C \infty\C -words , whose classes are called minimal classes and represent all the C \infty%DIFDELCMD < }%%% -words, we define a vertical coding of these words based on a three-letter alphabet, and a set of functions operating over this representation.
Revised: We present a new framework for studying C \infty\C -words, that are arbitrarily many times differentiable words over the alphabet \Sigma%DIFDELCMD < } %%% 1,2%DIFDELCMD < }%%% . After introducing an equivalence relation on C \infty\C -words , whose classes are called minimal classes and represent all the C \infty%DIFDELCMD < }%%% -words, we define a vertical coding of these words based on a three-letter alphabet, and a set of functions operating over this representation.
Edit intent: coherence
Original: We present a new framework for studying C \infty\C -words, that are arbitrarily many times differentiable words over the alphabet \Sigma%DIFDELCMD < } %%% = 1,2%DIFDELCMD < }%%% . After introducing an equivalence relation on C \infty\C -words , whose classes are called minimal classes and represent all the C \infty%DIFDELCMD < }%%% -words, we define a vertical coding of these words based on a three-letter alphabet, and a set of functions operating over this representation.
Revised: We present a new framework for studying C \infty\C -words, that are arbitrarily many times differentiable words over the alphabet \Sigma%DIFDELCMD < } %%% = %DIFDELCMD < }%%% . After introducing an equivalence relation on C \infty\C -words , whose classes are called minimal classes and represent all the C \infty%DIFDELCMD < }%%% -words, we define a vertical coding of these words based on a three-letter alphabet, and a set of functions operating over this representation.
Edit intent: coherence
Original: We present a new framework for studying C \infty\C -words, that are arbitrarily many times differentiable words over the alphabet \Sigma%DIFDELCMD < } %%% = 1,2%DIFDELCMD < }%%% . After introducing an equivalence relation on C \infty\C -words , whose classes are called minimal classes and represent all the C \infty%DIFDELCMD < }%%% -words, we define a vertical coding of these words based on a three-letter alphabet, and a set of functions operating over this representation.
Revised: We present a new framework for studying C \infty\C -words, that are arbitrarily many times differentiable words over the alphabet \Sigma%DIFDELCMD < } %%% = 1,2%DIFDELCMD < }%%% . After introducing an equivalence relation on C \infty\C -words , whose classes are called minimal classes and represent all the C %DIFDELCMD < }%%% -words, we define a vertical coding of these words based on a three-letter alphabet, and a set of functions operating over this representation.
Edit intent: fluency
Original:  We show that the minimal classes of C \infty%DIFDELCMD < }%%% -words can be represented on an infinite directed acyclic graph , which, as well as all the functions introduced for the vertical coding,\C can be defined recursively  with no explicit reference to C \infty\C -words.
Revised:  We show that the minimal classes of C %DIFDELCMD < }%%% -words can be represented on an infinite directed acyclic graph , which, as well as all the functions introduced for the vertical coding,\C can be defined recursively with no explicit reference to C \infty\C -words.
Edit intent: fluency
Original:  We show that the minimal classes of C \infty%DIFDELCMD < }%%% -words can be represented on an infinite directed acyclic graph , which, as well as all the functions introduced for the vertical coding,\C can be defined recursively  with no explicit reference to C \infty\C -words.
Revised:  We show that the minimal classes of C \infty%DIFDELCMD < }%%% through an infinite directed acyclic graph , which, as well as all the functions introduced for the vertical coding,\C can be defined recursively with no explicit reference to C \infty\C -words.
Edit intent: clarity
Original:  We show that the minimal classes of C \infty%DIFDELCMD < }%%% -words can be represented on an infinite directed acyclic graph , which, as well as all the functions introduced for the vertical coding,\C can be defined recursively  with no explicit reference to C \infty\C -words.
Revised:  We show that the minimal classes of C \infty%DIFDELCMD < }%%% -words can be represented on an infinite directed acyclic graph , which, as well as all the functions introduced for the vertical coding,\C can be defined recursively and with no explicit reference to C \infty\C -words.
