wanyu/IteraTeR_v2 huggingface dataset:
Original:  Our goal is to provide a theoretical explanation of the aforementionned experimental behaviour and to propose new techniques to speed up the Gr\"obner basis computations by using the multi-homogeneous structure of those systems.
Revised:  Our goal is to provide a theoretical explanation of the aforementionned experimental behaviour and to propose new techniques to speed up the Gr\"obner basis computations by using the multihomogeneous structure of those systems.
Edit intent: fluency
Original:  First, we adapt the classical F5 criterion to avoid reductions to zero  when the input is a set of bilinear polynomials.
Revised:  First, we adapt the classical F5 criterion to avoid reductions to zero which occur when the input is a set of bilinear polynomials.
Edit intent: clarity
Original:  We propose also a variant of the F5 Algorithm dedicated to multi-homogeneous systems which exploits a structural property of the Macaulay matrix which occurs on such inputs.
Revised:  We propose also a variant of the F5 Algorithm dedicated to multihomogeneous systems which exploits a structural property of the Macaulay matrix which occurs on such inputs.
Edit intent: fluency
Original:  We formally prove that our adaptation is (i) optimal, i.e., it always generates a feasible schedule as long as such a schedule exists,  (ii) valid, i.e., it complies with the all the requirements.
Revised:  We formally prove that our adaptation is (i) optimal, i.e., it always generates a feasible schedule as long as such a schedule exists, and (ii) valid, i.e., it complies with the all the requirements.
Edit intent: coherence
Original:  The models have stochastic population dynamics with "mutation, " "migration, " and extinction of species.
Revised:  The models have stochastic population dynamics with mutation, migration, and extinction of species.
Edit intent: fluency
Original:  The latter is well fitted by a q-exponential functionderived from an age-dependent mortality function.
Revised:  The latter has a longer tail than a simple exponential function, which indicates an age-dependent mortality function.
Edit intent: clarity
Original:  In this paper, we incorporate the ADT network model  within the algebraic network coding framework introduced by Koetter and Medard.
Revised:  In this paper, we show that the ADT network model within the algebraic network coding framework introduced by Koetter and Medard.
Edit intent: coherence
Original:  Our approach does not only provide an algebraic interpretation of the Min-cut Max-flow theorem  , but also extends to more general types of connections , such as two-level multicast, disjoint multicast, and multiple source multicast .
Revised:  Our approach does not only provide an algebraic interpretation of the Min-cut Max-flow theorem , but also extends to non-multicast connections such as multiple multicast, disjoint multicast, and multiple source multicast .
Edit intent: clarity
Original: Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic with quantification over edge sets (MSO) can be decided in linear time on any class of graphs of bounded tree-width , or in other words, MSO is fixed-parameter tractable  on any such class of graphs.
Revised: Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic (MSO) can be decided in linear time on any class of graphs of bounded tree-width , or in other words, MSO is fixed-parameter tractable on any such class of graphs.
Edit intent: clarity
Original: Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic with quantification over edge sets (MSO) can be decided in linear time on any class of graphs of bounded tree-width , or in other words, MSO is fixed-parameter tractable  on any such class of graphs.
Revised: Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic with quantification over edge sets (MSO) can be decided in linear time on any class of graphs of bounded treewidth , or in other words, MSO is fixed-parameter tractable on any such class of graphs.
Edit intent: clarity
Original: Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic with quantification over edge sets (MSO) can be decided in linear time on any class of graphs of bounded tree-width , or in other words, MSO is fixed-parameter tractable  on any such class of graphs.
Revised: Courcelle's famous theorem from 1990 states that any property of graphs definable in monadic second-order logic with quantification over edge sets (MSO) can be decided in linear time on any class of graphs of bounded tree-width , or in other words, MSO is fixed-parameter tractable in linear time on any such class of graphs.
Edit intent: clarity
Original:  In this paper we establish a strong lower bound for the complexity of MSO . In particular, we show that if C is any class of graphs which is closed under taking sub-graphs and whose tree-width is not bounded by a poly-logarithmic function (in fact, log^c n , for some small c suffices) then MSO-model checking is intractable on C (under a suitable assumption from complexity theory ).
