wanyu/IteraTeR_v2 huggingface dataset:
Original:  Owing to the complexity of multilayer models, an implicit assumption in Linsker's and subsequent papers has been that propagation delay is homogeneous and , consequently, plays little functional role in neural behaviour.
Revised:  Owing to the complexity of multilayer models, an implicit assumption in Linsker's and subsequent papers has been that propagation delay is homogeneous and plays little functional role in neural behaviour.
Edit intent: coherence
Original:  In this paper, we relax this assumption to examine the impact of axonal distance dependent propagation delay on neural learning.
Revised:  We relax this assumption to examine the impact of axonal distance dependent propagation delay on neural learning.
Edit intent: coherence
Original:  In this paper, we relax this assumption to examine the impact of axonal distance dependent propagation delay on neural learning.
Revised:  In this paper, we relax this assumption to examine the impact of distance-dependent axonal propagation delay on neural learning.
Edit intent: fluency
Original:  We show that propagation delay induces lowpass filtering by dispersing the arrival times of spikes from presynaptic neurons, providing a natural correlation cancellation mechanism for distal connections.
Revised:  We show that propagation delay induces low-pass filtering by dispersing the arrival times of spikes from presynaptic neurons, providing a natural correlation cancellation mechanism for distal connections.
Edit intent: fluency
Original:  The cutoff frequency decreases as the radial propagation delay within a layer , relative to propagation delay between the layers, increases, introducing an upper limit on temporal resolution.
Revised:  The cut-off frequency decreases as the radial propagation delay within a layer , relative to propagation delay between the layers, increases, introducing an upper limit on temporal resolution.
Edit intent: fluency
Original:  The cutoff frequency decreases as the radial propagation delay within a layer , relative to propagation delay between the layers, increases, introducing an upper limit on temporal resolution.
Revised:  The cutoff frequency decreases as the radial propagation delay within a layer increases relative to propagation delay between the layers, increases, introducing an upper limit on temporal resolution.
Edit intent: fluency
Original:  The cutoff frequency decreases as the radial propagation delay within a layer , relative to propagation delay between the layers, increases, introducing an upper limit on temporal resolution.
Revised:  The cutoff frequency decreases as the radial propagation delay within a layer , relative to propagation delay between the layers, introducing an upper limit on temporal resolution.
Edit intent: clarity
Original:  Given that the postsynaptic potential also acts as a lowpass filter, we show that the effective time constant of each should enable the processing of similar scales of temporal information.
Revised:  Given that the PSP also acts as a lowpass filter, we show that the effective time constant of each should enable the processing of similar scales of temporal information.
Edit intent: clarity
Original:  Given that the postsynaptic potential also acts as a lowpass filter, we show that the effective time constant of each should enable the processing of similar scales of temporal information.
Revised:  Given that the postsynaptic potential also acts as a low-pass filter, we show that the effective time constant of each should enable the processing of similar scales of temporal information.
Edit intent: fluency
Original:  This concords with frequency dependent contrast sensitivity in the visual system, which changes with eccentricity and receptive field size.
Revised:  This concords with frequency-dependent contrast sensitivity in the visual system, which changes with eccentricity and receptive field size.
Edit intent: fluency
Original:  We show that the addition of propagation delay stabilizes the leading eigenfunction to changes in homeostatic parameters , and hence stabilizes the resulting receptive field structure .
Revised:  We show that the addition of propagation delay stabilizes the leading eigenfunction to changes in homeostatic parameters .
Edit intent: clarity
Original:  These findings will provide a glimpse into the proprietary blackbox Knowledge Graph and what is meant by Google's mission to " " organize the world's information and make it universally accessible and useful " ".
Revised:  These findings will provide a glimpse into the proprietary blackbox Knowledge Graph and what is meant by Google's mission to " organize the world's information and make it universally accessible and useful " ".
Edit intent: fluency
Original:  These findings will provide a glimpse into the proprietary blackbox Knowledge Graph and what is meant by Google's mission to " " organize the world's information and make it universally accessible and useful " ".