Edit intent: coherence
Original:  We show that the minimal classes of C \infty%DIFDELCMD < }%%% -words can be represented on an infinite directed acyclic graph , which, as well as all the functions introduced for the vertical coding,\C can be defined recursively  with no explicit reference to C \infty\C -words.
Revised:  We show that the minimal classes of C \infty%DIFDELCMD < }%%% -words can be represented on an infinite directed acyclic graph , which, as well as all the functions introduced for the vertical coding,\C can be defined recursively with no explicit \infty\C -words.
Edit intent: clarity
Original:  This new representation adequately expresses the combinatorial structure of C \infty\C -words , and brings new perspectives in the study of the Kolakoski word and its factors .
Revised:  \infty\C -words , and brings new perspectives in the study of the Kolakoski word and its factors .
Edit intent: clarity
Original:  While the problem stays NP-complete, it allows algorithms in FPT for biologically relevant parameterizations.
Revised:  While the problem remains NP-complete, it allows algorithms in FPT for biologically relevant parameterizations.
Edit intent: fluency
Original: Graphs and networks are used to model interactions in a variety of contexts, and there is a growing need to be able to quickly assess the qualities of a graph in order to understand its underlying structure . Some of the most useful metrics are triangle based and give a measure of the connectedness of "friends of friends.
Revised: Graphs are used to model interactions in a variety of contexts, and there is a growing need to be able to quickly assess the qualities of a graph in order to understand its underlying structure . Some of the most useful metrics are triangle based and give a measure of the connectedness of "friends of friends.
Edit intent: clarity
Original: Graphs and networks are used to model interactions in a variety of contexts, and there is a growing need to be able to quickly assess the qualities of a graph in order to understand its underlying structure . Some of the most useful metrics are triangle based and give a measure of the connectedness of "friends of friends.
Revised: Graphs and networks are used to model interactions in a variety of contexts, and there is a growing need to quickly assess the qualities of a graph in order to understand its underlying structure . Some of the most useful metrics are triangle based and give a measure of the connectedness of "friends of friends.
Edit intent: clarity
Original: Graphs and networks are used to model interactions in a variety of contexts, and there is a growing need to be able to quickly assess the qualities of a graph in order to understand its underlying structure . Some of the most useful metrics are triangle based and give a measure of the connectedness of "friends of friends.
Revised: Graphs and networks are used to model interactions in a variety of contexts, and there is a growing need to be able to quickly assess the structure of a graph in order to understand its underlying structure . Some of the most useful metrics are triangle based and give a measure of the connectedness of "friends of friends.
Edit intent: clarity
Original: Graphs and networks are used to model interactions in a variety of contexts, and there is a growing need to be able to quickly assess the qualities of a graph in order to understand its underlying structure . Some of the most useful metrics are triangle based and give a measure of the connectedness of "friends of friends.
Revised: Graphs and networks are used to model interactions in a variety of contexts, and there is a growing need to be able to quickly assess the qualities of a graph . Some of the most useful metrics are triangle based and give a measure of the connectedness of "friends of friends.
Edit intent: coherence
Original: Verified boot is an interesting feature of Chromium OS that should detect any modification in the firmware, kernel or the root file system (rootfs) by a dedicated adversary.
Revised: Verified boot is an interesting feature of Chromium OS that supposedly can detect any modification in the firmware, kernel or the root file system (rootfs) by a dedicated adversary.
Edit intent: clarity
Original: Verified boot is an interesting feature of Chromium OS that should detect any modification in the firmware, kernel or the root file system (rootfs) by a dedicated adversary.
Revised: Verified boot is an interesting feature of Chromium OS that should detect any modification in the root file system (rootfs) by a dedicated adversary.
Edit intent: clarity
Original:  The exploit is based on the fact that although a kernel partition is paired with a rootfs , verification of kernel partition and rootfs are independent of each other .
Revised:  The exploit is based on the fact that a dedicated adversary can replace the rootfs and the corresponding verification information in the bootloader .
Edit intent: clarity
Original:  We experimentally demonstrate an attack using both the base and developer version of Chromium OS in which the adversary installs a spyware in the target system to send cached user data to the attacker machine in plain text which are otherwise inaccessible in encryptedform .
