wanyu/IteraTeR_full_sent huggingface dataset:
Original:  These two results stand in sharp contrast to computing the Tutte polynomial of a k-vertex graph (that is, the Tutte polynomial of a graphic {\em  matroid of dimension k---which is representable in dimension k over the binary field so that every vector has exactly two nonzero coordinates), which is known to be computable in 2^k k^{O(1)} time [Bj\"orklund {\em et al. , FOCS 2008].
Revised:  This is in sharp contrast to computing the Tutte polynomial of a k-vertex graph (that is, the Tutte polynomial of a graphic {\em matroid of dimension k---which is representable in dimension k over the binary field so that every vector has exactly two nonzero coordinates), which is known to be computable in 2^k k^{O(1)} time [Bj\"orklund {\em et al. , FOCS 2008].
Edit intent: clarity
Original:  These two results stand in sharp contrast to computing the Tutte polynomial of a k-vertex graph (that is, the Tutte polynomial of a graphic {\em  matroid of dimension k---which is representable in dimension k over the binary field so that every vector has exactly two nonzero coordinates), which is known to be computable in 2^k k^{O(1)} time [Bj\"orklund {\em et al. , FOCS 2008].
Revised:  These two results stand in sharp contrast to computing the Tutte polynomial of a k-vertex graph (that is, the Tutte polynomial of a {\em matroid of dimension k---which is representable in dimension k over the binary field so that every vector has exactly two nonzero coordinates), which is known to be computable in 2^k k^{O(1)} time [Bj\"orklund {\em et al. , FOCS 2008].
Edit intent: clarity
Original:  These two results stand in sharp contrast to computing the Tutte polynomial of a k-vertex graph (that is, the Tutte polynomial of a graphic {\em  matroid of dimension k---which is representable in dimension k over the binary field so that every vector has exactly two nonzero coordinates), which is known to be computable in 2^k k^{O(1)} time [Bj\"orklund {\em et al. , FOCS 2008].
Revised:  These two results stand in sharp contrast to computing the Tutte polynomial of a k-vertex graph (that is, the Tutte polynomial of a graphic {\em graphic matroid of dimension k---which is representable in dimension k over the binary field so that every vector has exactly two nonzero coordinates), which is known to be computable in 2^k k^{O(1)} time [Bj\"orklund {\em et al. , FOCS 2008].
Edit intent: meaning-changed
Original:  These two results stand in sharp contrast to computing the Tutte polynomial of a k-vertex graph (that is, the Tutte polynomial of a graphic {\em  matroid of dimension k---which is representable in dimension k over the binary field so that every vector has exactly two nonzero coordinates), which is known to be computable in 2^k k^{O(1)} time [Bj\"orklund {\em et al. , FOCS 2008].
Revised:  These two results stand in sharp contrast to computing the Tutte polynomial of a k-vertex graph (that is, the Tutte polynomial of a graphic {\em matroid of dimension k---which is representable in dimension k over the binary field so that every vector has two nonzero coordinates), which is known to be computable in 2^k k^{O(1)} time [Bj\"orklund {\em et al. , FOCS 2008].
Edit intent: clarity
Original:  Our lower-bound proofs proceed in three steps: (i) a classic connection due to Crapo and Rota [1970] between the number of tuples of codewords of full support and the Tutte polynomial of the matroid associated with the code;
Revised:  Our lower-bound proofs proceed via (i) a classic connection due to Crapo and Rota [1970] between the number of tuples of codewords of full support and the Tutte polynomial of the matroid associated with the code;
Edit intent: clarity
Original:  Our lower-bound proofs proceed in three steps: (i) a classic connection due to Crapo and Rota [1970] between the number of tuples of codewords of full support and the Tutte polynomial of the matroid associated with the code;
Revised:  Our lower-bound proofs proceed in three steps: (i) a connection due to Crapo and Rota [1970] between the number of tuples of codewords of full support and the Tutte polynomial of the matroid associated with the code;
Edit intent: clarity
Original:  We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension  k on k^{O(1)} points in k^{O(k)} arithmetic operationsin the base field{\em   .