Revised:  These findings will provide a glimpse into the proprietary blackbox Knowledge Graph and what is meant by Google's mission to " " organize the world's information and make it universally accessible and useful ".
Edit intent: fluency
Original: We analyze total, asymmetric and frequency connectedness between the oil and forex markets using high-frequency  intra-day data over  2007 -- 2015.
Revised: We analyze total, asymmetric and frequency connectedness between oil and forex markets using high-frequency intra-day data over 2007 -- 2015.
Edit intent: fluency
Original: We analyze total, asymmetric and frequency connectedness between the oil and forex markets using high-frequency  intra-day data over  2007 -- 2015.
Revised: We analyze total, asymmetric and frequency connectedness between the oil and forex markets using high-frequency , intra-day data over 2007 -- 2015.
Edit intent: fluency
Original: We analyze total, asymmetric and frequency connectedness between the oil and forex markets using high-frequency  intra-day data over  2007 -- 2015.
Revised: We analyze total, asymmetric and frequency connectedness between the oil and forex markets using high-frequency intra-day data over the period 2007 -- 2015.
Edit intent: clarity
Original:  Hence, the long-term connectedness reflects worrisome beliefs of investors and correlates with the increased market uncertainty .
Revised:  Hence, the long-term connectedness the most and lead to its dramatic increase during periods of distress .
Edit intent: clarity
Original:  Furthermore, we review some applications of marginal likelihoods to phylogenetics, highlighting how they can be used to learn about models of evolution from biological data  .
Revised:  Furthermore, we review some empirical studies that demonstrate how marginal likelihoods can be used to learn about models of evolution from biological data .
Edit intent: clarity
Original:  In this work, we consider the problem of PIR for uncoded storage constrained databases.
Revised:  In this work, we consider the problem of PIR from storage constrained databases.
Edit intent: clarity
Original:  On one extreme, \mu=1 is the replicated databases case previously settled .
Revised:  On one extreme, \mu=1 is the replicated databases case .
Edit intent: clarity
Original:  In particular, for any (N,K), we show that the optimal trade-off between storage, \mu, and the download cost, D(\mu), is given by the lower convex hull of the N pairs \left( \frac{t{N},D(\mu) = } \left(1+ 1{t}+ 1{t^{2}}+ \cdots + 1{t^{K-1}}\right)\right) for t=1,2,\ldots, N  .
Revised:  In particular, for any (N,K), we show that the optimal trade-off between storage, \mu, and the download cost, D(\mu), is given by the lower convex hull of the N pairs \left( \mu= \frac{t{N},D(\mu) = } \left(1+ 1{t}+ 1{t^{2}}+ \cdots + 1{t^{K-1}}\right)\right) for t=1,2,\ldots, N .
Edit intent: clarity
Original:  All recent improvements on the path TSP crucially exploit a structural property shown by An, Kleinberg, and Shmoys [Journal of the ACM, 2015], namely that narrow cuts with respect to a Held-Karp solution form a chain.
Revised:  All recent improvements on path TSP crucially exploit a structural property shown by An, Kleinberg, and Shmoys [Journal of the ACM, 2015], namely that narrow cuts with respect to a Held-Karp solution form a chain.
Edit intent: fluency
Original:  We significantly deviate from these approaches by showing the benefit to deal with larger s-t cuts, even though they are much less structured.
Revised:  We significantly deviate from these approaches by showing the benefit of dealing with larger s-t cuts, even though they are much less structured.
Edit intent: fluency
Original:  This avoids a recursive application of dynamic programming as used by Traub and Vygen, and leads to a considerable simpler algorithm avoiding an additional error term in the approximation guarantee.
Revised:  This avoids a recursive application of dynamic programming as used by Traub and Vygen, and leads to a considerably simpler algorithm avoiding an additional error term in the approximation guarantee.
Edit intent: fluency
Original:  Because we match the still unbeaten 1.5-approximation guarantee of Christofides' algorithm for TSP  , any further progress on the approximability of the path TSP will also lead to an improvement for TSP.
