wanyu/IteraTeR_full_sent huggingface dataset:
Original:  This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), \big(\big) the standard deviation of the number of tests per individual is of the same order as the cost .
Revised:  This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), and estimate that the cost of the best scheme of this type for p, determined by the choice of k=k_3(p), is of order O\big(\big) the standard deviation of the number of tests per individual is of the same order as the cost .
Edit intent: meaning-changed
Original:  This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), \big(\big) the standard deviation of the number of tests per individual is of the same order as the cost .
Revised:  This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), \big(p\log(1/p)\big) the standard deviation of the number of tests per individual is of the same order as the cost .
Edit intent: meaning-changed
Original:  This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), \big(\big) the standard deviation of the number of tests per individual is of the same order as the cost .
Revised:  This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), \big(\big). This is the standard deviation of the number of tests per individual is of the same order as the cost .
Edit intent: coherence
Original:  This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), \big(\big) the standard deviation of the number of tests per individual is of the same order as the cost .
Revised:  This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), \big(\big) the same order as the cost .
Edit intent: clarity
Original:  This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), \big(\big) the standard deviation of the number of tests per individual is of the same order as the cost .
Revised:  This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), \big(\big) the standard deviation of the number of tests per individual is of the same order as that of the cost of the optimal scheme, and the difference of these costs is explicitly bounded .
Edit intent: meaning-changed
Original:  As an example, when p=0.02 , the optimal strategy is k=3, m=(27,9,3) . The cost of this strategy is 0.20 , that is, the mean number of tests required to screen 100 individuals is 20.
Revised:  As an example, when p=0.02 , the optimal strategy is k=3, m=(27,9,3) , with cost 0.20 , that is, the mean number of tests required to screen 100 individuals is 20.
Edit intent: clarity
Original:  We propose a new algorithm to compute that transform of an image represented by unit (pixel/voxel) basis functions , for various two-dimensional (2D)/three-dimensional (3D) scanning geometries, such as 2D/3D parallel beam, 2D fan beam, 3D circular/helical cone beam, etc.
Revised:  We propose a new algorithm to compute the X-ray transform of an image represented by unit (pixel/voxel) basis functions , for various two-dimensional (2D)/three-dimensional (3D) scanning geometries, such as 2D/3D parallel beam, 2D fan beam, 3D circular/helical cone beam, etc.
Edit intent: meaning-changed
Original:  We propose a new algorithm to compute that transform of an image represented by unit (pixel/voxel) basis functions , for various two-dimensional (2D)/three-dimensional (3D) scanning geometries, such as 2D/3D parallel beam, 2D fan beam, 3D circular/helical cone beam, etc. Since the transform is acting as a line integral, the fundamental task is to calculate this integral of the unit basis functions, which is equivalently the intersection length of the ray with the associated unit.
Revised:  We propose a new algorithm to compute that transform of an image represented by unit (pixel/voxel) basis functions . The fundamental issue is equivalently calculating the intersection lengths of the ray with the associated unit.
Edit intent: clarity
Original:  Since the transform is acting as a line integral, the fundamental task is to calculate this integral of the unit basis functions, which is equivalently the intersection length of the ray with the associated unit. For a given ray, using the support of unit basis function, we derive the sufficient and necessary condition of non-vanishing intersectability , and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units  .
Revised:  Since the transform is acting as a line integral, the fundamental task is to calculate this integral of the unit basis functions, which is equivalently the intersection length of the ray with associated units. For any given ray, using the support of unit basis function, we derive the sufficient and necessary condition of non-vanishing intersectability , and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units .
Edit intent: fluency
Original:  For a given ray, using the support of unit basis function, we derive the sufficient and necessary condition of non-vanishing intersectability , and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units  .
Revised:  For a given ray, we first derive the sufficient and necessary condition of non-vanishing intersectability , and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units .
Edit intent: coherence
Original:  For a given ray, using the support of unit basis function, we derive the sufficient and necessary condition of non-vanishing intersectability , and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units  .
Revised:  For a given ray, using the support of unit basis function, we derive the sufficient and necessary condition for non-vanishing intersectability , and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units .
Edit intent: fluency
Original:  For a given ray, using the support of unit basis function, we derive the sufficient and necessary condition of non-vanishing intersectability , and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units  .
Revised:  For a given ray, using the support of unit basis function, we derive the sufficient and necessary condition of non-vanishing intersectability . By this condition, we then distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units .