Revised:  In this paper we establish a strong lower bound for the complexity of MSO . In particular, we show that if C is any class of graphs which is closed under taking subgraphs and whose treewidth is not bounded by a poly-logarithmic function (in fact, log^c n , for some small c suffices) then MSO-model checking is intractable on C (under a suitable assumption from complexity theory ).
Edit intent: fluency
Original:  In this paper we establish a strong lower bound for the complexity of MSO . In particular, we show that if C is any class of graphs which is closed under taking sub-graphs and whose tree-width is not bounded by a poly-logarithmic function (in fact, log^c n , for some small c suffices) then MSO-model checking is intractable on C (under a suitable assumption from complexity theory ).
Revised:  In this paper we establish a strong lower bound for the complexity of MSO . In particular, we show that if C is any class of graphs which is closed under taking sub-graphs and whose tree-width is not bounded by a polylogarithmic function (in fact, log^c n , for some small c suffices) then MSO-model checking is intractable on C (under a suitable assumption from complexity theory ).
Edit intent: fluency
Original:  In this paper we establish a strong lower bound for the complexity of MSO . In particular, we show that if C is any class of graphs which is closed under taking sub-graphs and whose tree-width is not bounded by a poly-logarithmic function (in fact, log^c n , for some small c suffices) then MSO-model checking is intractable on C (under a suitable assumption from complexity theory ).
Revised:  In this paper we establish a strong lower bound for the complexity of MSO . In particular, we show that if C is any class of graphs which is closed under taking sub-graphs and whose tree-width is not bounded by a poly-logarithmic function (in fact, \log^c n for some small c suffices) then MSO-model checking is intractable on C (under a suitable assumption from complexity theory ).
Edit intent: fluency
Original:  Error control is thus reduced to a coding theoretic problem on flats of a matroid , where two distinct metrics between flats are introduced .
Revised:  Error control is thus reduced to a coding theoretic problem on flats of a matroid , where two distinct metrics between flats for error correction .
Edit intent: clarity
Original:  Third, we introduce a novel combination technique for network coding, referred to as random affine network coding, based on affine combinations of packets.
Revised:  Third, we introduce a novel type of network coding, referred to as random affine network coding, based on affine combinations of packets.
Edit intent: clarity
Original:  Our matroid viewpoint also allows us to determine the true rate and combination power of random linear network coding .
Revised:  We thus show that RANC outperforms RLNC in terms of data rate and throughput thanks to a more efficient encoding of messages into packets .
Edit intent: clarity
Original:  Second, we model the possible alterations of a message by the network as an operator channel, which generalizes the channels proposed for random linear network coding and store and forward . Error control is thus reduced to a coding theoretic problem on flats of a matroid .
Revised:  Second, we model the possible alterations of a message by the network as an operator channel, which generalizes the channels proposed for RLNC and SAF . Error control is thus reduced to a coding theoretic problem on flats of a matroid .
Edit intent: clarity
Original:  Second, we model the possible alterations of a message by the network as an operator channel, which generalizes the channels proposed for random linear network coding and store and forward . Error control is thus reduced to a coding theoretic problem on flats of a matroid .
Revised:  Second, we model the possible alterations of a message by the network as an operator channel, which generalizes the channels proposed for random linear network coding and store and forward . Error control is thus reduced to a coding-theoretic problem on flats of a matroid .
Edit intent: clarity
Original:  We also determine two distinct metrics between flats for error correction.
Revised:  We also determine two distinct metrics can be used for error correction.
Edit intent: clarity
Original:  These are used to generalize neural and swarm architectures , namely artificial neural networks, ant colony optimization, and particle swarm optimization  .
Revised:  These are used to generalize neural and swarm architectures . Artificial neural networks, ant colony optimization, and particle swarm optimization .
Edit intent: coherence
Original:  These are used to generalize neural and swarm architectures , namely artificial neural networks, ant colony optimization, and particle swarm optimization  .
Revised:  These are used to generalize neural and swarm architectures , namely artificial neural networks, ant colony optimization, particle swarm optimization .