Revised:  We match the still unbeaten 1.5-approximation guarantee of Christofides' algorithm for TSP , any further progress on the approximability of the path TSP will also lead to an improvement for TSP.
Edit intent: coherence
Original:  Because we match the still unbeaten 1.5-approximation guarantee of Christofides' algorithm for TSP  , any further progress on the approximability of the path TSP will also lead to an improvement for TSP.
Revised:  Because we match the still unbeaten 1.5-approximation guarantee of Christofides' algorithm for TSP . Hence , any further progress on the approximability of the path TSP will also lead to an improvement for TSP.
Edit intent: coherence
Original:  Because we match the still unbeaten 1.5-approximation guarantee of Christofides' algorithm for TSP  , any further progress on the approximability of the path TSP will also lead to an improvement for TSP.
Revised:  Because we match the still unbeaten 1.5-approximation guarantee of Christofides' algorithm for TSP , any further progress on the approximability of path TSP will also lead to an improvement for TSP.
Edit intent: fluency
Original: We present a 1.5-approximation for the Metric Path Traveling Salesman Problem ( path TSP).
Revised: We present a 1.5-approximation for the Metric Path Traveling Salesman Problem ( Path TSP).
Edit intent: fluency
Original:  All recent improvements on path TSP crucially exploit a structural property shown by An, Kleinberg, and Shmoys [Journal of the ACM, 2015], namely that narrow cuts with respect to a Held-Karp solution form a chain.
Revised:  All recent improvements on Path TSP crucially exploit a structural property shown by An, Kleinberg, and Shmoys [Journal of the ACM, 2015], namely that narrow cuts with respect to a Held-Karp solution form a chain.
Edit intent: fluency
Original:  Hence, any further progress on the approximability of path TSP will also lead to an improvement for TSP.
Revised:  Hence, any further progress on the approximability of Path TSP will also lead to an improvement for TSP.
Edit intent: fluency
Original:  Key results about the polyhedral structure include that the number of vertices of this polytope is a Fibonacci number, the facets of the polytope can be described using the combinatorial `` cluster" structure of the underlying rooted tree, and the volume is equal to an Euler zig-zag number.
Revised:  Key results about the polyhedral structure include that the number of vertices of this polytope is a Fibonacci number, the facets of the polytope can be described using the combinatorial " cluster" structure of the underlying rooted tree, and the volume is equal to an Euler zig-zag number.
Edit intent: fluency
Original: We propose an Hybridized Discontinuous Galerkin method on polygonal tessellation, with a stabilization term penalizing locally in each element K  a residual term involving the fluxes  in the norm of the dual of H^1(K).
Revised: We propose an Hybridized Discontinuous Galerkin method on polygonal tessellations, stabilized by penalizing, locally in each element K a residual term involving the fluxes in the norm of the dual of H^1(K).
Edit intent: clarity
Original: We propose an Hybridized Discontinuous Galerkin method on polygonal tessellation, with a stabilization term penalizing locally in each element K  a residual term involving the fluxes  in the norm of the dual of H^1(K).
Revised: We propose an Hybridized Discontinuous Galerkin method on polygonal tessellation, with a stabilization term penalizing locally in each element K , a residual term involving the fluxes in the norm of the dual of H^1(K).
Edit intent: fluency
Original: We propose an Hybridized Discontinuous Galerkin method on polygonal tessellation, with a stabilization term penalizing locally in each element K  a residual term involving the fluxes  in the norm of the dual of H^1(K).
Revised: We propose an Hybridized Discontinuous Galerkin method on polygonal tessellation, with a stabilization term penalizing locally in each element K a residual term involving the fluxes , measured in the norm of the dual of H^1(K).
Edit intent: clarity
Original: We propose an Hybridized Discontinuous Galerkin method  on polygonal tessellations  , stabilized by penalizing, locally in each element K, a residual term involving the fluxes, measured in the norm of the dual of H^1(K).