Edit intent: clarity
Original:  For a given ray, using the support of unit basis function, we derive the sufficient and necessary condition of non-vanishing intersectability , and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units  .
Revised:  For a given ray, using the support of unit basis function, we derive the sufficient and necessary condition of non-vanishing intersectability , and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the ray. Only for those units .
Edit intent: coherence
Original:  For a given ray, using the support of unit basis function, we derive the sufficient and necessary condition of non-vanishing intersectability , and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units  .
Revised:  For a given ray, using the support of unit basis function, we derive the sufficient and necessary condition of non-vanishing intersectability , and obtain the analytic formula for the intersection length, which can be used to distinguish the units that produce valid intersections with the given ray, and then perform simple calculations only for those units rather than all the individuals, we calculate the intersection lengths by the obtained analytic formula. The proposed algorithm is adapted to 2D/3D parallel beam and 2D fan beam. Particularly, we derive the transformation formulas and generalize the algorithm to 3D circular and helical cone beams .
Edit intent: meaning-changed
Original: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix  C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if  possible.
Revised: For general complex or real 1-parameter matrix flows A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if possible.
Edit intent: fluency
Original: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix  C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if  possible.
Revised: For general complex or real 1-parameter matrix flow A(t)_{n,n} and for time-invariant static matrices A \in\CC this paper considers ways to decompose flows globally via one constant matrix C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if possible.
Edit intent: meaning-changed
Original: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix  C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if  possible.
Revised: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC_{n,n this paper considers ways to decompose flows globally via one constant matrix C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if possible.
Edit intent: fluency
Original: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix  C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if  possible.
Revised: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose matrix flows and single matrices globally via one constant matrix C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if possible.
Edit intent: meaning-changed
Original: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix  C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if  possible.
Revised: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix similarity C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if possible.
Edit intent: meaning-changed
Original: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix  C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if  possible.
Revised: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C or A = C^{-1\cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if possible.
Edit intent: meaning-changed
Original: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix  C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if  possible.
Revised: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal block A _k(t) square and the number of blocks \ell > 1 if possible.
Edit intent: fluency
Original: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix  C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if  possible.
Revised: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) or A_k square and their number \ell > 1 if possible.
Edit intent: clarity
Original: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix  C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if  possible.
Revised: For general complex or real 1-parameter matrix flow A(t)_{n,n} \CC this paper considers ways to decompose flows globally via one constant matrix C_{n,n} as A(t) = C ^{-1} \cdot diag(A_1(t), ..., A_\ell(t)) \cdot C \cdot diag(A_1,...,A_\ell)\cdot C } with each diagonal blockA _k(t) square and the number of blocks \ell > 1 if this is possible.
Edit intent: coherence
Original:  The theory behind our  algorithm is elementary and uses the concept of invariant subspaces for the Matlab {\tt eig} computed 'eigenvectors' of one flow matrix A (t_a) to find the coarsest simultaneous block structure for all flow matrices A (t_b).
Revised:  The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the Matlab {\tt eig} computed 'eigenvectors' of one flow matrix A (t_a) to find the coarsest simultaneous block structure for all flow matrices A (t_b).
Edit intent: clarity
Original:  The theory behind our  algorithm is elementary and uses the concept of invariant subspaces for the Matlab {\tt eig} computed 'eigenvectors' of one flow matrix A (t_a) to find the coarsest simultaneous block structure for all flow matrices A (t_b).
Revised:  The theory behind our algorithm is elementary and uses the concept of invariant subspaces for the Matlab {\tt eig} computed 'eigenvectors' of one associated flow matrix B (t_a) to find the coarsest simultaneous block structure for all flow matrices A (t_b).
Edit intent: meaning-changed
Original:  The theory behind our  algorithm is elementary and uses the concept of invariant subspaces for the Matlab {\tt eig} computed 'eigenvectors' of one flow matrix A (t_a) to find the coarsest simultaneous block structure for all flow matrices A (t_b).
Revised:  The theory behind our algorithm is elementary and uses the concept of invariant subspaces for the Matlab {\tt eig} computed 'eigenvectors' of one flow matrix A (t_a) to find the coarsest simultaneous block structure for all flow matrices B (t_b).
