wanyu/IteraTeR_v2 huggingface dataset:
Original:  (It is known that \max_{\|x\|=1 f(x) }%DIFDELCMD < \lesssim %%% \sqrt{n\,q\,\log q - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \]  When f is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% (\frac{n^{3/4}{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% (\frac{n{\sqrt{q}}) respectively.
Revised:  f(x) }%DIFDELCMD < \lesssim %%% \sqrt{n\,q\,\log q - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \] When f is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% (\frac{n^{3/4}{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% (\frac{n{\sqrt{q}}) respectively.
Edit intent: coherence
Original:  (It is known that \max_{\|x\|=1 f(x) }%DIFDELCMD < \lesssim %%% \sqrt{n\,q\,\log q - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \]  When f is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% (\frac{n^{3/4}{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% (\frac{n{\sqrt{q}}) respectively.
Revised:  (It is known that \max_{\|x\|=1 f(x) }%DIFDELCMD < \lesssim %%% - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \] When f is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% (\frac{n^{3/4}{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% (\frac{n{\sqrt{q}}) respectively.
Edit intent: coherence
Original:  (It is known that \max_{\|x\|=1 f(x) }%DIFDELCMD < \lesssim %%% \sqrt{n\,q\,\log q - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \]  When f is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% (\frac{n^{3/4}{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% (\frac{n{\sqrt{q}}) respectively.
Revised:  (It is known that \max_{\|x\|=1 f(x) }%DIFDELCMD < \lesssim %%% \sqrt{n\,q\,\log q - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \] - When f is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% (\frac{n^{3/4}{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% (\frac{n{\sqrt{q}}) respectively.
Edit intent: fluency
Original:  (It is known that \max_{\|x\|=1 f(x) }%DIFDELCMD < \lesssim %%% \sqrt{n\,q\,\log q - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \]  When f is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% (\frac{n^{3/4}{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% (\frac{n{\sqrt{q}}) respectively.
Revised:  (It is known that \max_{\|x\|=1 f(x) }%DIFDELCMD < \lesssim %%% \sqrt{n\,q\,\log q - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \] When A is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% (\frac{n^{3/4}{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% (\frac{n{\sqrt{q}}) respectively.
Edit intent: clarity
Original:  (It is known that \max_{\|x\|=1 f(x) }%DIFDELCMD < \lesssim %%% \sqrt{n\,q\,\log q - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \]  When f is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% (\frac{n^{3/4}{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% (\frac{n{\sqrt{q}}) respectively.
Revised:  (It is known that \max_{\|x\|=1 f(x) }%DIFDELCMD < \lesssim %%% \sqrt{n\,q\,\log q - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \] When f is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% }{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% (\frac{n{\sqrt{q}}) respectively.
Edit intent: fluency
Original:  (It is known that \max_{\|x\|=1 f(x) }%DIFDELCMD < \lesssim %%% \sqrt{n\,q\,\log q - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \]  When f is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% (\frac{n^{3/4}{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% (\frac{n{\sqrt{q}}) respectively.
Revised:  (It is known that \max_{\|x\|=1 f(x) }%DIFDELCMD < \lesssim %%% \sqrt{n\,q\,\log q - \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \] When f is a random degree-3 or 4 polynomial , we prove that with high probability, the degree-q SoS-Value is at most%DIFDELCMD < \tilde{O}%%% (\frac{n^{3/4}{q^{1/4}}), }%DIFDELCMD < \tilde{O}%%% {\sqrt{q}}) respectively.
Edit intent: fluency
Original: Unambiguous automata , i.e., nondeterministic automata with the restriction of having at most one accepting run over a word, have the potential to be used instead of deterministic automata in settings where nondeterministic automata can not be applied in general .
Revised: Unambiguous automata are nondeterministic automata in which every word has at most one accepting run over a word, have the potential to be used instead of deterministic automata in settings where nondeterministic automata can not be applied in general .
Edit intent: clarity
Original: Unambiguous automata , i.e., nondeterministic automata with the restriction of having at most one accepting run over a word, have the potential to be used instead of deterministic automata in settings where nondeterministic automata can not be applied in general .
Revised: Unambiguous automata , i.e., nondeterministic automata with the restriction of having at most one accepting run .
Edit intent: clarity
Original:  In this paper , we provide a polynomially time-bounded algorithm for probabilistic model checking of discrete-time Markov chains against unambiguous B\"uchi automataspecifications and report on our implementation and experiments .
Revised:  In this paper we give a polynomial-time algorithm for model checking discrete-time Markov chains against unambiguous B\"uchi automataspecifications and report on our implementation and experiments .
Edit intent: clarity
Original: We consider a number of graph kernels and proximity measures : commute time kernel, regularized Laplacian kernel, heat kernel, communicability , etc., and the corresponding distances as applied to clustering nodes in random graphs  .