Revised: We propose a Discontinuous Galerkin method on polygonal tessellations , stabilized by penalizing, locally in each element K, a residual term involving the fluxes, measured in the norm of the dual of H^1(K).
Edit intent: clarity
Original:  The scalar product corresponding to such a norm is numerically realized via the introduction of a (minimal) auxiliary space of VEM type . Stability and optimal error estimates in the broken H^1 norm are proven under a weak shape regularity assumption allowing the presence of very small edges.
Revised:  The scalar product corresponding to such a norm is numerically realized via the introduction of a (minimal) auxiliary space inspired by the Virtual Element Method . Stability and optimal error estimates in the broken H^1 norm are proven under a weak shape regularity assumption allowing the presence of very small edges.
Edit intent: clarity
Original:  That is, it is capable of automatically generating, merging and tuning the hyperplane based fuzzy rule in the single pass manner  .
Revised:  That is, it is capable of automatically generating, merging and tuning the hyperplane-based fuzzy rule in the single pass manner .
Edit intent: fluency
Original:  for example, for h(z) = z a(z)b(z) + c \in \mathbb{C}^{r \times r} from linearizations for \mathbf{a}(z) and \mathbf{b}(z).
Revised:  for example, for h(z) = z a(z)b(z) + \mathbf{c \in \mathbb{C}^{r \times r} from linearizations for \mathbf{a}(z) and \mathbf{b}(z).
Edit intent: fluency
Original: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%%  X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Revised: Standard triples\mathbf{X C_{1} - C_{0}, Y of regular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%% X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Edit intent: fluency
Original: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%%  X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Revised: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ n \times r } have the property%DIFDELCMD < \newline %%% X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Edit intent: clarity
Original: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%%  X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Revised: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times n } have the property%DIFDELCMD < \newline %%% X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Edit intent: clarity
Original: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%%  X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Revised: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } %DIFDELCMD < \newline %%% X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Edit intent: clarity
Original: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%%  X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Revised: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%% in order to use the representation X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Edit intent: clarity
Original: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%%  X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Revised: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%% X(z C_{1} - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Edit intent: fluency
Original: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%%  X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Revised: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%% X(z C_{1} ~ - \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Edit intent: fluency
Original: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%%  X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Revised: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%% X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Edit intent: fluency
Original: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%%  X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = ~ \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Revised: Standard triples\mathbf{X C_{1} - C_{0}, Y of nonsingular matrix polynomials P(z) \in C^{ r \times r } have the property%DIFDELCMD < \newline %%% X(z C_{1} ~ - ~ \mathbf{C}_{0})^{-1}\mathbf{Y} ~ = \mathbf{P}^{-1}(z) for z \notin \Lambda(\mathbf{P}(z)).
Edit intent: fluency
Original:  They can be used in constructing algebraic linearizations;
Revised:  This representation can be used in constructing algebraic linearizations;
Edit intent: clarity
Original:  for example, for \mathbf{h (z) = z \mathbf{a (z) \mathbf{b (z) + \mathbf{c \in \mathbb{C}}  \in \mathbb{C}} ^{ r \times r } from linearizations for \mathbf{a (z) and \mathbf{b (z).
Revised:  for example, for \mathbf{h (z) = z \mathbf{a (z) \mathbf{b (z) + \mathbf{c \in \mathbb{C}} \mathbf{C \in \mathbb{C}} ^{ r \times r } from linearizations for \mathbf{a (z) and \mathbf{b (z).
Edit intent: fluency
Original:  for example, for \mathbf{h (z) = z \mathbf{a (z) \mathbf{b (z) + \mathbf{c \in \mathbb{C}}  \in \mathbb{C}} ^{ r \times r } from linearizations for \mathbf{a (z) and \mathbf{b (z).
Revised:  for example, for \mathbf{h (z) = z \mathbf{a (z) \mathbf{b (z) + \mathbf{c \in \mathbb{C}} \in \mathbb{C}} ^{ r \times r } from linearizations for \mathbf{A (z) and \mathbf{b (z).
Edit intent: fluency
Original:  We give  proofs for the less familiar bases.