Edit intent: fluency
Original:  In this paper , the Computing Networks (CNs) framework is presented . CNs are used to generalize neural and swarm architectures.
Revised:  In this paper , the Computing Networks (CNs) framework . CNs are used to generalize neural and swarm architectures.
Edit intent: clarity
Original:  We then present an efficient encoding scheme which enables the communication of X_i for the three sources, three terminals case, given that each source terminal pair is connected by two {\em  edge disjoint paths .
Revised:  We then present an efficient encoding scheme which enables the communication of X_i for the three sources, three terminals case, given that each source terminal pair is connected by {\em edge disjoint paths .
Edit intent: clarity
Original: This paper is the second part of a serie of two papers dealing with bulking: a  quasi-order on cellular automata  comparing space-time diagrams up to some rescaling.
Revised: This paper is the second part of a series of two papers dealing with bulking: a quasi-order on cellular automata comparing space-time diagrams up to some rescaling.
Edit intent: fluency
Original: This paper is the second part of a serie of two papers dealing with bulking: a  quasi-order on cellular automata  comparing space-time diagrams up to some rescaling.
Revised: This paper is the second part of a serie of two papers dealing with bulking: a way to define quasi-order on cellular automata comparing space-time diagrams up to some rescaling.
Edit intent: clarity
Original: This paper is the second part of a serie of two papers dealing with bulking: a  quasi-order on cellular automata  comparing space-time diagrams up to some rescaling.
Revised: This paper is the second part of a serie of two papers dealing with bulking: a quasi-order on cellular automata by comparing space-time diagrams up to some rescaling.
Edit intent: clarity
Original:  In the present paper, we introduce 3 notions of simulation between cellular automata and study the quasi-order structures induced by these simulation relations on the whole set of cellular automata.
Revised:  In the present paper, we introduce three notions of simulation between cellular automata and study the quasi-order structures induced by these simulation relations on the whole set of cellular automata.
Edit intent: fluency
Original:  Various aspects of these quasi-orders are considered (induced equivalence relations, maximum elements, induced orders, etc) providing several formal tools  to classify cellular automata.
Revised:  Various aspects of these quasi-orders are considered (induced equivalence relations, maximum elements, induced orders, etc) providing several formal tools allowing to classify cellular automata.
Edit intent: clarity
Original: We study the problem of identity testing for depth-3 circuits , over the field of reals, of top fanin k and degree d (called sps(k,d) identities) .
Revised: We study the problem of identity testing for depth-3 circuits of top fanin k and degree d (called sps(k,d) identities) .
Edit intent: clarity
Original: We study the problem of identity testing for depth-3 circuits , over the field of reals, of top fanin k and degree d (called sps(k,d) identities) .
Revised: We study the problem of identity testing for depth-3 circuits , over the field of reals, of top fanin k and degree d .
Edit intent: clarity
Original:  We give a new structure theorem for such identities and improve the known deterministic d^{k^k}-time black-box identity test (Kayal Saraf , FOCS 2009) to one that takes d^{k^2}-time.
Revised:  We give a new structure theorem for such identities and improve the known deterministic d^{k^k}-time black-box identity test Saraf , FOCS 2009) to one that takes d^{k^2}-time.
Edit intent: clarity
Original:  We achieve this exponential improvement by settling affirmatively the "stronger " rank conjectures posed by Dvir Shpilka (STOC 2005) and Kayal Saraf (FOCS 2009).
Revised:  We achieve this exponential improvement by settling affirmatively the "stronger " rank conjectures posed by Shpilka (STOC 2005) and Kayal Saraf (FOCS 2009).
Edit intent: clarity
Original:  We achieve this exponential improvement by settling affirmatively the "stronger " rank conjectures posed by Dvir Shpilka (STOC 2005) and Kayal Saraf (FOCS 2009).
Revised:  We achieve this exponential improvement by settling affirmatively the "stronger " rank conjectures posed by Dvir Dvir-Shpilka (STOC 2005) and Kayal Saraf (FOCS 2009).
