wanyu/IteraTeR_v2 huggingface dataset:
Original:  The generalized theta graph \Theta { \iota } 1,...,{ \iota }p consists in two end-vertices joined by p \ge 2 internally vertex-disjoint paths with respective lengths 1 \le { \iota } 1 \le . . . \le { \iota } p. We prove that the packing chromatic number of any undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n 3 = |{i / 1 \le i \le p, { \iota } i = 3}|, and that both these bounds are tight.
Revised:  The generalized theta graph \Theta { \iota } 1,...,{ \ell }p consists in two end-vertices joined by p \ge 2 internally vertex-disjoint paths with respective lengths 1 \le { \iota } 1 \le . . . \le { \iota } p. We prove that the packing chromatic number of any undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n 3 = |{i / 1 \le i \le p, { \iota } i = 3}|, and that both these bounds are tight.
Edit intent: clarity
Original:  The generalized theta graph \Theta { \iota } 1,...,{ \iota }p consists in two end-vertices joined by p \ge 2 internally vertex-disjoint paths with respective lengths 1 \le { \iota } 1 \le . . . \le { \iota } p. We prove that the packing chromatic number of any undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n 3 = |{i / 1 \le i \le p, { \iota } i = 3}|, and that both these bounds are tight.
Revised:  The generalized theta graph \Theta { \iota } 1,...,{ \iota }p consists in two end-vertices joined by p \ge 2 internally vertex-disjoint paths with respective lengths 1 \le { \ell } 1 \le . . . \le { \iota } p. We prove that the packing chromatic number of any undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n 3 = |{i / 1 \le i \le p, { \iota } i = 3}|, and that both these bounds are tight.
Edit intent: fluency
Original:  The generalized theta graph \Theta { \iota } 1,...,{ \iota }p consists in two end-vertices joined by p \ge 2 internally vertex-disjoint paths with respective lengths 1 \le { \iota } 1 \le . . . \le { \iota } p. We prove that the packing chromatic number of any undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n 3 = |{i / 1 \le i \le p, { \iota } i = 3}|, and that both these bounds are tight.
Revised:  The generalized theta graph \Theta { \iota } 1,...,{ \iota }p consists in two end-vertices joined by p \ge 2 internally vertex-disjoint paths with respective lengths 1 \le { \iota } 1 \le . . . \le { \ell } p. We prove that the packing chromatic number of any undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n 3 = |{i / 1 \le i \le p, { \iota } i = 3}|, and that both these bounds are tight.
Edit intent: clarity
Original:  The generalized theta graph \Theta { \iota } 1,...,{ \iota }p consists in two end-vertices joined by p \ge 2 internally vertex-disjoint paths with respective lengths 1 \le { \iota } 1 \le . . . \le { \iota } p. We prove that the packing chromatic number of any undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n 3 = |{i / 1 \le i \le p, { \iota } i = 3}|, and that both these bounds are tight.
Revised:  The generalized theta graph \Theta { \iota } 1,...,{ \iota }p consists in two end-vertices joined by p \ge 2 internally vertex-disjoint paths with respective lengths 1 \le { \iota } 1 \le . . . \le { \iota } p. We prove that the packing chromatic number of any undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n 3 = |{i / 1 \le i \le p, { \ell } i = 3}|, and that both these bounds are tight.
Edit intent: fluency
Original:  The proposed duality makes it possible to apply energy-based intuition from associative memory to analyze computational properties of neural networks with unusual activation functions - the higher rectified polynomials which until now have not been used for training neural networks .
Revised:  The proposed duality makes it possible to apply energy-based intuition from associative memory to analyze computational properties of neural networks with unusual activation functions - the higher rectified polynomials which until now have not been used in deep learning .
Edit intent: clarity
Original: Network generation models provide an understanding of the dynamics behind the formation and evolution of different networks including social networks, technological networks and biological networks .
Revised: Network generation models provide an understanding of how real world networks evolve over time .
Edit intent: clarity
Original:  Two important applications of these models are to study the evolution dynamics of networkformation and to generate benchmark networks with known community structures.
Revised:  Two important applications of these models are to study the evolution dynamics and processes that shape a network, and to generate benchmark networks with known community structures.
Edit intent: clarity
Original:  Research has been conducted in both these directions  relatively independent of the other application area .
Revised:  Research has been conducted in both these directions , relatively independent of the other application area .
Edit intent: fluency
Original:  Research has been conducted in both these directions  relatively independent of the other application area .
Revised:  Research has been conducted in both these directions relatively independent of the other .