Revised:  We experimentally demonstrate an attack using both the base and developer version of Chromium OS in which the adversary installs a spyware in the target system to send cached user data to the attacker machine in plain text which are otherwise encrypted, and thus inaccessible .
Edit intent: clarity
Original:  We also discuss possible directions to mitigate the vulnerability.
Revised:  We also demonstrate techniques to mitigate this vulnerability.
Edit intent: clarity
Original: In this article we present an elitism levels traverse mechanism that we designed to find bounds on population-based Evolutionary algorithms solving unimodal functions.
Revised: In this article we present an Elitism Levels Traverse Mechanism that we designed to find bounds on population-based Evolutionary algorithms solving unimodal functions.
Edit intent: fluency
Original: Steganography is a general term referring to all methods for the embedding of additional  content into some form of carrier, with the aim of concealment of the introduced alterations.
Revised: Steganography is a general term referring to all methods for the embedding of additional secret content into some form of carrier, with the aim of concealment of the introduced alterations.
Edit intent: clarity
Original:  Present day steganographic methods are far more sophisticated than their ancient predecessors, but the main principles had remained unchanged.
Revised:  Present day steganographic methods are far more sophisticated than their ancient predecessors, but the main principles have remained unchanged.
Edit intent: fluency
Original:  A consequence of multiple TFs interacting at overlapping binding sites is that their binding noise becomes correlated.
Revised:  A consequence of multiple TFs interacting at overlapping binding-sites is that their binding noise becomes correlated.
Edit intent: clarity
Original:  We explicitly demonstrate these ideas for the toy model of two TFs competing for the same binding site.
Revised:  We demonstrate these ideas for the toy model of two TFs competing for the same binding site.
Edit intent: clarity
Original:  We suggest that this mechanism potentially explains the motif of a coherent feed-forward loop terminating in overlapping binding sites commonly found in developmental networks , and discuss three specific examples.
Revised:  We suggest that this mechanism potentially explains the motif of a coherent feed-forward gene regulatory modules , and discuss three specific examples.
Edit intent: clarity
Original:  The systematic method proposed herein can be used to shed light on TF cross-regulation networks either from direct measurements of binding noise, or bioinformatic analysis of overlapping binding sites .
Revised:  The systematic method proposed herein can be used to shed light on TF cross-regulation networks either from direct measurements of binding noise, or bioinformatic analysis of overlapping binding-sites .
Edit intent: clarity
Original:  We establish an equivalent to the Fundamental Theorem of Arithmetic by showing that arbitrary walks on G uniquely factorize into nestings of prime walks, which we show are the simple paths and simple cycles of G. Thus the algebra KG_O=(KG, +,O) is a noncommutative nonassociative unique factorization algebra with zero divisors, which we call the quiver nesting algebra.
Revised:  We establish an equivalent to the fundamental theorem of arithmetic: we demonstrate that arbitrary walks on G uniquely factorize into nestings of prime walks, which we show are the simple paths and simple cycles of G. Thus the algebra KG_O=(KG, +,O) is a noncommutative nonassociative unique factorization algebra with zero divisors, which we call the quiver nesting algebra.
Edit intent: clarity
Original:  We establish an equivalent to the Fundamental Theorem of Arithmetic by showing that arbitrary walks on G uniquely factorize into nestings of prime walks, which we show are the simple paths and simple cycles of G. Thus the algebra KG_O=(KG, +,O) is a noncommutative nonassociative unique factorization algebra with zero divisors, which we call the quiver nesting algebra.
Revised:  We establish an equivalent to the Fundamental Theorem of Arithmetic by showing that arbitrary walks on G uniquely factorize into nesting products of simple paths and simple cycles of G. Thus the algebra KG_O=(KG, +,O) is a noncommutative nonassociative unique factorization algebra with zero divisors, which we call the quiver nesting algebra.
Edit intent: clarity
Original:  By using this last result , we recently developed a new approach to matrix functions in [ P.-L. Giscard, S. J. Thwaite, D. Jaksch, arXiv:1112.1588 v2 (2011) ]  .
