wanyu/IteraTeR_v2 huggingface dataset:
Original: For a nonnegative matrix A and real diagonal matrix D, two known inequalities on the spectral radius, r(A^2 D^2) >= r(AD)^2 and r(A) , r(A D^2) >= r(AD)^2, leave open the question of what determines the order of r(A^2 D^2) with respect to r(A) , r(A D^2).
Revised: For a nonnegative matrix A and real diagonal matrix D, two known inequalities on the spectral radius, r(A^2 D^2) >= r(AD)^2 and r(A) , r(A D^2) >= r(AD)^2, leave open the question of what determines the order of r(A^2 D^2) with respect to r(A) r(A D^2).
Edit intent: fluency
Original:  For a diagonally symmetrizable nonnegative matrix A with all positive eigenvalues and nonnegative D, r(A^2 D) <= r(A) , r(AD).
Revised:  For a diagonally symmetrizable nonnegative matrix A with all positive eigenvalues and nonnegative D, r(A^2 D) <= r(A) r(AD).
Edit intent: clarity
Original:  This is a particular case of the more general result that r(A [(1-m) , r(B) , I + m B] D) is monotonic in m when all non-Perron eigenvalues of A have the same sign --- decreasing for positive signs and increasing for negative signs, for symmetrizable nonnegative A and B that commute.
Revised:  This is a particular case of the more general result that r(A [(1-m) r(B) , I + m B] D) is monotonic in m when all non-Perron eigenvalues of A have the same sign --- decreasing for positive signs and increasing for negative signs, for symmetrizable nonnegative A and B that commute.
Edit intent: fluency
Original:  This is a particular case of the more general result that r(A [(1-m) , r(B) , I + m B] D) is monotonic in m when all non-Perron eigenvalues of A have the same sign --- decreasing for positive signs and increasing for negative signs, for symmetrizable nonnegative A and B that commute.
Revised:  This is a particular case of the more general result that r(A [(1-m) , r(B) I + m B] D) is monotonic in m when all non-Perron eigenvalues of A have the same sign --- decreasing for positive signs and increasing for negative signs, for symmetrizable nonnegative A and B that commute.
Edit intent: fluency
Original: Given an infinite word x over an alphabet A , a letter b \in A , and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure S_x of x, if it exists, an extremal word of x . In this paper we consider the extremal words of morphic words.
Revised: Given an infinite word X over an alphabet A , a letter b \in A , and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure S_x of x, if it exists, an extremal word of x . In this paper we consider the extremal words of morphic words.
Edit intent: fluency
Original: Given an infinite word x over an alphabet A , a letter b \in A , and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure S_x of x, if it exists, an extremal word of x . In this paper we consider the extremal words of morphic words.
Revised: Given an infinite word x over an alphabet A a letter b \in A , and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure S_x of x, if it exists, an extremal word of x . In this paper we consider the extremal words of morphic words.
Edit intent: fluency
Original: Given an infinite word x over an alphabet A , a letter b \in A , and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure S_x of x, if it exists, an extremal word of x . In this paper we consider the extremal words of morphic words.
Revised: Given an infinite word x over an alphabet A , a letter b occurring in X , and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure S_x of x, if it exists, an extremal word of x . In this paper we consider the extremal words of morphic words.
Edit intent: clarity
Original: Given an infinite word x over an alphabet A , a letter b \in A , and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure S_x of x, if it exists, an extremal word of x . In this paper we consider the extremal words of morphic words.
Revised: Given an infinite word x over an alphabet A , a letter b \in A , and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure of X an extremal word of x . In this paper we consider the extremal words of morphic words.
Edit intent: clarity
Original: Given an infinite word x over an alphabet A , a letter b \in A , and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure S_x of x, if it exists, an extremal word of x . In this paper we consider the extremal words of morphic words.
Revised: Given an infinite word x over an alphabet A , a letter b \in A , and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure S_x of x, if it exists, an extremal word of X . In this paper we consider the extremal words of morphic words.
Edit intent: fluency
Original:  If x = g(f^{\omega}(a)) for some morphisms f and g, we give a simple condition on f and g that guarantees that all extremal words are morphic.