Edit intent: clarity
Original:  In this paper, we propose to study both these application areas together i. e.\ introduce a network generation model  based on evolution dynamics of real world networks and  generate networks with community structures that can be used as benchmark graphs to study community detection algorithms.
Revised:  In this paper, we propose to study both these application areas together . We introduce a network generation model based on evolution dynamics of real world networks and generate networks with community structures that can be used as benchmark graphs to study community detection algorithms.
Edit intent: coherence
Original:  In this paper, we propose to study both these application areas together i. e.\ introduce a network generation model  based on evolution dynamics of real world networks and  generate networks with community structures that can be used as benchmark graphs to study community detection algorithms.
Revised:  In this paper, we propose to study both these application areas together i. e.\ introduce a network generation model which is based on evolution dynamics of real world networks and generate networks with community structures that can be used as benchmark graphs to study community detection algorithms.
Edit intent: clarity
Original:  In this paper, we propose to study both these application areas together i. e.\ introduce a network generation model  based on evolution dynamics of real world networks and  generate networks with community structures that can be used as benchmark graphs to study community detection algorithms.
Revised:  In this paper, we propose to study both these application areas together i. e.\ introduce a network generation model based on evolution dynamics of real world networks and , it can generate networks with community structures that can be used as benchmark graphs to study community detection algorithms.
Edit intent: coherence
Original:  Results suggest that the networks generated using the proposed model present tougher challenges for community detection algorithms due to the topological structure introduced by evolution dynamics .
Revised:  Results suggest that the proposed model can generate networks which are not only structurally similar to real world networks but can be used to generate networks with varying community sizes and topologies .
Edit intent: clarity
Original:  With a computer search this allows to show  that the asymptotic density of runs in binary words is less than 183/193\approx 0.9482.
Revised:  This leads to an algorithm that allows to show that the asymptotic density of runs in binary words is less than 183/193\approx 0.9482.
Edit intent: clarity
Original:  collection of piecewise smooth surface patches and curve constraints , (ii)
Revised:  collection of piecewise smooth surface patches and curve segments , (ii)
Edit intent: clarity
Original:  a `protection ' strategy for domains containing 1-dimensional features that meet at sharply acute angles, and (iii) a new class of ` off-centre ' refinement rules designed to achieve high-quality point-placement along embedded 1-dimensional constraints . Experimental comparisons show that the new method outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Revised:  a protection strategy for domains containing 1-dimensional features that meet at sharply acute angles, and (iii) a new class of ` off-centre ' refinement rules designed to achieve high-quality point-placement along embedded 1-dimensional constraints . Experimental comparisons show that the new method outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Edit intent: fluency
Original:  a `protection ' strategy for domains containing 1-dimensional features that meet at sharply acute angles, and (iii) a new class of ` off-centre ' refinement rules designed to achieve high-quality point-placement along embedded 1-dimensional constraints . Experimental comparisons show that the new method outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Revised:  a `protection ' strategy for domains containing curve segments that subtend sharply acute angles, and (iii) a new class of ` off-centre ' refinement rules designed to achieve high-quality point-placement along embedded 1-dimensional constraints . Experimental comparisons show that the new method outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Edit intent: clarity
Original:  a `protection ' strategy for domains containing 1-dimensional features that meet at sharply acute angles, and (iii) a new class of ` off-centre ' refinement rules designed to achieve high-quality point-placement along embedded 1-dimensional constraints . Experimental comparisons show that the new method outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Revised:  a `protection ' strategy for domains containing 1-dimensional features that meet at sharply acute angles, and (iii) a new class of off-centre ' refinement rules designed to achieve high-quality point-placement along embedded 1-dimensional constraints . Experimental comparisons show that the new method outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Edit intent: fluency
Original:  a `protection ' strategy for domains containing 1-dimensional features that meet at sharply acute angles, and (iii) a new class of ` off-centre ' refinement rules designed to achieve high-quality point-placement along embedded 1-dimensional constraints . Experimental comparisons show that the new method outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Revised:  a `protection ' strategy for domains containing 1-dimensional features that meet at sharply acute angles, and (iii) a new class of ` off-centre refinement rules designed to achieve high-quality point-placement along embedded 1-dimensional constraints . Experimental comparisons show that the new method outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Edit intent: fluency
Original:  a `protection ' strategy for domains containing 1-dimensional features that meet at sharply acute angles, and (iii) a new class of ` off-centre ' refinement rules designed to achieve high-quality point-placement along embedded 1-dimensional constraints . Experimental comparisons show that the new method outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Revised:  a `protection ' strategy for domains containing 1-dimensional features that meet at sharply acute angles, and (iii) a new class of ` off-centre ' refinement rules designed to achieve high-quality point-placement along embedded curve features . Experimental comparisons show that the new method outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Edit intent: clarity
Original:  a `protection ' strategy for domains containing 1-dimensional features that meet at sharply acute angles, and (iii) a new class of ` off-centre ' refinement rules designed to achieve high-quality point-placement along embedded 1-dimensional constraints . Experimental comparisons show that the new method outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Revised:  a `protection ' strategy for domains containing 1-dimensional features that meet at sharply acute angles, and (iii) a new class of ` off-centre ' refinement rules designed to achieve high-quality point-placement along embedded 1-dimensional constraints . Experimental comparisons show that the new Frontal-Delaunay algorithm outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.