Revised: We consider a number of graph kernels and proximity measures including commute time kernel, regularized Laplacian kernel, heat kernel, communicability , etc., and the corresponding distances as applied to clustering nodes in random graphs .
Edit intent: clarity
Original:  It turns out that in most cases, logarithmic measures (i.e., measures resulting after taking logarithm of the proximities) perform much better while distinguishing  classes than the "plain" measures.
Revised:  It turns out that in most cases, logarithmic measures (i.e., measures resulting after taking logarithm of the proximities) perform better while distinguishing classes than the "plain" measures.
Edit intent: clarity
Original:  A possible explanation of this fact is that most kernels have a multiplicative nature, while the nature of distances used in cluster algorithms is an additive one (cf. the triangle inequality).
Revised:  A similar conclusion can be made for several well-known datasets. A possible origin of this effect is that most kernels have a multiplicative nature, while the nature of distances used in cluster algorithms is an additive one (cf. the triangle inequality).
Edit intent: clarity
Original:  The logarithmic transformation is just a tool to transform one nature to another . Moreover, some distances corresponding to the logarithmic measures possess a meaningful cutpoint additivity property.
Revised:  The logarithmic transformation is a tool to transform one nature to another . Moreover, some distances corresponding to the logarithmic measures possess a meaningful cutpoint additivity property.
Edit intent: coherence
Original:  The logarithmic transformation is just a tool to transform one nature to another . Moreover, some distances corresponding to the logarithmic measures possess a meaningful cutpoint additivity property.
Revised:  The logarithmic transformation is just a tool to transform the first nature to the second one . Moreover, some distances corresponding to the logarithmic measures possess a meaningful cutpoint additivity property.
Edit intent: clarity
Original: A method is proposed to identify target states that optimize a metastability index amongst a set of trial states and use these target states as milestones  to build Markov State Models  .
Revised: A method is proposed to identify target states that optimize a metastability index amongst a set of trial states and use these target states as milestones to build Markov State Models (MSMs) .
Edit intent: clarity
Original:  If the optimized metastability index is small, this automatically guarantees the accuracy of the MSM  in the sense that the transitions between the target milestones is indeed approximately Markovian.
Revised:  If the optimized metastability index is small, this automatically guarantees the accuracy of the MSM , in the sense that the transitions between the target milestones is indeed approximately Markovian.
Edit intent: fluency
Original:  The process often requires not only to choose pertinent ontologies, but also to develop new models in order to preserve the maximum of information and to express the semantic power of cultural heritage data.
Revised:  The process often requires not only to choose pertinent ontologies, but also to develop new models that preserve the maximum of information and to express the semantic power of cultural heritage data.
Edit intent: coherence
Original:  The process often requires not only to choose pertinent ontologies, but also to develop new models in order to preserve the maximum of information and to express the semantic power of cultural heritage data.
Revised:  The process often requires not only to choose pertinent ontologies, but also to develop new models in order to preserve the maximum of information and express the semantic power of cultural heritage data.
Edit intent: fluency
Original:  Data managed in archives, libraries and museums are themselves complex objects that need a deep reflection on even non conventional conceptual models.
Revised:  Indeed, data managed in archives, libraries and museums are themselves complex objects that need a deep reflection on even non conventional conceptual models.
Edit intent: coherence
Original:  Data managed in archives, libraries and museums are themselves complex objects that need a deep reflection on even non conventional conceptual models.
Revised:  Data managed in archives, libraries and museums are complex objects themselves, that need a deep reflection on even non conventional conceptual models.
Edit intent: fluency
Original:  Starting from these considerations, this paper describes a research project: to expose the richness of one of the most important collections in the European cultural heritage scenario , the Zeri Photo Archive , as Linked Open Data.
Revised:  Starting from these considerations, this paper describes a research project: to expose as Linked Open Data the vastness of one of the most important collections in the European cultural heritage scenario , the Zeri Photo Archive , as Linked Open Data.
Edit intent: clarity
Original:  Starting from these considerations, this paper describes a research project: to expose the richness of one of the most important collections in the European cultural heritage scenario , the Zeri Photo Archive , as Linked Open Data.
Revised:  Starting from these considerations, this paper describes a research project: to expose the richness of one of the most important collections of European cultural heritage scenario , the Zeri Photo Archive , as Linked Open Data.
Edit intent: clarity
Original:  Starting from these considerations, this paper describes a research project: to expose the richness of one of the most important collections in the European cultural heritage scenario , the Zeri Photo Archive , as Linked Open Data.
Revised:  Starting from these considerations, this paper describes a research project: to expose the richness of one of the most important collections in the European cultural heritage , the Zeri Photo Archive , as Linked Open Data.