Edit intent: meaning-changed
Original:  The method works very efficiently for all  matrix flows, be they differentiable, continuous or discontinuous in t, and for all  types of square matrix flows  such as hermitean, real symmetric, normal or general complex and real flows A(t)  , with or without Jordan block structures and with or without repeated eigenvalues.
Revised:  The method works very efficiently for all time-varying matrix flows, be they differentiable, continuous or discontinuous in t, and for all types of square matrix flows such as hermitean, real symmetric, normal or general complex and real flows A(t) , with or without Jordan block structures and with or without repeated eigenvalues.
Edit intent: meaning-changed
Original:  The method works very efficiently for all  matrix flows, be they differentiable, continuous or discontinuous in t, and for all  types of square matrix flows  such as hermitean, real symmetric, normal or general complex and real flows A(t)  , with or without Jordan block structures and with or without repeated eigenvalues.
Revised:  The method works very efficiently for all matrix flows, be they differentiable, continuous or discontinuous in t, and for all fixed entry matrices A; as well as for all types of square matrix flows such as hermitean, real symmetric, normal or general complex and real flows A(t) , with or without Jordan block structures and with or without repeated eigenvalues.
Edit intent: meaning-changed
Original:  The method works very efficiently for all  matrix flows, be they differentiable, continuous or discontinuous in t, and for all  types of square matrix flows  such as hermitean, real symmetric, normal or general complex and real flows A(t)  , with or without Jordan block structures and with or without repeated eigenvalues.
Revised:  The method works very efficiently for all matrix flows, be they differentiable, continuous or discontinuous in t, and for all types of square matrix flows or fixed entry matrices such as hermitean, real symmetric, normal or general complex and real flows A(t) , with or without Jordan block structures and with or without repeated eigenvalues.
Edit intent: meaning-changed
Original:  The method works very efficiently for all  matrix flows, be they differentiable, continuous or discontinuous in t, and for all  types of square matrix flows  such as hermitean, real symmetric, normal or general complex and real flows A(t)  , with or without Jordan block structures and with or without repeated eigenvalues.
Revised:  The method works very efficiently for all matrix flows, be they differentiable, continuous or discontinuous in t, and for all types of square matrix flows such as hermitean, real symmetric, normal or general complex and real flows A(t) or static matrices A , with or without Jordan block structures and with or without repeated eigenvalues.
Edit intent: meaning-changed
Original:  Our intended aim is to discover  decomposable flows as they originate in sensor given outputs for time-varying matrix problems and thereby  reduce the complexities of their numerical treatment .
Revised:  Our intended aim is to discover decomposable flows as they originate in sensor given outputs for time-varying matrix problems and thereby reduce the complexities of their numerical treatments through adapting 'divide and conquer' methods for their diagonal sub-blocks. Our method is also applicable to standard fixed entry matrices of all structures and types. In the process we discover and study k-normal fixed entry matrix classes that can be decomposed under unitary similarities into various k-dimensional block-diagonal forms .
Edit intent: meaning-changed
Original:  We introduce a new class of volatility operators which map the square integrable noise into the Filipovi \'{c space of forward curves , and we specify a deterministic parametrized version of it.
Revised:  For this purpose we introduce a new class of volatility operators which map the square integrable noise into the Filipovi \'{c space of forward curves , and we specify a deterministic parametrized version of it.
Edit intent: coherence
Original:  We introduce a new class of volatility operators which map the square integrable noise into the Filipovi \'{c space of forward curves , and we specify a deterministic parametrized version of it.
Revised:  We introduce a new class of state-dependent volatility operators that map the square integrable noise into the Filipovi \'{c space of forward curves , and we specify a deterministic parametrized version of it.
Edit intent: meaning-changed
Original:  We introduce a new class of volatility operators which map the square integrable noise into the Filipovi \'{c space of forward curves , and we specify a deterministic parametrized version of it.
Revised:  We introduce a new class of volatility operators which map the square integrable noise into the Filipovi \'{c space of forward curves . For calibration, we specify a deterministic parametrized version of it.
Edit intent: coherence
Original:  We introduce a new class of volatility operators which map the square integrable noise into the Filipovi \'{c space of forward curves , and we specify a deterministic parametrized version of it. For calibration purposes, we train a neural network to approximate the  option price as a function of the model parameters.
Revised:  We introduce a new class of volatility operators which map the square integrable noise into the Filipovi \'{c space of forward curves , and we specify a deterministic parametrized version of our model and train a neural network to approximate the option price as a function of the model parameters.
Edit intent: coherence