Pub Date : 2020-07-01DOI: 10.1080/17459737.2020.1777592
E. Amiot
There are many ways to define and measure organization, or complexity, in music, most using the notion of informational entropy, as the opposite of organization. Some researchers prompted me to study whether it could be done from the magnitudes of Fourier coefficients of musical objects (pc-sets or rhythms) instead of addressing their atomic elements (pitches, rhythmic onsets). Indeed I found that it could be a promising new approach to measuring organization of musical material. This note only purports to expose this novel idea, leaving for future research the task of comparing it with the numerous other definitions. I also sketch the study of one relevant basis for such comparisons which has been little explored, the asymptotics of entropy of arithmetic sequences modulo n.
{"title":"Entropy of Fourier coefficients of periodic musical objects","authors":"E. Amiot","doi":"10.1080/17459737.2020.1777592","DOIUrl":"https://doi.org/10.1080/17459737.2020.1777592","url":null,"abstract":"There are many ways to define and measure organization, or complexity, in music, most using the notion of informational entropy, as the opposite of organization. Some researchers prompted me to study whether it could be done from the magnitudes of Fourier coefficients of musical objects (pc-sets or rhythms) instead of addressing their atomic elements (pitches, rhythmic onsets). Indeed I found that it could be a promising new approach to measuring organization of musical material. This note only purports to expose this novel idea, leaving for future research the task of comparing it with the numerous other definitions. I also sketch the study of one relevant basis for such comparisons which has been little explored, the asymptotics of entropy of arithmetic sequences modulo n.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73832800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-27DOI: 10.1080/17459737.2020.1785568
David R. W. Sears, G. Widmer
Recurrent voice-leading patterns like the Mi-Re-Do compound cadence (MRDCC) rarely appear on the musical surface in complex polyphonic textures, so finding these patterns using computational methods remains a tremendous challenge. The present study extends the canonical n-gram approach by using skip-grams, which include sub-sequences in an n-gram list if their constituent members occur within a certain number of skips. We compiled four data sets of Western tonal music consisting of symbolic encodings of the notated score and a recorded performance, created a model pipeline for defining, counting, filtering, and ranking skip-grams, and ranked the position of the MRDCC in every possible model configuration. We found that the MRDCC receives a higher rank in the list when the pipeline employs 5 skips, filters the list by excluding n-gram types that do not reflect a genuine harmonic change between adjacent members, and ranks the remaining types using a statistical association measure.
{"title":"Beneath (or beyond) the surface: Discovering voice-leading patterns with skip-grams","authors":"David R. W. Sears, G. Widmer","doi":"10.1080/17459737.2020.1785568","DOIUrl":"https://doi.org/10.1080/17459737.2020.1785568","url":null,"abstract":"Recurrent voice-leading patterns like the Mi-Re-Do compound cadence (MRDCC) rarely appear on the musical surface in complex polyphonic textures, so finding these patterns using computational methods remains a tremendous challenge. The present study extends the canonical n-gram approach by using skip-grams, which include sub-sequences in an n-gram list if their constituent members occur within a certain number of skips. We compiled four data sets of Western tonal music consisting of symbolic encodings of the notated score and a recorded performance, created a model pipeline for defining, counting, filtering, and ranking skip-grams, and ranked the position of the MRDCC in every possible model configuration. We found that the MRDCC receives a higher rank in the list when the pipeline employs 5 skips, filters the list by excluding n-gram types that do not reflect a genuine harmonic change between adjacent members, and ranks the remaining types using a statistical association measure.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87031626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-29DOI: 10.1080/17459737.2020.1760953
Michael D. Fowler
In this article, I build an actantial model, , of John Cage's 1962 indeterminate work (4 33 No. 2). To further investigate Greimas' actantial axes of desire, knowledge and power, I generate an ontology, , that records the facts of the model through hierarchies of relations and concepts. This allows for a conceptual graph (CG), , that describes the score's instructions and its actants. Extracting subgraphs of then allows for reasoned arguments about the implications of Cage's instructions in the score, and in particular, the composer's reference to “an obligation to others.” Through a conceptual graph rule , I offer a framework for generating the structure of a score-informed interpretation of (4 33 No. 2) that is based on a number of key conditionals that map the actants of the piece and their relation to the unfolding of the work's narrative.
