wanyu/IteraTeR_v2 huggingface dataset:
Original:  We investigate the e?ects of the foraging parameters (e.g. attack rate and handling time) on the mean time to extinction.
Revised:  We investigate the effects of the foraging parameters (e.g. attack rate and handling time) on the mean time to extinction.
Edit intent: fluency
Original:  Mean time to extinction varies by orders of magnitude when we alter the foraging parameters, even when we exclude the e?ects of these parameters on the equilibrium population size.
Revised:  Mean time to extinction varies by orders of magnitude when we alter the foraging parameters, even when we exclude the effects of these parameters on the equilibrium population size.
Edit intent: fluency
Original:  These ?ndings clearly show that accounting for the nature of interspeci?c interactions is likely to be critically important when estimating extinction risk.
Revised:  These findings clearly show that accounting for the nature of interspeci?c interactions is likely to be critically important when estimating extinction risk.
Edit intent: fluency
Original:  These ?ndings clearly show that accounting for the nature of interspeci?c interactions is likely to be critically important when estimating extinction risk.
Revised:  These ?ndings clearly show that accounting for the nature of interspecific interactions is likely to be critically important when estimating extinction risk.
Edit intent: fluency
Original:  With the help of a decomposition of the trees into subtrees and an inclusion-exclusion principle, we find a closed-form expression for the probabilities of the topologies of the ancestral trees.
Revised:  With the help of a decomposition of the trees into subtrees , we calculate the probabilities of the topologies of the ancestral trees.
Edit intent: clarity
Original:  But the culturally transmitted trait interacts with the sex ratio to determine the the ecological consequences of successful invasion.
Revised:  But the culturally transmitted trait interacts with the sex ratio to determine the ecological consequences of successful invasion.
Edit intent: fluency
Original: An important problem that commonly arises in areas such as internet traffic flow analysis, phylogenetics and electrical circuit design, is to find a representation of any given metric D on a finite set X by an edge-weighted graph, such that the total edge length of the graph is minimum over all such graphs.
Revised: An important problem that commonly arises in areas such as internet traffic-flow analysis, phylogenetics and electrical circuit design, is to find a representation of any given metric D on a finite set X by an edge-weighted graph, such that the total edge length of the graph is minimum over all such graphs.
Edit intent: fluency
Original: An important problem that commonly arises in areas such as internet traffic flow analysis, phylogenetics and electrical circuit design, is to find a representation of any given metric D on a finite set X by an edge-weighted graph, such that the total edge length of the graph is minimum over all such graphs.
Revised: An important problem that commonly arises in areas such as internet traffic flow analysis, phylogenetics and electrical circuit design, is to find a representation of any given metric D on a finite set by an edge-weighted graph, such that the total edge length of the graph is minimum over all such graphs.
Edit intent: clarity
Original:  Recently S. Varone presented a heuristic greedy algorithm for computing optimal realizations.
Revised:  Recently Varone presented a heuristic greedy algorithm for computing optimal realizations.
Edit intent: clarity
Original: A continuous-time Markov chain can be constructed as either a homogeneous or inhomogeneous process .
Revised: Continuous-time Markov chains are a standard tool in phylogenetic inference .
Edit intent: clarity
Original:  If homogeneity is assumed, the resulting chain is formulated by specifying time-independent rates of substitutions between states in the chain.
Revised:  If homogeneity is assumed, the chain is formulated by specifying time-independent rates of substitutions between states in the chain.
Edit intent: clarity
Original:  If a "model " is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with time-dependent rates satisfying the same constraints.
Revised:  If a model is formulated in this way, it is possible to generalise it and allow for an inhomogeneous process, with time-dependent rates satisfying the same constraints.
Edit intent: fluency
Original:  It is then useful to require that there exists a homogeneous "average " of this inhomogeneous process within the same model.
Revised:  It is then useful to require that there exists a homogeneous average of this inhomogeneous process within the same model.
Edit intent: fluency
Original: Given a collection of n opaque unit disks in the plane, we want to stack them on top of each other in an order so as to maximize the total length of all pieces of their boundaries visible from above.
Revised: Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter---the total length of all pieces of their boundaries visible from above.
Edit intent: clarity
Original:  We prove that if the centers of the disks form a dense point set, i.e., the ratio of the maximum distance to the minimum distance between them is O(n^1/2), then there is a stacking order for which the visible perimeter is Omega(n^2/3).
Revised:  We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is O(n^1/2), then there is a stacking order for which the visible perimeter is Omega(n^2/3).
Edit intent: clarity
Original:  We also show that the order of magnitude of this bound cannot be improved in the case of the n^1/2 by n^1/2 piece of a sufficiently small square grid.
Revised:  We also show that this bound cannot be improved in the case of the n^1/2 by n^1/2 piece of a sufficiently small square grid.
Edit intent: clarity
Original:  We also show that the order of magnitude of this bound cannot be improved in the case of the n^1/2 by n^1/2 piece of a sufficiently small square grid.
