wanyu/IteraTeR_v2 huggingface dataset:
Original:  These channels are partly motivated by settings in which the communication is done with rather unconventional physical carriers, such as particles, molecules, items, etc.
Edit intent: coherence
Original: The existence of stationary Markov perfect equilibria in stochastic games is shown in several contexts under a general condition called "  coarser transition kernels".
Revised: The existence of stationary Markov perfect equilibria in stochastic games is shown under a general condition called " coarser transition kernels".
Edit intent: clarity
Original:  These results include various earlier existence results on correlated equilibria, noisy stochastic games,  stochastic games with mixtures of constant transition kernels as special cases.
Revised:  This result includes various earlier existence results on correlated equilibria, noisy stochastic games, stochastic games with mixtures of constant transition kernels as special cases.
Edit intent: fluency
Original:  These results include various earlier existence results on correlated equilibria, noisy stochastic games,  stochastic games with mixtures of constant transition kernels as special cases.
Revised:  These results include various earlier existence results on correlated equilibria, noisy stochastic games, and stochastic games with mixtures of constant transition kernels as special cases.
Edit intent: fluency
Original:  The results here also shed some new light on a recent example on the nonexistence of stationary equilibrium.
Revised:  The results here also shed some new light on a recent example about the nonexistence of stationary equilibrium.
Edit intent: fluency
Original:  This result includes various earlier existence results on correlated equilibria, noisy stochastic games, and stochastic games with mixtures of constant transition kernels as special cases.
Revised:  This result covers various earlier existence results on correlated equilibria, noisy stochastic games, and stochastic games with mixtures of constant transition kernels as special cases.
Edit intent: clarity
Original:  As a consequence of our main result, we obtain an \exp\left(-\Omega(n^{2/5})\right) upper bound on the the discrepancy of a function in AC^0, and an \exp\left( - \Omega(n^{2/5})\right) lower bound on the threshold weight of AC^0, improving over the previous best results of \exp\left(-\Omega(n^{1/3})\right) and \exp\left(\Omega(n^{1/3})\right) respectively.
Revised:  As a consequence of our main result, we obtain an \exp\left(-\Omega(n^{2/5})\right) upper bound on the the discrepancy of a function in AC^0, and an \exp\left( \Omega(n^{2/5})\right) lower bound on the threshold weight of AC^0, improving over the previous best results of \exp\left(-\Omega(n^{1/3})\right) and \exp\left(\Omega(n^{1/3})\right) respectively.
Edit intent: fluency
Original:  We believe that the methodology developed herein is applicable to other biomedical domains where microcirculation can be imaged but experimental flow measurements are unavailable or difficult to obtain.
Revised:  We believe that the methodology developed herein is applicable to other biomedical domains where microvasculature can be imaged but experimental flow measurements are unavailable or difficult to obtain.
Edit intent: clarity
Original: A quantum gate with one or two qubits may be described as  element of Spin group due to isomorphisms Spin(3) \simeq SU(2) and Spin(6) \simeq SU(4).
Revised: A quantum gate with one or two qubits may be described as an element of Spin group due to isomorphisms Spin(3) \simeq SU(2) and Spin(6) \simeq SU(4).
Edit intent: fluency
Original:  A quantum circuit with one- and two-gates may be used for construction of arbitrary unitary transformation SU(2^n).
Revised:  A quantum circuit with one- and two-qubit gates may be used for construction of arbitrary unitary transformation SU(2^n).
Edit intent: clarity
Original:  In this work are discussed `Spin(3n) circuits' composed from one- and two-gates represented by elements of Spin groups.
Revised:  `Spin(3n) circuits' composed from one- and two-gates represented by elements of Spin groups.
Edit intent: coherence
Original:  Remote resemblance with well-known sets of non-universal quantum gates (e.g., matchgates) related with Spin(2n) and possibility of classical simulation are also discussed.
Revised:  Remote resemblance with well-known sets of non-universal quantum gates (e.g., matchgates) related with Spin(2n) and possibility of effective simulation on a classical computer are also discussed.