Edit intent: clarity
Original:  Empirical evidence on quantum state tomography and sparse phase retrieval applications support our speedup findings .
Revised:  Empirical evidence on quantum state tomography and sparse phase retrieval applications .
Edit intent: clarity
Original: We propose a simple and scalable non-convex method for low-rank PSD matrix problems with a generic (strongly ) convex objectivef, and additional matrix norm constraints .
Revised: We propose a simple and scalable non-convex method for low-rank matrix problems with a generic (strongly ) convex objectivef, and additional matrix norm constraints .
Edit intent: clarity
Original:  nevertheless, we show that our method favors local linear convergence rate in the non-convex factored space, for a class of convex norm-constrained problems .
Revised:  nevertheless, we show that our method favors local linear convergence in the factored space .
Edit intent: clarity
Original:  When the number of new servers is large, we propose an efficient 3-approximate algorithm that .
Revised:  When the number of new servers is large, we propose an efficient 3-approximate algorithm .
Edit intent: clarity
Original:  Our recipes for the approximation of the remaining roots within the same asymptotic computational cost should answer the authors' challenge.
Revised:  The challenge of fast approximation of the remaining roots within the same asymptotic computational cost should answer the authors' challenge.
Edit intent: clarity
Original:  Our work can be of independent interest as an example of synergistic variation and combination of polynomial root-finding methods towards enhancing their power  .
Revised:  Our work can be of independent interest as an example of synergy of the combination of various methods of polynomial root-finding methods towards enhancing their power .
Edit intent: clarity
Original:  Our work can be of independent interest as an example of synergistic variation and combination of polynomial root-finding methods towards enhancing their power  .
Revised:  Our work can be of independent interest as an example of synergistic variation and combination of polynomial root-finding towards enhancing their power .
Edit intent: clarity
Original: We were initially motivated by the paper  by Schleicher and Stoll of 2017 about the initialization of Newton's iterations  . Given a black box subroutine for the evaluation of the Newton's ratio of a polynomial and its derivative, their algorithm very fast approximates all roots of a univariate polynomial except for a small fraction of them.
Revised: We were initially motivated by the paper by Schleicher and Stoll about initialization of Newton's iterations . Given a black box subroutine for the evaluation of the Newton's ratio of a polynomial and its derivative, their algorithm very fast approximates all roots of a univariate polynomial except for a small fraction of them.
Edit intent: coherence
Original: We were initially motivated by the paper  by Schleicher and Stoll of 2017 about the initialization of Newton's iterations  . Given a black box subroutine for the evaluation of the Newton's ratio of a polynomial and its derivative, their algorithm very fast approximates all roots of a univariate polynomial except for a small fraction of them.
Revised: We were initially motivated by the paper by Schleicher and Stoll of 2017 about the initialization of Newton's iterations for univariate polynomial root-finding . Given a black box subroutine for the evaluation of the Newton's ratio of a polynomial and its derivative, their algorithm very fast approximates all roots of a univariate polynomial except for a small fraction of them.
Edit intent: clarity
Original:  The challenge of fast approximation of the remaining roots motivated our present work, but our recipes for this task should have independent and much broader interest for implicit deflation in polynomial root-finding .
Revised:  The challenge of fast approximation of the remaining roots motivated our present work, but our recipes for this task should have much broader independent interest for implicit deflation in polynomial root-finding .
Edit intent: clarity
Original:  The challenge of fast approximation of the remaining roots motivated our present work, but our recipes for this task should have independent and much broader interest for implicit deflation in polynomial root-finding .
Revised:  The challenge of fast approximation of the remaining roots motivated our present work, but our recipes for this task should have independent and much broader interest for implicit deflation in root-finding .
Edit intent: clarity
Original:  They can be also an example of synergy of the combination of various methods of polynomial root-finding towards enhancing their power, in particular towards faster convergence of functional iterations in a larger domain.