Revised:  We also show that the order of magnitude of this bound cannot be improved in the case of a sufficiently small n^1/2 by n^1/2 piece of a sufficiently small square grid.
Edit intent: fluency
Original:  We also show that the order of magnitude of this bound cannot be improved in the case of the n^1/2 by n^1/2 piece of a sufficiently small square grid.
Revised:  We also show that the order of magnitude of this bound cannot be improved in the case of the n^1/2 by n^1/2 uniform grid.
Edit intent: clarity
Original:  The above results partially answer some questions raised by Cabello, Haverkort, van Kreveld, and Speckmann.
Revised:  The above results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.
Edit intent: clarity
Original:  We prove new lower bounds that imply the rank is at least 3n^2-4n^{3/2 + 3n  .
Revised:  I prove that the rank is at least 3n^2-4n^{3/2 + 3n .
Edit intent: clarity
Original:  The previous lower bound, due to Blaser, was (5 / 2)n^2-3n .
Revised:  The previous lower bound, due to Blaser, was 5n^2 / 2)n^2-3n .
Edit intent: clarity
Original:  We give an alge- braic characterization of LMO, showing that the syntactic monoids of the languages in LMO are exactly the  pseudovariety of J-trivial literally idempotent monoids, where J is the Green's relation determined by two-sided ideals.
Revised:  We give an algebraic characterization of LMO, showing that the syntactic monoids of the languages in LMO are exactly the pseudovariety of J-trivial literally idempotent monoids, where J is the Green's relation determined by two-sided ideals.
Edit intent: fluency
Original:  We give an alge- braic characterization of LMO, showing that the syntactic monoids of the languages in LMO are exactly the  pseudovariety of J-trivial literally idempotent monoids, where J is the Green's relation determined by two-sided ideals.
Revised:  We give an alge- braic characterization of LMO, showing that the syntactic monoids of the languages in LMO are exactly the literal pseudovariety of J-trivial literally idempotent monoids, where J is the Green's relation determined by two-sided ideals.
Edit intent: fluency
Original:  We also prove that LMO coincides with the class of literally idempotent piecewise testable regular languages.
Revised:  We also prove that LMO coincides with the literal variety of literally idempotent piecewise testable regular languages.
Edit intent: clarity
Original: Deformation of single stranded DNA  before reaching the pore in translocation process is investigated.
Revised: Deformation of single stranded DNA in translocation process before reaching the pore in translocation process is investigated.
Edit intent: fluency
Original: Deformation of single stranded DNA  before reaching the pore in translocation process is investigated.
Revised: Deformation of single stranded DNA before reaching the pore is investigated.
Edit intent: clarity
Original:  By solving the Laplace equation in a suitable coordinate system and with appropriate boundary conditions, an approximate solution for  electric field inside and outside of a narrow pore is obtained.
Revised:  By solving the Laplace equation in a suitable coordinate system and with appropriate boundary conditions, an approximate solution for the electric field inside and outside of a narrow pore is obtained.
Edit intent: fluency
Original:  With an analysis based on "electrohydrodynamic equivalence" we determine the possibility of extension of a charged polymer due to the presence of  electric field gradient in the vicinity of the pore entrance.
Revised:  With an analysis based on "electrohydrodynamic equivalence" we determine the possibility of extension of a charged polymer due to the presence of an electric field gradient in the vicinity of the pore entrance.
Edit intent: fluency
Original:  I present a new approach/concept of dimensional enhancement - 2.5D "a 2D display of a virtual 3D environment", which can be implemented in social networking sites and further in other system applications.
Revised:  A new approach/concept of dimensional enhancement - 2.5D "a 2D display of a virtual 3D environment", which can be implemented in social networking sites and further in other system applications.
Edit intent: coherence
Original: Let n, r, k be positive integers such that 3\leq k < n and 2\leq r \leq k - 1. Let m(n, r, k) denote the maximum number of edges an r-uniform hypergraph on n vertices can have under the condition that any collection of i edges, 1 \leq i \leq k, span at least i vertices.
Revised: Let n, r, k be positive integers such that 3\leq k < n and 2\leq r \leq k-1. Let m(n, r, k) denote the maximum number of edges an r-uniform hypergraph on n vertices can have under the condition that any collection of i edges, 1 \leq i \leq k, span at least i vertices.
Edit intent: fluency
Original:  This problem is related to the forbidden hypergraph problem introduced by Brown, Erdos, and Soss, and S\'{o}s } and very recently discussed in the context of combinatorial batch codes.
Revised:  This problem is related to the forbidden hypergraph problem introduced by Brown, Erd\H{os, and S\'{o}s } and very recently discussed in the context of combinatorial batch codes.
Edit intent: coherence
Original:  In this short article, using a result of Erdos , we provide asymptotic improvement of the  bound on m(n, r, k) for all 3 \leq r \leq k - 4 , for 7 \leq k < n ) for all possible values of k.
Revised:  In this short article, using a result of Erdos , we provide asymptotic improvement of the bound on m(n, r, k) for all 3 \leq r \leq k-4 , for 7 \leq k < n ) for all possible values of k.