Edit intent: clarity
Original:  We achieve this exponential improvement by settling affirmatively the "stronger " rank conjectures posed by Dvir Shpilka (STOC 2005) and Kayal Saraf (FOCS 2009).
Revised:  We achieve this exponential improvement by settling affirmatively the "stronger " rank conjectures posed by Dvir Shpilka (STOC 2005) and Saraf (FOCS 2009).
Edit intent: clarity
Original:  For identities over R, the latter paper had shown a rank bound of k^k (note the independence from d) for simple, minimal sps(k,d) identities, which we improve to k^2 . Our main theorem (almost optimally) pins down the connection between higher dimensional Sylvester-Gallai theorems and the rank of depth-3 identities in a very transparent manner.
Revised:  For identities over R, the latter paper had shown a rank bound of k^k (note the independence from d) for simple, minimal sps(k,d) identities, which we improve to k^2 . Our main theorem (almost optimally) pins down the relation between higher dimensional Sylvester-Gallai theorems and the rank of depth-3 identities in a very transparent manner.
Edit intent: fluency
Original:  Next, we combine algebraic lemmas with some combinatorics to argue about the remainder of the circuit (the non-nucleus part), bounding the rank of this portion through higher dimensional versions of the Sylvester-Gallai Theorem.
Edit intent: coherence
Original:  Our proof methods explain the structure of any depth-3 identity C: the nucleus of C is a low rank identity, while the remainder is a high dimensional Sylvester-Gallai configuration.
Revised:  Our proof methods explain the structure of any depth-3 identity C: there is a nucleus of C is a low rank identity, while the remainder is a high dimensional Sylvester-Gallai configuration.
Edit intent: clarity
Original:  Our proof methods explain the structure of any depth-3 identity C: the nucleus of C is a low rank identity, while the remainder is a high dimensional Sylvester-Gallai configuration.
Revised:  Our proof methods explain the structure of any depth-3 identity C: the nucleus of C that forms a low rank identity, while the remainder is a high dimensional Sylvester-Gallai configuration.
Edit intent: clarity
Original:  The implied algorithm uses no divisions in D or in D x%DIFDELCMD < ] %%% and has worst-case complexity5%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% . Over a field, this reduces to 3%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% , and the B\'{e non-zero scalar.
Revised:  The x%DIFDELCMD < ] %%% and has worst-case complexity5%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% . Over a field, this reduces to 3%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% , and the B\'{e non-zero scalar.
Edit intent: coherence
Original:  The implied algorithm uses no divisions in D or in D x%DIFDELCMD < ] %%% and has worst-case complexity5%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% . Over a field, this reduces to 3%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% , and the B\'{e non-zero scalar.
Revised:  The implied algorithm uses no divisions in D or in D %DIFDELCMD < ] %%% and has worst-case complexity5%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% . Over a field, this reduces to 3%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% , and the B\'{e non-zero scalar.
Edit intent: clarity
Original:  The implied algorithm uses no divisions in D or in D x%DIFDELCMD < ] %%% and has worst-case complexity5%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% . Over a field, this reduces to 3%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% , and the B\'{e non-zero scalar.
Revised:  The implied algorithm uses no divisions in D or in D x%DIFDELCMD < ] %%% %DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% . Over a field, this reduces to 3%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% , and the B\'{e non-zero scalar.
Edit intent: coherence
Original:  The implied algorithm uses no divisions in D or in D x%DIFDELCMD < ] %%% and has worst-case complexity5%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% . Over a field, this reduces to 3%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% , and the B\'{e non-zero scalar.
Revised:  The implied algorithm uses no divisions in D or in D x%DIFDELCMD < ] %%% and has worst-case complexity5%DIFDELCMD < \lfloor %%% {4}}%DIFDELCMD < \rfloor%%% . Over a field, this reduces to 3%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% , and the B\'{e non-zero scalar.
Edit intent: fluency
Original:  The implied algorithm uses no divisions in D or in D x%DIFDELCMD < ] %%% and has worst-case complexity5%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% . Over a field, this reduces to 3%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% , and the B\'{e non-zero scalar.