Revised:  We complement these lower bounds with two algorithm designs. The first design computes the Tutte polynomial of a linear matroid of dimension k on k^{O(1)} points in k^{O(k)} arithmetic operationsin the base field{\em .
Edit intent: meaning-changed
Original:  We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension  k on k^{O(1)} points in k^{O(k)} arithmetic operationsin the base field{\em   .
Revised:  We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension ~ k on k^{O(1)} points in k^{O(k)} arithmetic operationsin the base field{\em .
Edit intent: meaning-changed
Original:  We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension  k on k^{O(1)} points in k^{O(k)} arithmetic operationsin the base field{\em   .
Revised:  We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension k on k^{O(1)} points in k^{O(k)} operations. The second design generalizes the Bj\"orklund~{\em .
Edit intent: meaning-changed
Original:  We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension  k on k^{O(1)} points in k^{O(k)} arithmetic operationsin the base field{\em   .
Revised:  We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension k on k^{O(1)} points in k^{O(k)} arithmetic operationsin the base field{\em et al. .
Edit intent: meaning-changed
Original:  We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension  k on k^{O(1)} points in k^{O(k)} arithmetic operationsin the base field{\em   .
Revised:  We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension k on k^{O(1)} points in k^{O(k)} arithmetic operationsin the base field{\em algorithm and runs in q^{k+1 .
Edit intent: meaning-changed
Original: In this article we propose and calculate, under the Black Scholes option pricing model, a measure of the relative value of a delta- Symmetric Strangle . The proposed measure accounts for the price of the strangle, relative to the (present value of the ) spread between the two strikes, all expressed, after a natural re-parameterization, in terms of delta and a volatility parameter.
Revised: In this article we propose and calculate, under the Black Scholes option pricing model, a measure of the relative value of a delta-Symmetric Strangle and compute it under the standard Black-Scholes option pricing model. This new measure accounts for the price of the strangle, relative to the (present value of the ) spread between the two strikes, all expressed, after a natural re-parameterization, in terms of delta and a volatility parameter.
Edit intent: meaning-changed
Original:  The proposed measure accounts for the price of the strangle, relative to the (present value of the ) spread between the two strikes, all expressed, after a natural re-parameterization, in terms of delta and a volatility parameter.
Revised:  The proposed measure accounts for the price of the strangle, relative to the Present Value of the spread between the two strikes, all expressed, after a natural re-parameterization, in terms of delta and a volatility parameter.
Edit intent: fluency
Original:  We show the startling main result that under the standard BS option pricing model, this measure of relative value is a function of delta only and is independent of the time to expiry, the price of the underlying security or the prevailing volatility used in the pricing model.
Revised:  We show that under the standard BS option pricing model, this measure of relative value is a function of delta only and is independent of the time to expiry, the price of the underlying security or the prevailing volatility used in the pricing model.
Edit intent: coherence
Original:  We show the startling main result that under the standard BS option pricing model, this measure of relative value is a function of delta only and is independent of the time to expiry, the price of the underlying security or the prevailing volatility used in the pricing model.
Revised:  We show the startling main result that under the standard BS option pricing model, this measure of relative value is bounded by a simple function of delta only and is independent of the time to expiry, the price of the underlying security or the prevailing volatility used in the pricing model.
Edit intent: clarity
Original:  {\it   In fact, the simple and intuitively appealing expression for this measure  allows us to study the strangle's exit strategy and the corresponding optimal choices for values {\it   of delta.
Revised:  We demonstrate how this bound can be used as a quick{\it In fact, the simple and intuitively appealing expression for this measure allows us to study the strangle's exit strategy and the corresponding optimal choices for values {\it of delta.
Edit intent: meaning-changed
Original:  {\it   In fact, the simple and intuitively appealing expression for this measure  allows us to study the strangle's exit strategy and the corresponding optimal choices for values {\it   of delta.
Revised:  {\it to assess, regardless the market volatility, the duration of the contract or the price of the underlying security, the market (relative) value of the \delta-strangle in comparison to its BS (relative) price. In fact, the simple and intuitively appealing expression for this measure allows us to study the strangle's exit strategy and the corresponding optimal choices for values {\it of delta.