Revised:  If X = g(f^{\omega}(a)) for some morphisms f and g, we give a simple condition on f and g that guarantees that all extremal words are morphic.
Edit intent: fluency
Original:  Our technique also yields easy characterizations of extremal words of the Period-doubling and Chacon words and a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin-Shapiro word.
Revised:  Our technique also yields easy characterizations of extremal words of the Period-doubling word and the Chacon word and give a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin-Shapiro word.
Edit intent: clarity
Original:  The following questions are addressed : the role (on the genus of the representing automata) of the size of the alphabet, the topological cost of the powerset determinization procedure and the minimization procedure.
Revised:  The following questions are addressed . First, depending on the size of the alphabet, the topological cost of the powerset determinization procedure and the minimization procedure.
Edit intent: coherence
Original:  For this general model, the main contributions are  (i) ordering results for both coverage probability and per user rate in closed-form for any BS distribution for the three considered techniques, using novel tools from stochastic orders, (ii) upper bounds on the coverage probability assuming a Poisson BS distribution, and (iii) a comparison of the area spectral efficiency (ASE).
Revised:  For this general model, the main contributions are : (i) ordering results for both coverage probability and per user rate in closed-form for any BS distribution for the three considered techniques, using novel tools from stochastic orders, (ii) upper bounds on the coverage probability assuming a Poisson BS distribution, and (iii) a comparison of the area spectral efficiency (ASE).
Edit intent: fluency
Original:  For this general model, the main contributions are  (i) ordering results for both coverage probability and per user rate in closed-form for any BS distribution for the three considered techniques, using novel tools from stochastic orders, (ii) upper bounds on the coverage probability assuming a Poisson BS distribution, and (iii) a comparison of the area spectral efficiency (ASE).
Revised:  For this general model, the main contributions are (i) ordering results for both coverage probability and per user rate in closed form for any BS distribution for the three considered techniques, using novel tools from stochastic orders, (ii) upper bounds on the coverage probability assuming a Poisson BS distribution, and (iii) a comparison of the area spectral efficiency (ASE).
Edit intent: fluency
Original:  Later, a novel improved approximation algorithm with ratio (1+\beta, \max{2, 1+\ln\frac{1{\beta}}} {\beta}\}} ) is developed by constructing interesting auxiliary graphs and employing the cycle cancelation method.
Revised:  Later, a novel improved approximation algorithm with ratio (1+\beta, \max{2, 1+\ln\frac{1{\beta}}} \,\max\{2,\,1+\ln\frac{1{\beta}\}} ) is developed by constructing interesting auxiliary graphs and employing the cycle cancelation method.
Edit intent: coherence
Original:  Later, a novel improved approximation algorithm with ratio (1+\beta, \max{2, 1+\ln\frac{1{\beta}}} {\beta}\}} ) is developed by constructing interesting auxiliary graphs and employing the cycle cancelation method.
Revised:  Later, a novel improved approximation algorithm with ratio (1+\beta, \max{2, 1+\ln\frac{1{\beta}}} {\beta}\}} ) is developed by constructing interesting auxiliary graphs and employing the cycle cancellation method.
Edit intent: fluency
Original:  Besides, when \beta=0, by slightly modifying our algorithm, an approximation algorithm with ratio (1, (1+\epsilon)(\ln n+\ln\frac{1{\epsilon}}  )), i.e. an algorithm with only a single factor ratio O(\ln n) on cost, can be immediately obtained by setting the delay of each edge e to%DIFDELCMD < \lfloor %%% \frac{d(e){\frac{\epsilon D}{n}}}%DIFDELCMD < \rfloor %DIFDELCMD < %%% for a given fixed \epsilon>0.
Revised:  Besides, by setting \beta=0, by slightly modifying our algorithm, an approximation algorithm with ratio (1, (1+\epsilon)(\ln n+\ln\frac{1{\epsilon}} )), i.e. an algorithm with only a single factor ratio O(\ln n) on cost, can be immediately obtained by setting the delay of each edge e to%DIFDELCMD < \lfloor %%% \frac{d(e){\frac{\epsilon D}{n}}}%DIFDELCMD < \rfloor %DIFDELCMD < %%% for a given fixed \epsilon>0.