Revised:  This last result has already found applications in the field of matrix computations [ P.-L. Giscard, S. J. Thwaite, D. Jaksch, arXiv:1112.1588 v2 (2011) ] .
Edit intent: clarity
Original:  By using this last result , we recently developed a new approach to matrix functions in [ P.-L. Giscard, S. J. Thwaite, D. Jaksch, arXiv:1112.1588 v2 (2011) ]  .
Revised:  By using this last result , we recently developed a new approach to matrix functions in [ arXiv:1112.1588 v2 (2011) ] .
Edit intent: coherence
Original: We show that  characteristic series of walks (paths) on any finite quiver G are given by continued fractions of finite depth over the simple paths and simple cycles of G.  We obtain an explicit formula giving these continued fractions.
Revised: We show that the characteristic series of walks (paths) on any finite quiver G are given by continued fractions of finite depth over the simple paths and simple cycles of G. We obtain an explicit formula giving these continued fractions.
Edit intent: fluency
Original: We show that  characteristic series of walks (paths) on any finite quiver G are given by continued fractions of finite depth over the simple paths and simple cycles of G.  We obtain an explicit formula giving these continued fractions.
Revised: We show that characteristic series of walks (paths) on any finite quiver G are given by continued fractions of finite depth involving the simple paths and simple cycles of G. We obtain an explicit formula giving these continued fractions.
Edit intent: clarity
Original:  This last result has already found applications in the field of matrix computations arXiv:1112.1588 .
Revised:  This last result has already found applications in the field of matrix computations .
Edit intent: clarity
Original:  We give further examples of applications of our results.
Revised:  We give examples illustrating our results.
Edit intent: clarity
Original: We show that the characteristic series of walks (paths) between any two vertices of any finite digraph or weighteddigraph G is given by a universal continued fraction of finite depth  involving the simple paths and simple cycles of G . A simple path is a walk forbidden to visit any vertex more than once.
Revised: We show that the series of all walks between any two vertices of any finite digraph or weighteddigraph G is given by a universal continued fraction of finite depth involving the simple paths and simple cycles of G . A simple path is a walk forbidden to visit any vertex more than once.
Edit intent: clarity
Original:  We show that the simple paths and simple cycles are the prime elements of the ensemble of all walks on G equipped with the nesting product.
Revised:  We show that the simple paths and simple cycles are the prime elements of the set of all walks on G equipped with the nesting product.
Edit intent: clarity
Original:  We obtain a recursive formula producing the prime factorization of ensembles of walks.
Revised:  We obtain a recursive formula producing the prime factorization of sets of walks.
Edit intent: clarity
Original:  We describe a WTP quadrature algorithm based on an algorithm of Hegland, and also formulate a version of Wasilkowski and Wo\'zniakowski 's WTP algorithm, here called the WW algorithm.
Revised:  We describe a WTP quadrature algorithm based on an algorithm of Hegland, and also formulate a version of Wasilkowski and Wozniakowski 's WTP algorithm, here called the WW algorithm.
Edit intent: fluency
Original:  We prove that our algorithm is optimal and therefore lower in cost than the WW algorithm, and therefore both algorithms have the optimal asymptotic rate of convergence given by Theorem 3 of Wasilkowski and Wo\'zniakowski .
Revised:  We prove that our algorithm is optimal and therefore lower in cost than the WW algorithm, and therefore both algorithms have the optimal asymptotic rate of convergence given by Theorem 3 of Wasilkowski and Wozniakowski .
Edit intent: fluency
Original: This paper examines sparse grid quadrature on weighted tensor products (WTP) of reproducing kernel Hilbert spaces on products of the unit sphere  .
Revised: We examine sparse grid quadrature on weighted tensor products (WTP) of reproducing kernel Hilbert spaces on products of the unit sphere .
Edit intent: clarity
Original:  We prove that our algorithm is optimal and therefore lower in cost than the WW algorithm , and therefore both algorithms have the optimal asymptotic rate of convergence given by Theorem 3 of Wasilkowski and Wozniakowski .