Edit intent: meaning-changed
Original:  {\it   In fact, the simple and intuitively appealing expression for this measure  allows us to study the strangle's exit strategy and the corresponding optimal choices for values {\it   of delta.
Revised:  {\it In fact, the simple and intuitively appealing expression for this measure and bound allows us to study the strangle's exit strategy and the corresponding optimal choices for values {\it of delta.
Edit intent: meaning-changed
Original:  {\it   In fact, the simple and intuitively appealing expression for this measure  allows us to study the strangle's exit strategy and the corresponding optimal choices for values {\it   of delta.
Revised:  {\it In fact, the simple and intuitively appealing expression for this measure allows us to also study in detail the strangle's exit strategy and the corresponding optimal choices for values {\it of delta.
Edit intent: meaning-changed
Original:  {\it   In fact, the simple and intuitively appealing expression for this measure  allows us to study the strangle's exit strategy and the corresponding optimal choices for values {\it   of delta.
Revised:  {\it In fact, the simple and intuitively appealing expression for this measure allows us to study the strangle's exit strategy and the corresponding optimal choices for values {\it choice for a value of delta.
Edit intent: meaning-changed
Original: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality .
Revised:  Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality .
Edit intent: clarity
Original: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality .
Revised: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality .
Edit intent: fluency
Original: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality .
Revised: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality .
Edit intent: clarity
Original: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality .
Revised: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G . They related \lambda_\infty to the {\em vertex expansion} of G via a Cheeger-type inequality .
Edit intent: coherence
Original: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality .
Revised: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of the graph via a Cheeger-type inequality .
Edit intent: fluency
Original: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality . This is analogous to the Cheeger-type inequality relating the spectral gap  , \lambda_2 (G), of the graph to its {\em edge expansion}.
Revised: Twenty years ago, Bobkov, Houdr\'e, and the last author introduced a Poincar\'e-type functional graph parameter, \lambda_\infty (G) , of a graph G , and related it to the {\em vertex expansion} of G via a Cheeger-type inequality , analogous to the Cheeger-type inequality relating the spectral gap , \lambda_2 (G), of the graph to its {\em edge expansion}.
Edit intent: coherence
Original:  Following the work of the second author with Raghavendra and Vempala, wherein the complexity of \lambda_\infty was related to the so-called Small-Set Expansion (SSE) problem, it has been believed that computing \lambda_\infty is a hard problem.
Revised:  Following the work of the second author with Raghavendra and Vempala, wherein the complexity of \lambda_\infty was related to the so-called small-set expansion (SSE) problem, it has been believed that computing \lambda_\infty is a hard problem.
Edit intent: fluency
Original:  We settle this question by proving that computing \lambda_\infty is indeed NP-hard . Additionally, we use our techniques to prove NP-hardness of computing the spread constant (of a graph) \emph{  , a geometric measure  introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs.
Revised:  We settle this question by proving that computing \lambda_\infty is indeed NP-hard , even for weighted trees. Our gadget further proves NP-hardness of computing the spread constant (of a graph) \emph{ , a geometric measure introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs.
Edit intent: meaning-changed
Original:  Additionally, we use our techniques to prove NP-hardness of computing the spread constant (of a graph) \emph{  , a geometric measure  introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs.
Revised:  Additionally, we use our techniques to prove NP-hardness of computing \emph{ , a geometric measure introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs.
Edit intent: clarity
Original:  Additionally, we use our techniques to prove NP-hardness of computing the spread constant (of a graph) \emph{  , a geometric measure  introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs.
Revised:  Additionally, we use our techniques to prove NP-hardness of computing the spread constant (of a graph) \emph{spread constant , a geometric measure introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs.
Edit intent: meaning-changed
Original:  Additionally, we use our techniques to prove NP-hardness of computing the spread constant (of a graph) \emph{  , a geometric measure  introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs.
Revised:  Additionally, we use our techniques to prove NP-hardness of computing the spread constant (of a graph) \emph{ of a weighted tree; i.e. , a geometric measure introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs.
Edit intent: meaning-changed
Original:  Additionally, we use our techniques to prove NP-hardness of computing the spread constant (of a graph) \emph{  , a geometric measure  introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs.