Edit intent: coherence
Original:  Besides, when \beta=0, by slightly modifying our algorithm, an approximation algorithm with ratio (1, (1+\epsilon)(\ln n+\ln\frac{1{\epsilon}}  )), i.e. an algorithm with only a single factor ratio O(\ln n) on cost, can be immediately obtained by setting the delay of each edge e to%DIFDELCMD < \lfloor %%% \frac{d(e){\frac{\epsilon D}{n}}}%DIFDELCMD < \rfloor %DIFDELCMD < %%% for a given fixed \epsilon>0.
Revised:  Besides, when \beta=0, an approximation algorithm with ratio (1, (1+\epsilon)(\ln n+\ln\frac{1{\epsilon}} )), i.e. an algorithm with only a single factor ratio O(\ln n) on cost, can be immediately obtained by setting the delay of each edge e to%DIFDELCMD < \lfloor %%% \frac{d(e){\frac{\epsilon D}{n}}}%DIFDELCMD < \rfloor %DIFDELCMD < %%% for a given fixed \epsilon>0.
Edit intent: clarity
Original:  Besides, when \beta=0, by slightly modifying our algorithm, an approximation algorithm with ratio (1, (1+\epsilon)(\ln n+\ln\frac{1{\epsilon}}  )), i.e. an algorithm with only a single factor ratio O(\ln n) on cost, can be immediately obtained by setting the delay of each edge e to%DIFDELCMD < \lfloor %%% \frac{d(e){\frac{\epsilon D}{n}}}%DIFDELCMD < \rfloor %DIFDELCMD < %%% for a given fixed \epsilon>0.
Revised:  Besides, when \beta=0, by slightly modifying our algorithm, an approximation algorithm with ratio (1, {\epsilon}} )), i.e. an algorithm with only a single factor ratio O(\ln n) on cost, can be immediately obtained by setting the delay of each edge e to%DIFDELCMD < \lfloor %%% \frac{d(e){\frac{\epsilon D}{n}}}%DIFDELCMD < \rfloor %DIFDELCMD < %%% for a given fixed \epsilon>0.
Edit intent: coherence
Original:  To the best of our knowledge, this is the first non-trivial approximation algorithm for the kBCP problem which strictly obeys the delay constraint .
Revised:  To the best of our knowledge, this is the first non-trivial approximation algorithm for the kBCP problem that strictly obeys the delay constraint .
Edit intent: fluency
Original:  Our developed algorithms can be directly used to solve some related problems, in particular, the k-disjoint restricted shortest path problem (kRSP) 10%DIFDELCMD < ]%%% , resulting in the same ratio (1+\beta, \max{2, 1+\ln\frac{1{\beta}}), which improves currently the best result of ratio (2, 2) in }%DIFDELCMD < [%%% 6 .
Revised:  Our developed algorithms can be directly used to solve some related problems, in particular, the k-disjoint restricted shortest path problem (kRSP) %DIFDELCMD < ]%%% , resulting in the same ratio (1+\beta, \max{2, 1+\ln\frac{1{\beta}}), which improves currently the best result of ratio (2, 2) in }%DIFDELCMD < [%%% 6 .
Edit intent: clarity
Original:  Our developed algorithms can be directly used to solve some related problems, in particular, the k-disjoint restricted shortest path problem (kRSP) 10%DIFDELCMD < ]%%% , resulting in the same ratio (1+\beta, \max{2, 1+\ln\frac{1{\beta}}), which improves currently the best result of ratio (2, 2) in }%DIFDELCMD < [%%% 6 .
Revised:  Our developed algorithms can be directly used to solve some related problems, in particular, the k-disjoint restricted shortest path problem (kRSP) 10%DIFDELCMD < ]%%% {\beta}}), which improves currently the best result of ratio (2, 2) in }%DIFDELCMD < [%%% 6 .
Edit intent: coherence
Original: Previous ILU-based preconditioning methods  have their own parameters to improve their performances.
Revised: The ILU-based preconditioning methods have their own parameters to improve their performances.