{"title":"From actantial model to conceptual graph: Thematized action in John Cage's 0′00′(4′33′′No. 2)","authors":"Michael D. Fowler","doi":"10.1080/17459737.2020.1760953","DOIUrl":"https://doi.org/10.1080/17459737.2020.1760953","url":null,"abstract":"In this article, I build an actantial model, , of John Cage's 1962 indeterminate work (4 33 No. 2). To further investigate Greimas' actantial axes of desire, knowledge and power, I generate an ontology, , that records the facts of the model through hierarchies of relations and concepts. This allows for a conceptual graph (CG), , that describes the score's instructions and its actants. Extracting subgraphs of then allows for reasoned arguments about the implications of Cage's instructions in the score, and in particular, the composer's reference to “an obligation to others.” Through a conceptual graph rule , I offer a framework for generating the structure of a score-informed interpretation of (4 33 No. 2) that is based on a number of key conditionals that map the actants of the piece and their relation to the unfolding of the work's narrative.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74099713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-24DOI: 10.1080/17459737.2020.1763488
Peter beim Graben, Maria Mannone
How can discrete pitches and chords emerge from the continuum of sound? Using a quantum cognition model of tonal music, we prove that the associated Schrödinger equation in Fourier space is invariant under continuous pitch transpositions. However, this symmetry is broken in the case of transpositions of chords, entailing a discrete cyclic group as transposition symmetry. Our research relates quantum mechanics with music and is consistent with music theory and seminal insights by Hermann von Helmholtz.
{"title":"Musical pitch quantization as an eigenvalue problem","authors":"Peter beim Graben, Maria Mannone","doi":"10.1080/17459737.2020.1763488","DOIUrl":"https://doi.org/10.1080/17459737.2020.1763488","url":null,"abstract":"How can discrete pitches and chords emerge from the continuum of sound? Using a quantum cognition model of tonal music, we prove that the associated Schrödinger equation in Fourier space is invariant under continuous pitch transpositions. However, this symmetry is broken in the case of transpositions of chords, entailing a discrete cyclic group as transposition symmetry. Our research relates quantum mechanics with music and is consistent with music theory and seminal insights by Hermann von Helmholtz.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83952727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-03DOI: 10.1080/17459737.2020.1786745
Mattia G. Bergomi, A. Baratè
Meaningful low-dimensional representations of dynamical processes are essential to better understand the mechanisms underlying complex systems, from music composition to learning in both biological and artificial intelligence. We suggest to describe time-varying systems by considering the evolution of their geometrical and topological properties in time, by using a method based on persistent homology. In the static case, persistent homology allows one to provide a representation of a manifold paired with a continuous function as a collection of multisets of points and lines called persistence diagrams. The idea is to fingerprint the change of a variable-geometry space as a time series of persistence diagrams, and afterwards compare such time series by using dynamic time warping. As an application, we express some music features and their time dependency by updating the values of a function defined on a polyhedral surface, called the Tonnetz. Thereafter, we use this time-based representation to automatically classify three collections of compositions according to their style, and discuss the optimal time-granularity for the analysis of different musical genres.
{"title":"Homological persistence in time series: an application to music classification","authors":"Mattia G. Bergomi, A. Baratè","doi":"10.1080/17459737.2020.1786745","DOIUrl":"https://doi.org/10.1080/17459737.2020.1786745","url":null,"abstract":"Meaningful low-dimensional representations of dynamical processes are essential to better understand the mechanisms underlying complex systems, from music composition to learning in both biological and artificial intelligence. We suggest to describe time-varying systems by considering the evolution of their geometrical and topological properties in time, by using a method based on persistent homology. In the static case, persistent homology allows one to provide a representation of a manifold paired with a continuous function as a collection of multisets of points and lines called persistence diagrams. The idea is to fingerprint the change of a variable-geometry space as a time series of persistence diagrams, and afterwards compare such time series by using dynamic time warping. As an application, we express some music features and their time dependency by updating the values of a function defined on a polyhedral surface, called the Tonnetz. Thereafter, we use this time-based representation to automatically classify three collections of compositions according to their style, and discuss the optimal time-granularity for the analysis of different musical genres.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91326696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-03DOI: 10.1080/17459737.2020.1799563
Dmitri Tymoczko
Music theorists have modeled voice leadings as paths through higher-dimensional configuration spaces. This paper uses topological techniques to construct two-dimensional diagrams capturing these spaces’ most important features. The goal is to enrich set theory’s contrapuntal power by simplifying the description of its geometry. Along the way, I connect homotopy theory to “transformational theory,” show how set-class space generalizes the neo-Riemannian transformations, extend the Tonnetz to arbitrary chords, and develop a simple contrapuntal “alphabet” for describing voice leadings. I mention several compositional applications and analyze short excerpts from Gesualdo, Mozart, Wagner, Stravinsky, Schoenberg, Schnittke, and Mahanthappa.