Revised:  We prove that our algorithm is optimal and therefore lower in cost than the WW algorithm . Both algorithms therefore have the optimal asymptotic rate of convergence given by Theorem 3 of Wasilkowski and Wozniakowski .
Edit intent: coherence
Original: A problem of much theoretical and practical interest is to determine or estimate the diameter of various families of Cayley graphs . Let \Gamma be a Cayley graph on n! vertices generated by a transposition tree on vertex set 1,2,...,n%DIFDELCMD < }%%% . In an oft-cited paper \mbox{ Akers:
Revised: A problem of practical and theoretical interest is to determine or estimate the diameter of various families of Cayley graphs . Let \Gamma be a Cayley graph on n! vertices generated by a transposition tree on vertex set 1,2,...,n%DIFDELCMD < }%%% . In an oft-cited paper \mbox{ Akers:
Edit intent: clarity
Original: A problem of much theoretical and practical interest is to determine or estimate the diameter of various families of Cayley graphs . Let \Gamma be a Cayley graph on n! vertices generated by a transposition tree on vertex set 1,2,...,n%DIFDELCMD < }%%% . In an oft-cited paper \mbox{ Akers:
Revised: A problem of much theoretical and practical interest is to determine or estimate the diameter of various families of Cayley networks . Let \Gamma be a Cayley graph on n! vertices generated by a transposition tree on vertex set 1,2,...,n%DIFDELCMD < }%%% . In an oft-cited paper \mbox{ Akers:
Edit intent: clarity
Original: A problem of much theoretical and practical interest is to determine or estimate the diameter of various families of Cayley graphs . Let \Gamma be a Cayley graph on n! vertices generated by a transposition tree on vertex set 1,2,...,n%DIFDELCMD < }%%% . In an oft-cited paper \mbox{ Akers:
Revised: A problem of much theoretical and practical interest is to determine or estimate the diameter of various families of Cayley graphs . Let \Gamma be a Cayley graph on n! vertices generated by a transposition tree on 1,2,...,n%DIFDELCMD < }%%% . In an oft-cited paper \mbox{ Akers:
Edit intent: clarity
Original: Krishnamurthy:19890pt (cf. also \mbox{ \cite[p. 188]{Hahn:Sabidussi:1997}}0pt ), it is shown that the diameter of \Gamma is bounded as }%DIFDELCMD < \diam%%% (\Gamma)%DIFDELCMD < \le %%% \max_{\pi \in S_n{c(\pi)-n+i=1^n \dist_T(i,\pi(i))}, where the maximum is over all permutations in S_n, c(\pi) denotes the number of cycles in the disjoint cycle representation of \pi, and }%DIFDELCMD < \dist%%% _T is the distance function on pairs of vertices of the tree.
Revised: Krishnamurthy:19890pt (cf. also \mbox{ \cite[p. 188]{Hahn:Sabidussi:1997}}0pt ), it is shown that the diameter of \Gamma is bounded as }%DIFDELCMD < \diam%%% %DIFDELCMD < \le %%% \max_{\pi \in S_n{c(\pi)-n+i=1^n \dist_T(i,\pi(i))}, where the maximum is over all permutations in S_n, c(\pi) denotes the number of cycles in the disjoint cycle representation of \pi, and }%DIFDELCMD < \dist%%% _T is the distance function on pairs of vertices of the tree.
Edit intent: coherence
Original:  Observe that evaluating this upper bound requires \Omega(n! n^2)computations since the quantity in braces above needs to be evaluated for each permutation in S_n . In this work,
Revised:  Observe that evaluating this upper bound requires \Omega(n! ) . In this work,
Edit intent: coherence
Original:  Let \Gamma be a Cayley graph of the symmetric group generated by a transposition tree on n vertices.
Edit intent: coherence
Original:  The previously known estimate for the diameter of this family of Cayley graphs is an upper bound given in the oft-cited paper of Akers and Krishnamurthy (1989).
Revised:  The previously known estimate for the diameter of Cayley graphs generated by transposition trees is an upper bound given in the oft-cited paper of Akers and Krishnamurthy (1989).
Edit intent: clarity