Edit intent: clarity
Original:   This paper proposes a novel ILU-based preconditioner named auto-accelerated ILU, or A2ILU.
Revised:  This paper proposes a novel ILU-based preconditioner called the auto-accelerated ILU, or A2ILU.
Edit intent: fluency
Original:  In order to improve the convergence, A2ILU introduces acceleration parameters which modifies the ILU factorized preconditioning matrix.
Revised:  In order to improve the convergence, A2ILU introduces acceleration parameters which modify the ILU factorized preconditioning matrix.
Edit intent: fluency
Original:  A2ILU needs no more operation than the original ILU because the acceleration parameters are optimized automatically by A2ILU itself.
Revised:  A2ILU needs no more operations than the original ILU because the acceleration parameters are optimized automatically by A2ILU itself.
Edit intent: fluency
Original:  Numerical tests reveal the superior performance of A2ILU  and robustness of parameter-sensitivities .
Revised:  Numerical tests reveal the performance of A2ILU and robustness of parameter-sensitivities .
Edit intent: clarity
Original:  Numerical tests reveal the superior performance of A2ILU  and robustness of parameter-sensitivities .
Revised:  Numerical tests reveal the superior performance of A2ILU and robustness of parameter sensitivities .
Edit intent: fluency
Original:  Our experimental results show that MT19937 has low-weight F_2-linear relations in dimensions higher than 623, and  that some output vectors with specific lags are rejected or have small p-values in the birthday spacings tests.
Revised:  Our experimental results show that MT19937 has low-weight F_2-linear relations in dimensions higher than 623, and show that some output vectors with specific lags are rejected or have small p-values in the birthday spacings tests.
Edit intent: fluency
Original:  An advantage of such generators is that we can assess them fast by using theoretical criteria, such as the dimension of equidistribution with v-bit accuracy.
Revised:  An advantage of such generators is that we can assess them quickly by using theoretical criteria, such as the dimension of equidistribution with v-bit accuracy.
Edit intent: coherence
Original: Integration of data from genomewide single nucleotide polymorphism (SNP) association studies of different traits should allow researchers to disentangle the genetics of potentially related traits within individually associated regions.
Revised: Integration of data from genome-wide single nucleotide polymorphism (SNP) association studies of different traits should allow researchers to disentangle the genetics of potentially related traits within individually associated regions.
Edit intent: fluency
Original:  Methods have ranged from visual comparison of association p values for each trait to formal statistical colocalisation testing of individual regions, which requires selection of a set of SNPs summarizing the association in a region.
Revised:  Formal statistical colocalisation testing of individual regions, which requires selection of a set of SNPs summarizing the association in a region.
Edit intent: clarity
Original:  We show that the SNP selection method greatly affects type 1 error rates, with all published studies to date having used SNP selection methods that result in substantially biased inference.
Revised:  We show that the SNP selection method greatly affects type 1 error rates, with published studies having used methods expected to result in substantially biased inference.
Edit intent: clarity
Original:  We show that unbiased inference can be made either by avoiding variable selection and instead testing the most informative principal components or by integrating over variable selection using Bayesian model averaging  .
Revised:  We show that either avoiding variable selection and instead testing the most informative principal components or by integrating over variable selection using Bayesian model averaging .
Edit intent: clarity
Original:  We show that unbiased inference can be made either by avoiding variable selection and instead testing the most informative principal components or by integrating over variable selection using Bayesian model averaging  .
Revised:  We show that unbiased inference can be made either by avoiding variable selection and instead testing the most informative principal components or integrating over variable selection using Bayesian model averaging .
Edit intent: fluency
Original:  Application to data from Graves' disease and Hashimoto's thyroiditis reveals a common genetic signature across seven regions shared between the diseases, and indicates that for five out of six regions which have been significantly associated with one disease and not the other, the lack of evidence in one disease represents genuine absence of association  rather than lack of power  .
Revised:  Application to data from Graves' disease and Hashimoto's thyroiditis reveals a common genetic signature across seven regions shared between the diseases, and indicates that in five of six regions which have been significantly associated with one disease and not the other, the lack of evidence in one disease represents genuine absence of association rather than lack of power .