{"title":"Why topology?","authors":"Dmitri Tymoczko","doi":"10.1080/17459737.2020.1799563","DOIUrl":"https://doi.org/10.1080/17459737.2020.1799563","url":null,"abstract":"Music theorists have modeled voice leadings as paths through higher-dimensional configuration spaces. This paper uses topological techniques to construct two-dimensional diagrams capturing these spaces’ most important features. The goal is to enrich set theory’s contrapuntal power by simplifying the description of its geometry. Along the way, I connect homotopy theory to “transformational theory,” show how set-class space generalizes the neo-Riemannian transformations, extend the Tonnetz to arbitrary chords, and develop a simple contrapuntal “alphabet” for describing voice leadings. I mention several compositional applications and analyze short excerpts from Gesualdo, Mozart, Wagner, Stravinsky, Schoenberg, Schnittke, and Mahanthappa.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73705497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-30DOI: 10.1080/17459737.2020.1753122
Caleb Mutch
Connections between mathematics and music have been recognized since the days of Ancient Greece. The Pythagoreans' association of musical intervals with integer ratios is so well known that it occludes the great variety of approaches to the music-mathematical relationship in Ancient Greece and Rome. The present article uncovers this diversity by examining how authors from Antiquity used one mathematical element – the number series 16, 17, 18 – to serve different ends. The number 17 provides the simplest way to divide the 9:8 whole tone into two parts, but the status of those two parts was debated, as was their relationship to the standard 256:243 semitone of Greek theory. I account for this diversity by appealing to the contexts in which the authors wrote – music treatises vs. commentaries on Plato's Timaeus – and to the importance placed on mathematics in Neoplatonist curricula. The article concludes by examining how confusion regarding the 16, 17, 18 series lingered even in the medieval period due to ambiguities in Boethius's De institutione musica, and how medieval authors eventually superseded the debate through a Euclid-inspired geometric division of the 9:8 ratio.
{"title":"Mathematical approaches to defining the semitone in antiquity","authors":"Caleb Mutch","doi":"10.1080/17459737.2020.1753122","DOIUrl":"https://doi.org/10.1080/17459737.2020.1753122","url":null,"abstract":"Connections between mathematics and music have been recognized since the days of Ancient Greece. The Pythagoreans' association of musical intervals with integer ratios is so well known that it occludes the great variety of approaches to the music-mathematical relationship in Ancient Greece and Rome. The present article uncovers this diversity by examining how authors from Antiquity used one mathematical element – the number series 16, 17, 18 – to serve different ends. The number 17 provides the simplest way to divide the 9:8 whole tone into two parts, but the status of those two parts was debated, as was their relationship to the standard 256:243 semitone of Greek theory. I account for this diversity by appealing to the contexts in which the authors wrote – music treatises vs. commentaries on Plato's Timaeus – and to the importance placed on mathematics in Neoplatonist curricula. The article concludes by examining how confusion regarding the 16, 17, 18 series lingered even in the medieval period due to ambiguities in Boethius's De institutione musica, and how medieval authors eventually superseded the debate through a Euclid-inspired geometric division of the 9:8 ratio.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79120564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-17DOI: 10.1080/17459737.2020.1732490
R. Gómez, L. Nasser
We address the broad idea of using mathematical models to inform music theory and composition by implementing them directly in the process of music creation. The mathematical model we will use is the Thue–Morse dynamical system. We briefly survey previously published works that were similarly motivated, and then discuss a new piece of music we composed, inspired by this symbolic dynamical system. In the course of our analysis, we also present an alternative proof of the well-know fact that the Thue–Morse subshift has topological entropy zero that motivated us to think of the map between binary sequences and musical scales as “binary keyboards.” Indeed, the binary representation allows to study musical scales through mathematical properties of the sequences that define them; here we present the set of standard Thue–Morse scales and compare them with other well-known musical scales.