Edit intent: clarity
Original:  Required steps for reaching this aim are here described: the mapping of the descriptive elements really used in the existing Zeri Photo Archive catalog ( based on the Italian content standards Scheda F and Scheda OA) to RDF;
Revised:  Required steps for reaching this aim are here described: the mapping of the descriptive elements really used in the current Zeri Photo Archive catalog ( based on the Italian content standards Scheda F and Scheda OA) to RDF;
Edit intent: clarity
Original:  Finally, in order to provide a result capable to give consistency to the complexity of such scenario, a first RDF dataset (both open and linked) was created, according to the models developed .
Revised:  Finally, in order to provide a result capable to demonstrate the complexity of such scenario, a first RDF dataset (both open and linked) was created, according to the models developed .
Edit intent: clarity
Original:  Finally, in order to provide a result capable to give consistency to the complexity of such scenario, a first RDF dataset (both open and linked) was created, according to the models developed .
Revised:  Finally, in order to provide a result capable to give consistency to the complexity of our scenario .
Edit intent: clarity
Original: The description of cultural heritage objects in the Linked Open Data (LOD) perspective is not a trivial task.
Revised: Describing cultural heritage objects in the Linked Open Data (LOD) perspective is not a trivial task.
Edit intent: fluency
Original: The description of cultural heritage objects in the Linked Open Data (LOD) perspective is not a trivial task.
Revised: The description of cultural heritage objects from the perspective of Linked Open Data (LOD) perspective is not a trivial task.
Edit intent: clarity
Original: The description of cultural heritage objects in the Linked Open Data (LOD) perspective is not a trivial task.
Revised: The description of cultural heritage objects in the Linked Open Data (LOD) is not a trivial task.
Edit intent: clarity
Original:  The process often requires not only to choose pertinent ontologies, but also to develop new models that preserve the maximum of information and express the semantic power of cultural heritage data.
Revised:  The process often requires not only choosing pertinent ontologies, but also to develop new models that preserve the maximum of information and express the semantic power of cultural heritage data.
Edit intent: fluency
Original:  The process often requires not only to choose pertinent ontologies, but also to develop new models that preserve the maximum of information and express the semantic power of cultural heritage data.
Revised:  The process often requires not only to choose pertinent ontologies, but also developing new models that preserve the maximum of information and express the semantic power of cultural heritage data.
Edit intent: fluency
Original:  The process often requires not only to choose pertinent ontologies, but also to develop new models that preserve the maximum of information and express the semantic power of cultural heritage data.
Revised:  The process often requires not only to choose pertinent ontologies, but also to develop new models that preserve the most information and express the semantic power of cultural heritage data.
Edit intent: clarity
Original:  Indeed, data managed in archives, libraries and museums are complex objects themselves, that need a deep reflection on even non conventional conceptual models.
Revised:  Indeed, data managed in archives, libraries and museums are complex objects themselves, which require a deep reflection on even non conventional conceptual models.
Edit intent: fluency
Original:  Indeed, data managed in archives, libraries and museums are complex objects themselves, that need a deep reflection on even non conventional conceptual models.
Revised:  Indeed, data managed in archives, libraries and museums are complex objects themselves, that need a deep reflection on even non-conventional conceptual models.
Edit intent: fluency
Original:  Starting from these considerations, this paper describes a research project: to expose as Linked Open Data the vastness of one of the most important collections of European cultural heritage, the Zeri Photo Archive  .
Revised:  Starting from these considerations, this paper describes a research project: to expose the vastness of one of the most important collections of European cultural heritage, the Zeri Photo Archive .
Edit intent: clarity
Original:  Starting from these considerations, this paper describes a research project: to expose as Linked Open Data the vastness of one of the most important collections of European cultural heritage, the Zeri Photo Archive  .
Revised:  Starting from these considerations, this paper describes a research project: to expose as Linked Open Data the vastness of one of the most important collections of European cultural heritage, the Zeri Photo Archive , as Linked Open Data .
Edit intent: fluency
Original:   we mapped into RDF the descriptive elements used in the current Zeri Photo Archive catalog into RDF , converting into CIDOC-CRM , SPAR Ontologies and the two new aforementioned models the source data based on the Italian content standards Scheda F (Photography Entry, in English) and Scheda OA (Work of art Entry, in English) .
Revised:  then we mapped into RDF the descriptive elements used in the current Zeri Photo Archive catalog into RDF , converting into CIDOC-CRM , SPAR Ontologies and the two new aforementioned models the source data based on the Italian content standards Scheda F (Photography Entry, in English) and Scheda OA (Work of art Entry, in English) .