Edit intent: clarity
Original: A quantum gate with one or two qubits may be described as an element of Spin group due to isomorphisms Spin(3) \simeq SU(2) and Spin(6) \simeq SU(4).
Revised: All quantum gates with one and two qubits may be described as an element of Spin group due to isomorphisms Spin(3) \simeq SU(2) and Spin(6) \simeq SU(4).
Edit intent: clarity
Original: A quantum gate with one or two qubits may be described as an element of Spin group due to isomorphisms Spin(3) \simeq SU(2) and Spin(6) \simeq SU(4).
Revised: A quantum gate with one or two qubits may be described by elements of Spin groups due to isomorphisms Spin(3) \simeq SU(2) and Spin(6) \simeq SU(4).
Edit intent: fluency
Original:  For n > 2 group of n-qubit gates SU(2^n)  has bigger dimension than Spin(3n).
Revised:  However, the group of n-qubit gates SU(2^n) has bigger dimension than Spin(3n).
Edit intent: coherence
Original:   `Spin(3n) circuits' are defined in this work as products of elements associated with one- and two-qubit gates due to above-mentioned isomorphisms.
Revised:  Analogously, the `Spin(3n) circuits' are defined in this work as products of elements associated with one- and two-qubit gates due to above-mentioned isomorphisms.
Edit intent: coherence
Original:   `Spin(3n) circuits' are defined in this work as products of elements associated with one- and two-qubit gates due to above-mentioned isomorphisms.
Revised:  `Spin(3n) circuits' are introduced in this work as products of elements associated with one- and two-qubit gates due to above-mentioned isomorphisms.
Edit intent: clarity
Original:   `Spin(3n) circuits' are defined in this work as products of elements associated with one- and two-qubit gates due to above-mentioned isomorphisms.
Revised:  `Spin(3n) circuits' are defined in this work as products of elements associated with one- and two-qubit gates with respect to the above-mentioned isomorphisms.
Edit intent: clarity
Original:  Certain resemblance with well-known sets of non-universal quantum gates ( e.g., matchgates  ) related with Spin(2n) and possibility of effective simulation on a classical computer are also discussed.
Revised:  Certain resemblance with well-known sets of non-universal quantum gates e.g., matchgates ) related with Spin(2n) and possibility of effective simulation on a classical computer are also discussed.
Edit intent: fluency
Original:  An alternative, more recent, model considers the reciprocal interaction of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes.
Revised:  An alternative, more recent, model considers the mutual inhibition of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes.
Edit intent: clarity
Original:  Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes can be classified as grazing bifurcations.
Revised:  Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes occur at grazing bifurcations.
Edit intent: clarity
Original:  The demonstration of the mathematical similarities of the two models is important because it brings a deeper insight to sleep-wake dynamics and allows some features of the two-process model to be interpreted physiologically  .
Revised:  The demonstration of the mathematical similarities of the two models is valuable because not only does it allow some features of the two-process model to be interpreted physiologically .
Edit intent: clarity
Original: We analyse the ubiquitous SPH pressure jump at a density gradient when solving the equations of inviscid hydrodynamics.
Revised: The spurious pressure jump at a density gradient when solving the equations of inviscid hydrodynamics.
Edit intent: clarity
Original:  This helps to explain many of the proposedcures as suggested in the literature.
Revised:  This helps to explain many of the proposed as suggested in the literature.
Edit intent: fluency
Original:  In this note  we show that all eulerian multigraphs with an odd number of edges have no interval coloring.
Revised:  In this note , we show that all eulerian multigraphs with an odd number of edges have no interval coloring.
Edit intent: fluency
Original:  In this note  we show that all eulerian multigraphs with an odd number of edges have no interval coloring.
Revised:  In this note we show that all Eulerian multigraphs with an odd number of edges have no interval coloring.
Edit intent: fluency
Original:  We also give some methods for constructing of interval non-edge-colorable eulerian multigraphs.