Revised:  Additionally, we use our techniques to prove NP-hardness of computing the spread constant (of a graph) \emph{ , a geometric measure of the graph, introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs.
Edit intent: clarity
Original:  Additionally, we use our techniques to prove NP-hardness of computing the spread constant (of a graph) \emph{  , a geometric measure  introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality on cartesian products of graphs.
Revised:  Additionally, we use our techniques to prove NP-hardness of computing the spread constant (of a graph) \emph{ , a geometric measure introduced by Alon, Boppana, and Spencer, in the context of deriving an asymptotic isoperimetric inequality of Cartesian products of graphs.
Edit intent: fluency
Original:  We complement our hardness results by providing approximation schemes for computing \lambda_\infty and the spread constant of star graphs, and investigate constant approximability for weighted trees. Finally{\em   , we provide improved approximation results for the maximum variance embedding problem for general graphs, by replacing the optimal orthogonal projection (PCA) with a randomized projection approach .
Revised:  We conclude this case by providing a fully polynomial time approximation scheme. We further study a generalization of spread constant in machine learning literature, namely the{\em , we provide improved approximation results for the maximum variance embedding problem for general graphs, by replacing the optimal orthogonal projection (PCA) with a randomized projection approach .
Edit intent: meaning-changed
Original:  Finally{\em   , we provide improved approximation results for the maximum variance embedding problem for general graphs, by replacing the optimal orthogonal projection (PCA) with a randomized projection approach .
Revised:  Finally{\em , we provide fast combinatorial algorithms that avoid solving a semidefinite relaxation of the problem. On the other hand, for general graphs, by replacing the optimal orthogonal projection (PCA) with a randomized projection approach .
Edit intent: meaning-changed
Original:  Finally{\em   , we provide improved approximation results for the maximum variance embedding problem for general graphs, by replacing the optimal orthogonal projection (PCA) with a randomized projection approach .
Revised:  Finally{\em , we provide improved approximation results for the maximum variance embedding problem for general graphs, we propose a randomized projection method that can outperform the optimal orthogonal projection (PCA) with a randomized projection approach .
Edit intent: meaning-changed
Original:  Finally{\em   , we provide improved approximation results for the maximum variance embedding problem for general graphs, by replacing the optimal orthogonal projection (PCA) with a randomized projection approach .
Revised:  Finally{\em , we provide improved approximation results for the maximum variance embedding problem for general graphs, by replacing the optimal orthogonal projection , i.e., PCA, classically used for rounding of the optimum lifted solution (to SDP relaxation) of the problem .
Edit intent: meaning-changed
Original:  People can use the framework on their smartphones for the virus detection purpose. Nowadays , smartphones are powerful with existing computation-rich processors, memory space, and large number of sensors including cameras, microphone, temperature sensor, inertial sensors, proximity, colour-sensor, humidity-sensor, and wireless chipsets/sensors.
Revised:  People can use the framework on their smartphones for the virus detection purposes. Nowadays Smartphones are powerful with existing computation-rich processors, memory space, and large number of sensors including cameras, microphone, temperature sensor, inertial sensors, proximity, colour-sensor, humidity-sensor, and wireless chipsets/sensors.
Edit intent: fluency
Original:  We identify that a successful approach gives an arrested phase regime following the Ostwald growth, where in the course of time  one phase transforms into another metastable phase with a similar free energy as observed in oxygen interstitials diffusion in quantum complex matter and in crystallization of proteins.
Revised:  We identify that a successful approach gives an arrested phase regime following the Ostwald growth, where , over the course of time one phase transforms into another metastable phase with a similar free energy as observed in oxygen interstitials diffusion in quantum complex matter and in crystallization of proteins.
Edit intent: clarity
Original:  We identify that a successful approach gives an arrested phase regime following the Ostwald growth, where in the course of time  one phase transforms into another metastable phase with a similar free energy as observed in oxygen interstitials diffusion in quantum complex matter and in crystallization of proteins.
Revised:  We identify that a successful approach gives an arrested phase regime following the Ostwald growth, where in the course of time , one phase transforms into another metastable phase with a similar free energy as observed in oxygen interstitials diffusion in quantum complex matter and in crystallization of proteins.
Edit intent: fluency