Revised:  The implied algorithm uses no divisions in D or in D x%DIFDELCMD < ] %%% and has worst-case complexity5%DIFDELCMD < \lfloor %%% \frac{n^2{4}}%DIFDELCMD < \rfloor%%% . Over a field, this reduces to 3%DIFDELCMD < \lfloor %%% {4}}%DIFDELCMD < \rfloor%%% , and the B\'{e non-zero scalar.
Edit intent: fluency
Original:  As an application , we give a new algorithm to compute a B\'{ezout identity for u,v \in D}%DIFDELCMD < [%%% x , with u monic and 0\leq \deg(v)\leq \deg(u), where D is a principal ideal domain.
Revised:  As an application , we give a new zout identity for u,v \in D}%DIFDELCMD < [%%% x , with u monic and 0\leq \deg(v)\leq \deg(u), where D is a principal ideal domain.
Edit intent: clarity
Original:  As an application , we give a new algorithm to compute a B\'{ezout identity for u,v \in D}%DIFDELCMD < [%%% x , with u monic and 0\leq \deg(v)\leq \deg(u), where D is a principal ideal domain.
Revised:  As an application , we give a new algorithm to compute a B\'{ezout identity for u,v \in D}%DIFDELCMD < [%%% , with u monic and 0\leq \deg(v)\leq \deg(u), where D is a principal ideal domain.
Edit intent: clarity
Original: In the problem Max Lin, we are given a system Az=b of m linear equations with n variables over F_2 in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables in order to maximize the total weight of satisfied equations . Max Lin Above Average ( MLAA ) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc.
Revised: In the problem Max Lin, we are given a system Az=b of m linear equations with n variables over F_2 in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables in order to maximize the total weight of satisfied equations . Max Lin Above Average ( Max Lin AA ) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc.
Edit intent: clarity
Original:  In MLAA all weights are integral and we wish to decide whether there is an assignment of values to the variables such that the total weight of satisfied equations minus the total weight of falsified equations is at least k, where k is the parameter.
Revised:  In MLAA all weights are integral and we are to decide whether there is an assignment of values to the variables such that the total weight of satisfied equations minus the total weight of falsified equations is at least k, where k is the parameter.
Edit intent: clarity
Original:  In MLAA all weights are integral and we wish to decide whether there is an assignment of values to the variables such that the total weight of satisfied equations minus the total weight of falsified equations is at least k, where k is the parameter.
Revised:  In MLAA all weights are integral and we wish to decide whether the maximum excess is at least k, where k is the parameter.
Edit intent: clarity
Original:  Mahajan et al. raised the question of determining the parameterized complexity of MLAA.
Edit intent: coherence
Original:  We also prove that MLAA is polynomial-time solvable for every fixed k and, moreover, MLAA is in the parameterized complexity class W P%DIFDELCMD < ]%%% . We show that maximization of arbitrary pseudo-boolean functions, i.e., functions f: \{-1, + 1\}^n%DIFDELCMD < \to %%% \mathbb{R clauses.
Revised:  We also prove that MLAA is polynomial-time solvable for every fixed k and, moreover, MLAA is in the parameterized complexity class W %DIFDELCMD < ]%%% . We show that maximization of arbitrary pseudo-boolean functions, i.e., functions f: \{-1, + 1\}^n%DIFDELCMD < \to %%% \mathbb{R clauses.
Edit intent: clarity
Original: In this work we analyze the topological and dynamical properties of a simple model of complex food webs, namely the niche model.
Revised: In this work we analyse the topological and dynamical properties of a simple model of complex food webs, namely the niche model.
Edit intent: fluency
Original:  After this we introduce " predators " and " prey " weighted graphs from which we underline patterns of competition among species.
Revised:  After this we introduce " prey " and " prey " weighted graphs from which we underline patterns of competition among species.
Edit intent: clarity
Original:  After this we introduce " predators " and " prey " weighted graphs from which we underline patterns of competition among species.
Revised:  After this we introduce " predators " and " predators " weighted graphs from which we underline patterns of competition among species.
Edit intent: clarity