Revised:  We also give some methods for constructing of interval non-edge-colorable Eulerian multigraphs.
Edit intent: fluency
Original:  It is widely observed in numerous bacterial species that individual bacterial cells will stochastically switch among multiple phenotypes  to survive in rapidly changing environments.
Revised:  It is observed in numerous bacterial species that individual bacterial cells will stochastically switch among multiple phenotypes to survive in rapidly changing environments.
Edit intent: clarity
Original:  It is widely observed in numerous bacterial species that individual bacterial cells will stochastically switch among multiple phenotypes  to survive in rapidly changing environments.
Revised:  It is widely observed in numerous bacterial species that individual cells can stochastically switch among multiple phenotypes to survive in rapidly changing environments.
Edit intent: clarity
Original:  It is widely observed in numerous bacterial species that individual bacterial cells will stochastically switch among multiple phenotypes  to survive in rapidly changing environments.
Revised:  It is widely observed in numerous bacterial species that individual bacterial cells will stochastically switch among multiple phenotypes for the population to survive in rapidly changing environments.
Edit intent: clarity
Original:  This  phenotypic heterogeneity with stochastic phenotypic switching is generally assumed to be an adaptive bet-hedging strategy.
Revised:  This kind of phenotypic heterogeneity with stochastic phenotypic switching is generally assumed to be an adaptive bet-hedging strategy.
Edit intent: clarity
Original:  This  phenotypic heterogeneity with stochastic phenotypic switching is generally assumed to be an adaptive bet-hedging strategy.
Revised:  This phenotypic heterogeneity with stochastic phenotype switching is generally assumed to be an adaptive bet-hedging strategy.
Edit intent: fluency
Original:  This  phenotypic heterogeneity with stochastic phenotypic switching is generally assumed to be an adaptive bet-hedging strategy.
Revised:  This phenotypic heterogeneity with stochastic phenotypic switching is generally understood to be an adaptive bet-hedging strategy.
Edit intent: clarity
Original:  To gain a deeper understanding how bet-hedging is achieved and the pattern and information behind experimental data , a mathematical model is needed .
Revised:  Mathematical models are essential to gain a deeper understanding how bet-hedging is achieved and the pattern and information behind experimental data , a mathematical model is needed .
Edit intent: clarity
Original:  To gain a deeper understanding how bet-hedging is achieved and the pattern and information behind experimental data , a mathematical model is needed .
Revised:  To gain a deeper insight into the principle behind bet-hedging is achieved and the pattern and information behind experimental data , a mathematical model is needed .
Edit intent: clarity
Original:  To gain a deeper understanding how bet-hedging is achieved and the pattern and information behind experimental data , a mathematical model is needed .
Revised:  To gain a deeper understanding how bet-hedging and the pattern and information behind experimental data , a mathematical model is needed .
Edit intent: clarity
Original:  To gain a deeper understanding how bet-hedging is achieved and the pattern and information behind experimental data , a mathematical model is needed .
Revised:  To gain a deeper understanding how bet-hedging is achieved and the pattern behind experimental data , a mathematical model is needed .
Edit intent: clarity
Original:  To gain a deeper understanding how bet-hedging is achieved and the pattern and information behind experimental data , a mathematical model is needed .
Revised:  To gain a deeper understanding how bet-hedging is achieved and the pattern and information behind experimental data .
Edit intent: clarity
Original:  In this article, we proposed a unified nonlinear stochastic model of multistable bacterial systems at the molecular level.
Revised:  In this paper, we propose a nonlinear stochastic model of multistable bacterial systems at the molecular level.
Edit intent: clarity
Original:  In addition, we also provided a quantitative characterization of the critical state during the transition among multiple phenotypes .
Revised:  In addition, we provide a quantitative characterization of the critical state during the transition among multiple phenotypes .
Edit intent: coherence
Original:  In addition, we also provided a quantitative characterization of the critical state during the transition among multiple phenotypes .
Revised:  In addition, we also provided a quantitative characterization of the critical state of multistable bacterial systems and develop an effective data-driven method to identify the critical state without resorting to specific mathematical models .