Revised:  They can be also an example of synergy of the combination of various methods for root-finding towards enhancing their power, in particular towards faster convergence of functional iterations in a larger domain.
Edit intent: clarity
Original: We were initially motivated by the paper SS17%DIFDELCMD < ] %%% by Schleicher and Stoll about initialization of Newton's iterations for univariate polynomial root-finding.
Revised:  SS17%DIFDELCMD < ] %%% by Schleicher and Stoll about initialization of Newton's iterations for univariate polynomial root-finding.
Edit intent: coherence
Original: We were initially motivated by the paper SS17%DIFDELCMD < ] %%% by Schleicher and Stoll about initialization of Newton's iterations for univariate polynomial root-finding.
Revised: We were initially motivated by the paper SS17%DIFDELCMD < ] %%% Functional iterations such as Newton's iterations for univariate polynomial root-finding.
Edit intent: clarity
Original:  At first we dramatically increase their local efficiency by means of the incorporation of fast algorithms for multipoint polynomial evaluation and Fast Multipole Method.
Edit intent: coherence
Original:  Then we assume that some roots have been approximated and try to approximate the remaining roots efficiently, which is a realistic assumption in various function iterations and in root-finding by means of subdivision.
Revised:  Then we assume that some roots have been approximated and try to approximate the remaining roots efficiently, which is a realistic assumption in various functional iterations and also in root-finding by means of subdivision.
Edit intent: clarity
Original:  Natural approach by using explicit deflation has been well developed, but we study the alternative techniques of implicit deflation and mapping the variable  .
Revised:  Natural approach by using explicit deflation has been well developed, but we study the alternative techniques of implicit deflation combined with mapping the variable .
Edit intent: clarity
Original:  Then we assume that some roots have been approximated and try to approximate the remaining roots efficiently, which is a realistic assumption in various functional iterations and also in root-finding by means of subdivision.
Revised:  Then we assume that some roots have been approximated and try to approximate the remaining roots . Such situation is also realistic for root-finding by means of subdivision.
Edit intent: clarity
Original:  Natural approach by using explicit deflation has been well developed, but we study the alternative techniques of implicit deflation combined with mapping the variable  .
Revised:  A natural approach of applying explicit deflation has been well developed, but we study the alternative techniques of implicit deflation combined with mapping the variable .
Edit intent: clarity
Original:  Starting from a simple abstract state machine, we capture the essence of such language aspects and synchronization primitives as release/acquire atomics, relaxed and non-atomic memory accesses and undefined racy behaviors through a series of gradual refinements of the abstract program state .
Revised:  Starting from a simple abstract state machine, through a series of gradual refinements of the abstract state, we capture such language aspects and synchronization primitives as release/acquire atomics, relaxed and non-atomic memory accesses and undefined racy behaviors through a series of gradual refinements of the abstract program state .
Edit intent: clarity
Original:  Unusual among mosquito borne diseases, ZIKV has been shown to also be sexually transmitted . Critical to the assessment of outbreak risk, estimation of the potential attack rates, and assessment of control measures, are estimates of the basic reproduction number, R0 , and estimates of the relative role of sexual transmission in outbreak dynamics .
Revised:  Unusual among mosquito-borne diseases, ZIKV has been shown to also be sexually transmitted . Critical to the assessment of outbreak risk, estimation of the potential attack rates, and assessment of control measures, are estimates of the basic reproduction number, R0 , and estimates of the relative role of sexual transmission in outbreak dynamics .
Edit intent: fluency
Original:  Unusual among mosquito borne diseases, ZIKV has been shown to also be sexually transmitted . Critical to the assessment of outbreak risk, estimation of the potential attack rates, and assessment of control measures, are estimates of the basic reproduction number, R0 , and estimates of the relative role of sexual transmission in outbreak dynamics .
Revised:  Unusual among mosquito borne diseases, ZIKV has been shown to also be sexually transmitted . Critical to the assessment of outbreak risk, estimation of the potential attack rates, and assessment of control measures, are estimates of the basic reproduction number, R0 .
Edit intent: clarity
Original:  Methods: We estimate R0 , and relative role of sexual transmission, of the 2015 ZIKV outbreak in Barranquilla, Colombia, through an analysis of the exponential rise in identified syndromic ZIKV cases (n= 1,470 to the end of December , 2015).
Revised:  Methods: We estimated the R0 , and relative role of sexual transmission, of the 2015 ZIKV outbreak in Barranquilla, Colombia, through an analysis of the exponential rise in identified syndromic ZIKV cases (n= 1,470 to the end of December , 2015).
Edit intent: fluency