wanyu/IteraTeR_v2 huggingface dataset:
Original:  Based only on the knowledge of average loss rates on the direct and overhearing links, intermediate nodes can make decisions for both intra-session ( %DIFDELCMD < \ie %%%  how much redundancy to add in each flow) and inter-session ( %DIFDELCMD < \ie %%%  what percentage of flows to code together) coding.
Revised:  Based only on the knowledge of the loss rates on the direct and overhearing links, intermediate nodes can make decisions for both intra-session ( %DIFDELCMD < \ie %%% how much redundancy to add in each flow) and inter-session ( %DIFDELCMD < \ie %%% what percentage of flows to code together) coding.
Edit intent: clarity
Original:  Based only on the knowledge of average loss rates on the direct and overhearing links, intermediate nodes can make decisions for both intra-session ( %DIFDELCMD < \ie %%%  how much redundancy to add in each flow) and inter-session ( %DIFDELCMD < \ie %%%  what percentage of flows to code together) coding.
Revised:  Based only on the knowledge of average loss rates on the direct and overhearing links, intermediate nodes can make decisions for both intra-session ( %DIFDELCMD < \ie %%% how much redundancy to add in each flow) and inter-session ( %DIFDELCMD < \ie %%% i.e., what percentage of flows to code together) coding.
Edit intent: clarity
Original:  We propose two practical schemes, I^2NC-state and I^2NC-stateless, that mimic the structure of the NUM optimal solution and implement the properties mentioned above .
Revised:  We propose two practical schemes, I2NC-state and I2NC-stateless, which mimic the structure of the NUM optimal solution and implement the properties mentioned above .
Edit intent: fluency
Original:  We propose two practical schemes, I^2NC-state and I^2NC-stateless, that mimic the structure of the NUM optimal solution and implement the properties mentioned above .
Revised:  We propose two practical schemes, I^2NC-state and I^2NC-stateless, that mimic the structure of the NUM optimal solution .
Edit intent: coherence
Original:  We demonstrate the benefits of our schemes (in terms of error resilience, and thereby throughput) compared to COPE, through simulation in%DIFDELCMD < {\tt %%% GloMoSim .
Revised:  We demonstrate the benefits of our schemes %DIFDELCMD < {\tt %%% GloMoSim .
Edit intent: clarity
Original:  We demonstrate the benefits of our schemes (in terms of error resilience, and thereby throughput) compared to COPE, through simulation in%DIFDELCMD < {\tt %%% GloMoSim .
Revised:  We demonstrate the benefits of our schemes (in terms of error resilience, and thereby throughput) compared to COPE, through simulation in%DIFDELCMD < {\tt %%% through simulations .
Edit intent: clarity
Original:  In particular we develop a combinatorial models for the categories of modular continuous time dynamical systems on manifolds .
Revised:  In particular , we construct combinatorial models for the categories of modular continuous time dynamical systems on manifolds .
Edit intent: clarity
Original:  In particular we develop a combinatorial models for the categories of modular continuous time dynamical systems on manifolds .
Revised:  In particular we develop a combinatorial models for categories of modular continuous time dynamical systems on manifolds .
Edit intent: fluency
Original: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{}  \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Revised: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: \mathrm{S\_k} (n)=\min _{1 \le t \le n \{} \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Edit intent: clarity
Original: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{}  \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Revised: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min \{} \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Edit intent: fluency
Original: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{}  \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Revised: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{} \_ \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Edit intent: fluency
Original: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{}  \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Revised: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{} 1\leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Edit intent: fluency
Original: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{}  \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Revised: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{} \leqslant t\leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Edit intent: clarity
Original: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{}  \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Revised: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{} \leqslant \leqslant n} \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Edit intent: fluency
Original: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{}  \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Revised: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{} \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{k-1 (t) \} \right\ (n) = 2^n -
Edit intent: fluency
Original: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{}  \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n -
Revised: For the multi-peg Tower of Hanoi problem with k %DIFDELCMD < \ge %%% \geqslant 4 pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: S_k\_k} (n)=\min _{1 \le t \le n \{} \leqslant \leqslant } \left\{ 2 \cdot S_k\_k} (n-t) + S_{k-1\_}{ (t) \} \right\ (n) = 2^n --
Edit intent: fluency
Original:  = \min _{1 \le t \le n \{p_k \cdot G_k}  \leqslant \leqslant }\left\{ \_k} (n-t)
Revised:  = \min _{1 \le t \le n \{p_k \cdot G_k} \leqslant \leqslant n}\left\{ \_k} (n-t)
Edit intent: clarity
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q \_3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant \left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: fluency
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers and (q_i)_{i \ge 3} } \left(\right) \geqslant \left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: fluency
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) i\geqslant \left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: clarity
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant 3 \left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: fluency
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant and\left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: coherence
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant \left(\right) \geqslant 3 and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: fluency
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant \left(\right) \geqslant and we show that the sequence of differences \left(\mathrm{G\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: fluency
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant \left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right)\_ \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: fluency
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant \left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) n\geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: clarity
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant \left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant 1 consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: fluency
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant \left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left(\prod\_ }q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: fluency
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant \left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( i=3}q\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: fluency
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant \left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }^{kq\_i}\right) \left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: fluency
Original:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant  \left(\right) \geqslant  and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant  consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \left(  ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Revised:  + q _3 , for two sequences of arbitrary positive integers (p_i)_{i \ge 3 and (q_i)_{i \ge 3} } \left(\right) \geqslant \left(\right) \geqslant and we show that the sequence of differences \left(\_k(n)- \mathrm{G}\_k} ( G_k(n)-G_k( n-1) )_{n \ge 1 \right) \geqslant consists of numbers of the form (\prod_{i=3^{k}q_i) \cdot (\prod_{i=3}} \left( }q\_i}\right) \cdot\left( ^{k}{p _i }^{\alpha _i } ) \right) , with \alpha _i%DIFDELCMD < \ge %%% \geqslant 0 for all i, lined in the increasing order.
Edit intent: fluency