Edit intent: clarity
Original:  Moreover, we prove a tight \Theta(n) monotone depth bound for a function in monotone P . This implies an average-case hierarchy theorem within monotone P similar to a result of Filmus et al. (FOCS 2013) . (2) Proof Complexity: We prove new rank lower bounds as well as obtain the first length--space lower bounds for semi-algebraic proof systems, including Lov\'asz--Schrijver and Lasserre (SOS) systems.
Revised:  Moreover, we prove a \Theta(n) monotone depth bound for a function in monotone P . This implies an average-case hierarchy theorem within monotone P similar to a result of Filmus et al. (FOCS 2013) . (2) Proof Complexity: We prove new rank lower bounds as well as obtain the first length--space lower bounds for semi-algebraic proof systems, including Lov\'asz--Schrijver and Lasserre (SOS) systems.
Edit intent: clarity
Original:  Moreover, we prove a tight \Theta(n) monotone depth bound for a function in monotone P . This implies an average-case hierarchy theorem within monotone P similar to a result of Filmus et al. (FOCS 2013) . (2) Proof Complexity: We prove new rank lower bounds as well as obtain the first length--space lower bounds for semi-algebraic proof systems, including Lov\'asz--Schrijver and Lasserre (SOS) systems.
Revised:  Moreover, we prove a tight \Theta(n) monotone depth bound for a function in monotone P . (2) Proof Complexity: We prove new rank lower bounds as well as obtain the first length--space lower bounds for semi-algebraic proof systems, including Lov\'asz--Schrijver and Lasserre (SOS) systems.
Edit intent: coherence
Original:  In particular, our most general theorem shows that it suffices to set m =%DIFDELCMD < \Omega%%% ( \gamma_2^2(T) + 1 ) and s =%DIFDELCMD < \Omega(1) %%%  as long as s,m additionally satisfy a certain tradeoff condition that is governed by the geometry of T (and as we show for several examples of interest, is easy to verify).
Revised:  Our most general theorem shows that it suffices to set m =%DIFDELCMD < \Omega%%% ( \gamma_2^2(T) + 1 ) and s =%DIFDELCMD < \Omega(1) %%% as long as s,m additionally satisfy a certain tradeoff condition that is governed by the geometry of T (and as we show for several examples of interest, is easy to verify).
Edit intent: coherence
Original:  In particular, our most general theorem shows that it suffices to set m =%DIFDELCMD < \Omega%%% ( \gamma_2^2(T) + 1 ) and s =%DIFDELCMD < \Omega(1) %%%  as long as s,m additionally satisfy a certain tradeoff condition that is governed by the geometry of T (and as we show for several examples of interest, is easy to verify).
Revised:  In particular, our most general theorem shows that it suffices to set m %DIFDELCMD < \Omega%%% ( \gamma_2^2(T) + 1 ) and s =%DIFDELCMD < \Omega(1) %%% as long as s,m additionally satisfy a certain tradeoff condition that is governed by the geometry of T (and as we show for several examples of interest, is easy to verify).
Edit intent: fluency
Original:  In particular, our most general theorem shows that it suffices to set m =%DIFDELCMD < \Omega%%% ( \gamma_2^2(T) + 1 ) and s =%DIFDELCMD < \Omega(1) %%%  as long as s,m additionally satisfy a certain tradeoff condition that is governed by the geometry of T (and as we show for several examples of interest, is easy to verify).
Revised:  In particular, our most general theorem shows that it suffices to set m =%DIFDELCMD < \Omega%%% ( \gamma_2^2(T) + 1 %DIFDELCMD < \Omega(1) %%% as long as s,m additionally satisfy a certain tradeoff condition that is governed by the geometry of T (and as we show for several examples of interest, is easy to verify).
Edit intent: coherence
Original:  Here \gamma_2 is Talagrand's functional, and we write f =%DIFDELCMD < \Omega%%% (g ) to mean f %DIFDELCMD < \ge %%% Cg \le  (\varepsilon^{-1}\log n)^c for some constants C,c>0.