Edit intent: fluency
Original:   we mapped into RDF the descriptive elements used in the current Zeri Photo Archive catalog into RDF , converting into CIDOC-CRM , SPAR Ontologies and the two new aforementioned models the source data based on the Italian content standards Scheda F (Photography Entry, in English) and Scheda OA (Work of art Entry, in English) .
Revised:  we mapped into RDF the descriptive elements used in the current Zeri Photo Archive catalog , converting into CIDOC-CRM , SPAR Ontologies and the two new aforementioned models the source data based on the Italian content standards Scheda F (Photography Entry, in English) and Scheda OA (Work of art Entry, in English) .
Edit intent: coherence
Original:   we mapped into RDF the descriptive elements used in the current Zeri Photo Archive catalog into RDF , converting into CIDOC-CRM , SPAR Ontologies and the two new aforementioned models the source data based on the Italian content standards Scheda F (Photography Entry, in English) and Scheda OA (Work of art Entry, in English) .
Revised:  we mapped into RDF the descriptive elements used in the current Zeri Photo Archive catalog into RDF , converting into CIDOC-CRM and into the two new aforementioned models the source data based on the Italian content standards Scheda F (Photography Entry, in English) and Scheda OA (Work of art Entry, in English) .
Edit intent: coherence
Original:   we mapped into RDF the descriptive elements used in the current Zeri Photo Archive catalog into RDF , converting into CIDOC-CRM , SPAR Ontologies and the two new aforementioned models the source data based on the Italian content standards Scheda F (Photography Entry, in English) and Scheda OA (Work of art Entry, in English) .
Revised:  we mapped into RDF the descriptive elements used in the current Zeri Photo Archive catalog into RDF , converting into CIDOC-CRM , SPAR Ontologies and the two new aforementioned models the source data based on the Italian content standards Scheda F (Photography Entry, in English) and Scheda OA (Work of Art Entry, in English) .
Edit intent: coherence
Original:  Finally , we created an RDF dataset of the output of the mapping that could show a result capable to demonstrate the complexity of our scenario.
Revised:  Finally , we created an RDF dataset of the output of the mapping that could show a result capable of demonstrating the complexity of our scenario.
Edit intent: fluency
Original:  Since the TF binding preferences ultimately define the  target genes, the motif family organization entails information about the structure of transcriptional regulation as it has been shaped by evolution.
Revised:  Since the TF binding preferences ultimately define the TF target genes, the motif family organization entails information about the structure of transcriptional regulation as it has been shaped by evolution.
Edit intent: fluency
Original:  One way to restrict instances of k-SAT is to limit the number of times a variable can occur .
Revised:  One way to restrict instances of k-SAT is to limit the number of times a variable can be occurred .
Edit intent: clarity
Original:  It was shown that for an instance of 4-SAT with the property that every literal appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for literals, such that every clause contains exactly 2 true literals and 2 false literals is NP-complete.
Revised:  It was shown that for an instance of 4-SAT with the property that every literal appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for variables such that every clause contains exactly 2 true literals and 2 false literals is NP-complete.
Edit intent: clarity
Original:  It was shown that for an instance of 4-SAT with the property that every literal appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for literals, such that every clause contains exactly 2 true literals and 2 false literals is NP-complete.
Revised:  It was shown that for an instance of 4-SAT with the property that every literal appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for literals, such that every clause contains exactly two true variables and two false variables is NP-complete.
Edit intent: clarity
Original:  In this work, we show that deciding the satisfiability of 3-SAT with the property that every literal appears in exactly 4 clauses (2 time negated and 2 times not negated),  is NP -complete.
Revised:  In this work, we show that deciding the satisfiability of 3-SAT with the property that every literal appears in exactly four clauses (two times negated and two times not negated), is NP -complete.
Edit intent: fluency
Original:  For a nonempty regular graph G = (V,E)  , it was asked by \mbox{ dehhh  to determine whether for a given independent set T , there is an independent dominating set D for T such that T \cap D =\varnothing ?
Revised:  For a nonempty regular graph G = (V,E) , it was asked in to determine whether for a given independent set T , there is an independent dominating set D for T such that T \cap D =\varnothing ?
Edit intent: clarity
Original:  For a nonempty regular graph G = (V,E)  , it was asked by \mbox{ dehhh  to determine whether for a given independent set T , there is an independent dominating set D for T such that T \cap D =\varnothing ?
Revised:  For a nonempty regular graph G = (V,E) , it was asked by \mbox{ dehhh to determine whether for a given independent set T there is an independent dominating set D for T such that T \cap D =\varnothing ?
Edit intent: fluency
Original:  As an application of (2/2/3)-SAT problem , we show that for every r\geq 3, this problem is \mathbf{NP} -complete.
Revised:  As an application of (2/2/3)-SAT problem we show that for every r\geq 3, this problem is \mathbf{NP} -complete.
Edit intent: fluency