Edit intent: fluency
Original:  In this short article, using a result of }[  Erdos, we provide asymptotic improvement of the upper bound on m(n, r, k) for all 3 \leq r \leq k-4 , for 7 \leq k< n.
Revised:  In this short article, using a result of }[ Erdos, we improve the existing upper bound on m(n, r, k) for all 3 \leq r \leq k-4 , for 7 \leq k< n.
Edit intent: clarity
Original:   We provide exact value of m(n, 2, 5) for n \geq 5.
Revised:  enumerate We provide exact value of m(n, 2, 5) for n \geq 5.
Edit intent: clarity
Original:  This problem is related to the forbidden hypergraph problem introduced by Brown, Erdos, and S\'{o and very recently discussed in the context of combinatorial batch codes.
Revised:  This problem is related to the forbidden hypergraph problem introduced by Brown, Erdos, and S\'os and very recently discussed in the context of combinatorial batch codes.
Edit intent: fluency
Original:  {enumerate}[(i)] Using a result due to Erdos , we improve the existing upper bound on m(n, k, r) (O( n^r) ) for 7\leq k, and 3 \leq r \leq k-4.
Revised:  {enumerate}[(i)] Using a result due to Erdos we are able to show m(n, k, r) (O( n^r) ) for 7\leq k, and 3 \leq r \leq k-4.
Edit intent: clarity
Original:  {enumerate}[(i)] Using a result due to Erdos , we improve the existing upper bound on m(n, k, r) (O( n^r) ) for 7\leq k, and 3 \leq r \leq k-4.
Revised:  {enumerate}[(i)] Using a result due to Erdos , we improve the existing upper bound on m(n, k, r) (O( n^r) for 7\leq k, and 3 \leq r \leq k-4.
Edit intent: fluency
Original:  We give an explicit construction to show\lceil\rceil\lceil \rceil  m(n, k-3 , k) = \Omega(n^{k-3 \Theta(n ^{k-3 .  For 2-uniform CBCs and obtain the following results.
Revised:  We give an explicit construction to show\lceil\rceil\lceil \rceil m(n, k-3 , k) = \Theta(n ^{k-3 . For 2-uniform CBCs and obtain the following results.
Edit intent: clarity
Original:  We give an explicit construction to show\lceil\rceil\lceil \rceil  m(n, k-3 , k) = \Omega(n^{k-3 \Theta(n ^{k-3 .  For 2-uniform CBCs and obtain the following results.
Revised:  We give an explicit construction to show\lceil\rceil\lceil \rceil m(n, k-3 , k) = \Omega(n^{k-3 \Theta(n ^{k-3 . For 2-uniform CBCs we obtain the following results.
Edit intent: coherence
Original:  We use a result of K\"{ovari }%DIFDELCMD < {\em %%%  et al. to show m(n, 2, k) = O(n ^{ 3/2 }) for k \geq 6.
Revised:  Using a result of K\"{ovari }%DIFDELCMD < {\em %%% et al. to show m(n, 2, k) = O(n ^{ 3/2 }) for k \geq 6.
Edit intent: coherence
Original:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Revised:  This is sharp for infinitely many values of n . We show m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Edit intent: coherence
Original:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Revised:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, k ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Edit intent: clarity
Original:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Revised:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = O(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Edit intent: clarity
Original:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Revised:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) %DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Edit intent: clarity
Original:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Revised:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Edit intent: clarity
Original:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Revised:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) = \Theta(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Edit intent: clarity
Original:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Revised:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) for k = 6, 7, 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Edit intent: clarity
Original:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Revised:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Edit intent: clarity
Original:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Revised:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. %DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Edit intent: clarity
Original:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% enumeratevari, S\'os, and Tur\'{a}n.}
Revised:  This is sharp for infinitely many values of n when k=7 or 8 and implies that m(n, 2, 6 ) = \Theta(n ^{ 3/2{\lfloor\frac{k}{4}\rfloor}} }) . Using a result of Lazebnik%DIFDELCMD < {\em %%% et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (\Omega(n ^{ \frac{k+1 }) ) for all k \geq 8 and infinitely many values of n. enumerate%DIFDELCMD < } {%%% by using a result of K\"{ovari, S\'os, and Tur\'{a}n.}
Edit intent: clarity
Original:  This model is known to robustly store collections of randomly generated binary patterns as stable-points of the network dynamics.
Revised:  This model is known to robustly store collections of randomly generated binary patterns as stable-states of the network dynamics.
Edit intent: clarity
Original:  In this note, we design elementary families of Little-Hopfield networks that  solve this problem affirmatively  .
Revised:  In this note, we design simple families of Little-Hopfield networks that solve this problem affirmatively .
Edit intent: clarity
Original:  In this note, we design elementary families of Little-Hopfield networks that  solve this problem affirmatively  .
Revised:  In this note, we design elementary families of Little-Hopfield networks that provably solve this problem affirmatively .
Edit intent: clarity