Revised:  Here \gamma_2 is Talagrand's functional, and we write f %DIFDELCMD < \Omega%%% (g ) to mean f %DIFDELCMD < \ge %%% Cg \le (\varepsilon^{-1}\log n)^c for some constants C,c>0.
Edit intent: clarity
Original:  Here \gamma_2 is Talagrand's functional, and we write f =%DIFDELCMD < \Omega%%% (g ) to mean f %DIFDELCMD < \ge %%% Cg \le  (\varepsilon^{-1}\log n)^c for some constants C,c>0.
Revised:  Here \gamma_2 is Talagrand's functional, and we write f =%DIFDELCMD < \Omega%%% (g ) to mean f %DIFDELCMD < \ge %%% \le (\varepsilon^{-1}\log n)^c for some constants C,c>0.
Edit intent: clarity
Original:  Our result can be seen as an extension to sparse \Phi of works of Klartag, Mendelson, J. Funct.
Revised:  Our result Klartag, Mendelson, J. Funct.
Edit intent: clarity
Original:  Anal. 2005%DIFDELCMD < ]%%% , Gordon, GAFA 1988%DIFDELCMD < ]%%% , and Mendelson, Pajor, Tomczak-Jaegermann, GAFA 2007%DIFDELCMD < ]%%% , which were concerned with dense \Phi having i.i.d. (sub) gaussian entries.
Revised:  Anal. 2005%DIFDELCMD < ]%%% Gordon, GAFA 1988%DIFDELCMD < ]%%% , and Mendelson, Pajor, Tomczak-Jaegermann, GAFA 2007%DIFDELCMD < ]%%% , which were concerned with dense \Phi having i.i.d. (sub) gaussian entries.
Edit intent: fluency
Original:  Anal. 2005%DIFDELCMD < ]%%% , Gordon, GAFA 1988%DIFDELCMD < ]%%% , and Mendelson, Pajor, Tomczak-Jaegermann, GAFA 2007%DIFDELCMD < ]%%% , which were concerned with dense \Phi having i.i.d. (sub) gaussian entries.
Revised:  Anal. 2005%DIFDELCMD < ]%%% , %DIFDELCMD < ]%%% , and Mendelson, Pajor, Tomczak-Jaegermann, GAFA 2007%DIFDELCMD < ]%%% , which were concerned with dense \Phi having i.i.d. (sub) gaussian entries.
Edit intent: clarity
Original:  Anal. 2005%DIFDELCMD < ]%%% , Gordon, GAFA 1988%DIFDELCMD < ]%%% , and Mendelson, Pajor, Tomczak-Jaegermann, GAFA 2007%DIFDELCMD < ]%%% , which were concerned with dense \Phi having i.i.d. (sub) gaussian entries.
Revised:  Anal. 2005%DIFDELCMD < ]%%% , Gordon, GAFA 1988%DIFDELCMD < ]%%% Mendelson, Pajor, Tomczak-Jaegermann, GAFA 2007%DIFDELCMD < ]%%% , which were concerned with dense \Phi having i.i.d. (sub) gaussian entries.
Edit intent: coherence
Original:  Our work introduces a theory that qualitatively unifies several results related to the Johnson-Lindenstrauss lemma, subspace embeddings, and Fourier-based methods for obtaining matrices satisfying the restricted isometry property .
Revised:  We qualitatively unify several results related to the Johnson-Lindenstrauss lemma, subspace embeddings, and Fourier-based methods for obtaining matrices satisfying the restricted isometry property .
Edit intent: clarity
Original:  For T a subset of the unit sphere, \varepsilon\in(0,1/2) given, we study settings for m,s required to ensure \mathbb{E}_\Phi x\in T \left|\|\Phi x\|_2^2 - 1 \right| < \varepsilon , i.e.  so that \Phi preserves the norm of every x\in T simultaneously and multiplicatively up to 1+\varepsilon.