Edit intent: fluency
Original:  Application to data from Graves' disease and Hashimoto's thyroiditis reveals a common genetic signature across seven regions shared between the diseases, and indicates that for five out of six regions which have been significantly associated with one disease and not the other, the lack of evidence in one disease represents genuine absence of association  rather than lack of power  .
Revised:  Application to data from Graves' disease and Hashimoto's thyroiditis reveals a common genetic signature across seven regions shared between the diseases, and indicates that for five out of six regions associated with Graves' disease and not the other, the lack of evidence in one disease represents genuine absence of association rather than lack of power .
Edit intent: clarity
Original:  Application to data from Graves' disease and Hashimoto's thyroiditis reveals a common genetic signature across seven regions shared between the diseases, and indicates that for five out of six regions which have been significantly associated with one disease and not the other, the lack of evidence in one disease represents genuine absence of association  rather than lack of power  .
Revised:  Application to data from Graves' disease and Hashimoto's thyroiditis reveals a common genetic signature across seven regions shared between the diseases, and indicates that for five out of six regions which have been significantly associated with one disease and not Hashimoto's thyroiditis, this more likely reflects genuine absence of association rather than lack of power .
Edit intent: clarity
Original:  Application to data from Graves' disease and Hashimoto's thyroiditis reveals a common genetic signature across seven regions shared between the diseases, and indicates that for five out of six regions which have been significantly associated with one disease and not the other, the lack of evidence in one disease represents genuine absence of association  rather than lack of power  .
Revised:  Application to data from Graves' disease and Hashimoto's thyroiditis reveals a common genetic signature across seven regions shared between the diseases, and indicates that for five out of six regions which have been significantly associated with one disease and not the other, the lack of evidence in one disease represents genuine absence of association with the latter rather than lack of power .
Edit intent: clarity
Original:  But we also obtain new knowledge about its behaviour: a better under- standing of the size of its wild-type attractor basin (larger than previously thought), the identification of novel minimal conditions and critical nodes that control wild-type behaviour, and the resilience of these to stochastic interventions.
Revised:  But we also obtain new knowledge about its behaviour: a better understanding of the size of its wild-type attractor basin (larger than previously thought), the identification of novel minimal conditions and critical nodes that control wild-type behaviour, and the resilience of these to stochastic interventions.
Edit intent: clarity
Original:  These simultaneous coded-multicasting opportunities are enabled by careful content overlap at the various caches in the network, created by a central coordinating server.
Revised:  These coded-multicasting opportunities are enabled by careful content overlap at the various caches in the network, created by a central coordinating server.
Edit intent: clarity
Original:  Despite this lack of coordination, the proposed scheme is nevertheless able to create simultaneous coded-multicasting opportunities , and hence achieves a rate close to the  centralized scheme.
Revised:  Despite this lack of coordination, the proposed scheme is nevertheless able to create coded-multicasting opportunities , and hence achieves a rate close to the centralized scheme.
Edit intent: clarity
Original:  Despite this lack of coordination, the proposed scheme is nevertheless able to create simultaneous coded-multicasting opportunities , and hence achieves a rate close to the  centralized scheme.
Revised:  Despite this lack of coordination, the proposed scheme is nevertheless able to create simultaneous coded-multicasting opportunities and achieves a rate close to the centralized scheme.
Edit intent: coherence
Original:  Such methods, however, are less efficient in regards to bandwidth usage than dedicated access mecanisms and less reliable in overloaded network conditions.
Edit intent: coherence
Original:  We investigate the potential to improve the performance of short flows in regards to transmission delay, over random access methods for DVB-RCS2 that is currently under development.
Revised:  We investigate the poten- tial to improve the performance of short flows in regards to transmission delay, over random access methods for DVB-RCS2 that is currently under development.
Edit intent: fluency
Original:  We investigate the potential to improve the performance of short flows in regards to transmission delay, over random access methods for DVB-RCS2 that is currently under development.
Revised:  We investigate the potential to improve the performance of short flows in regards to transmission delay, over random access methods for DVB- RCS2 that is currently under development.
Edit intent: fluency