Revised:  We give explicit proofs for the less familiar bases.
Edit intent: clarity
Original: We define "generalized standard triples" X, zC_{1} - C_{0}, Y of regular matrix polynomials P(z) \in C ^{n \times n in order to use the representation X(z C_{1}  -  C_{0})^{-1}Y  =  \mathbf{P}^{-1}(z) for z%DIFDELCMD < \notin %%% \Lambda(\mathbf{P .
Revised: We define generalized standard triples X, zC_{1} - C_{0}, Y of regular matrix polynomials P(z) \in C ^{n \times n in order to use the representation X(z C_{1} - C_{0})^{-1}Y = \mathbf{P}^{-1}(z) for z%DIFDELCMD < \notin %%% \Lambda(\mathbf{P .
Edit intent: fluency
Original: We define "generalized standard triples" X, zC_{1} - C_{0}, Y of regular matrix polynomials P(z) \in C ^{n \times n in order to use the representation X(z C_{1}  -  C_{0})^{-1}Y  =  \mathbf{P}^{-1}(z) for z%DIFDELCMD < \notin %%% \Lambda(\mathbf{P .
Revised: We define "generalized standard triples" X, zC_{1} - C_{0}, Y of regular matrix polynomials P(z) \in C ^{n \times n in order to use the representation X(z C_{1} - C_{0})^{-1}Y = \mathbf{P}^{-1}(z) for z%DIFDELCMD < \notin %%% \Lambda(\mathbf{P .
Edit intent: fluency
Original: We define "generalized standard triples" X, zC_{1} - C_{0}, Y of regular matrix polynomials P(z) \in C ^{n \times n in order to use the representation X(z C_{1}  -  C_{0})^{-1}Y  =  \mathbf{P}^{-1}(z) for z%DIFDELCMD < \notin %%% \Lambda(\mathbf{P .
Revised: We define "generalized standard triples" X, zC_{1} - C_{0}, Y of regular matrix polynomials P(z) \in C ^{n \times n in order to use the representation X(z C_{1} - C_{0})^{-1}Y = ~ \mathbf{P}^{-1}(z) for z%DIFDELCMD < \notin %%% \Lambda(\mathbf{P .
Edit intent: fluency
Original:  for example, for\emph{  \mathbf{H}(z) = z \mathbf{A}(z)\mathbf{B}(z) + \mathbf{C} \in \mathbb{C} ^{n \times n from linearizations for \mathbf{A}(z) and \mathbf{B}(z).
Revised:  for example, for\emph{ such as \mathbf{H}(z) = z \mathbf{A}(z)\mathbf{B}(z) + \mathbf{C} \in \mathbb{C} ^{n \times n from linearizations for \mathbf{A}(z) and \mathbf{B}(z).
Edit intent: clarity
Original:  for example, for\emph{  \mathbf{H}(z) = z \mathbf{A}(z)\mathbf{B}(z) + \mathbf{C} \in \mathbb{C} ^{n \times n from linearizations for \mathbf{A}(z) and \mathbf{B}(z).
Revised:  for example, for\emph{ \mathbf{H}(z) = z \mathbf{A}(z)\mathbf{B}(z) + \mathbf{C} \in \mathbb{C} ^{n \times n from linearizations for \mathbf{A}(z) and \mathbf{B}(z).
Edit intent: fluency
Original:  For convenience  we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases;
Revised:  For convenience , we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases;
Edit intent: fluency
Original: We define generalized standard triples X,  zC_{1} - C_{0}, \mathbf{Y P(z) \in C^{n \times n}[z]  in order to use the representation X(z C_{1}~-~C_{0})^{-1}Y~=~P^{-1}(z) except if z is an eigenvalue  .
Revised: We define generalized standard triples X, \mathbf{Y zC_{1} - C_{0}, \mathbf{Y P(z) \in C^{n \times n}[z] in order to use the representation X(z C_{1}~-~C_{0})^{-1}Y~=~P^{-1}(z) except if z is an eigenvalue .
Edit intent: fluency