Revised:  For a subset T of the unit sphere, \varepsilon\in(0,1/2) given, we study settings for m,s required to ensure \mathbb{E}_\Phi x\in T \left|\|\Phi x\|_2^2 - 1 \right| < \varepsilon , i.e. so that \Phi preserves the norm of every x\in T simultaneously and multiplicatively up to 1+\varepsilon.
Edit intent: fluency
Original:  For T a subset of the unit sphere, \varepsilon\in(0,1/2) given, we study settings for m,s required to ensure \mathbb{E}_\Phi x\in T \left|\|\Phi x\|_2^2 - 1 \right| < \varepsilon , i.e.  so that \Phi preserves the norm of every x\in T simultaneously and multiplicatively up to 1+\varepsilon.
Revised:  For T a subset of the unit sphere, \varepsilon\in(0,1/2) given, we study settings for m,s required to ensure \mathbb{E}_\Phi x\in T \left|\|\Phi x\|_2^2 - 1 \right| < \varepsilon , i.e. \ so that \Phi preserves the norm of every x\in T simultaneously and multiplicatively up to 1+\varepsilon.
Edit intent: fluency
Original:  Our most general theorem shows that it suffices to set m \mathbin{\substack{<\\ *} \gamma_2^2(T) + 1 and s \substack{<\\ *} 1 as long as s,m additionally satisfy a certain tradeoff condition that is governed by }  the geometry of T (and as we show for several examples of interest, is easy to verify).
Revised:  } \gamma_2^2(T) + 1 and s \substack{<\\ *} 1 as long as s,m additionally satisfy a certain tradeoff condition that is governed by } the geometry of T (and as we show for several examples of interest, is easy to verify).
Edit intent: coherence
Original:  Here \gamma_2 is Talagrand's functional, and we write f \substack{<\\ * g to mean f }%DIFDELCMD < \le %%% Cg \cdot (\varepsilon^{-1\log n)^c for some constants C,c>0. }  Our result is a sparse \Phi analog of Gordon's theorem, which was concerned with  dense \Phi having i.i.d. gaussian entries.
Revised:  Here \gamma_2 is Talagrand's functional, and we write f \substack{<\\ * g to mean f }%DIFDELCMD < \le %%% Cg \cdot (\varepsilon^{-1\log n)^c for some constants C,c>0. } Our result is a sparse analog of Gordon's theorem, which was concerned with dense \Phi having i.i.d. gaussian entries.
Edit intent: fluency
Original:  Here \gamma_2 is Talagrand's functional, and we write f \substack{<\\ * g to mean f }%DIFDELCMD < \le %%% Cg \cdot (\varepsilon^{-1\log n)^c for some constants C,c>0. }  Our result is a sparse \Phi analog of Gordon's theorem, which was concerned with  dense \Phi having i.i.d. gaussian entries.
Revised:  Here \gamma_2 is Talagrand's functional, and we write f \substack{<\\ * g to mean f }%DIFDELCMD < \le %%% Cg \cdot (\varepsilon^{-1\log n)^c for some constants C,c>0. } Our result is a sparse \Phi analog of Gordon's theorem, which was concerned with a dense \Phi having i.i.d. gaussian entries.
Edit intent: fluency
Original:  For a subset T of the unit sphere, \varepsilon\in(0,1/2) given, we study settings for m,s required to ensure \mathbb{E}_\Phi x\in T \left|\|\Phi x\|_2^2 - 1 \right| < \varepsilon , i.e. \ so that \Phi preserves the norm of every x\in T simultaneously and multiplicatively up to 1+\varepsilon.
Revised:  For a subset T of the unit sphere, \varepsilon\in(0,1/2) given, we study settings for m,s required to ensure \mathbb{E}_\Phi x\in T \left|\|\Phi x\|_2^2 - 1 \right| < \varepsilon , i.e. so that \Phi preserves the norm of every x\in T simultaneously and multiplicatively up to 1+\varepsilon.
Edit intent: fluency