{"title":"Symbolic structures in music theory and composition, binary keyboards, and the Thue–Morse shift","authors":"R. Gómez, L. Nasser","doi":"10.1080/17459737.2020.1732490","DOIUrl":"https://doi.org/10.1080/17459737.2020.1732490","url":null,"abstract":"We address the broad idea of using mathematical models to inform music theory and composition by implementing them directly in the process of music creation. The mathematical model we will use is the Thue–Morse dynamical system. We briefly survey previously published works that were similarly motivated, and then discuss a new piece of music we composed, inspired by this symbolic dynamical system. In the course of our analysis, we also present an alternative proof of the well-know fact that the Thue–Morse subshift has topological entropy zero that motivated us to think of the map between binary sequences and musical scales as “binary keyboards.” Indeed, the binary representation allows to study musical scales through mathematical properties of the sequences that define them; here we present the set of standard Thue–Morse scales and compare them with other well-known musical scales.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81577276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-05DOI: 10.1080/17459737.2020.1726690
R. Cubarsi
Abstract scales are formalized as a cyclic group of classes of projection functions related to iterations of the scale generator. Their representatives in the frequency domain are used to built cyclic sequences of tone iterates satisfying the closure condition. The refinement of cyclic sequences with regard to the best closure provides a constructive algorithm that allows to determine cyclic scales avoiding continued fractions. New proofs of the main properties are obtained as a consequence of the generating procedure. When the scale tones are generated from the two elementary factors associated with the generic widths of the step intervals we get the partition of the octave leading to the fundamental Bézout's identity relating several characteristic scale indices. This relationship is generalized to prove a new relationship expressing the partition that the frequency ratios associated with the two sizes composing the different step-intervals induce to a specific set of octaves.
{"title":"An alternative approach to generalized Pythagorean scales. Generation and properties derived in the frequency domain","authors":"R. Cubarsi","doi":"10.1080/17459737.2020.1726690","DOIUrl":"https://doi.org/10.1080/17459737.2020.1726690","url":null,"abstract":"Abstract scales are formalized as a cyclic group of classes of projection functions related to iterations of the scale generator. Their representatives in the frequency domain are used to built cyclic sequences of tone iterates satisfying the closure condition. The refinement of cyclic sequences with regard to the best closure provides a constructive algorithm that allows to determine cyclic scales avoiding continued fractions. New proofs of the main properties are obtained as a consequence of the generating procedure. When the scale tones are generated from the two elementary factors associated with the generic widths of the step intervals we get the partition of the octave leading to the fundamental Bézout's identity relating several characteristic scale indices. This relationship is generalized to prove a new relationship expressing the partition that the frequency ratios associated with the two sizes composing the different step-intervals induce to a specific set of octaves.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84580787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-02DOI: 10.1080/17459737.2020.1725667
Jason Yust
The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the 12 pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. This article examines how these duplications can relate to important musical concepts such as key or pitch height, and details a method of removing such redundancies and the resulting changes to the homology of the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth chords, scalar tetrachords, scales, etc., as well as Zeitnetze on common cyclic rhythms or timelines. Their different topologies – whether orientable, bounded, manifold, etc. – reveal some of the topological character of musical concepts.
{"title":"Generalized Tonnetze and Zeitnetze, and the topology of music concepts","authors":"Jason Yust","doi":"10.1080/17459737.2020.1725667","DOIUrl":"https://doi.org/10.1080/17459737.2020.1725667","url":null,"abstract":"The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the 12 pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. This article examines how these duplications can relate to important musical concepts such as key or pitch height, and details a method of removing such redundancies and the resulting changes to the homology of the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth chords, scalar tetrachords, scales, etc., as well as Zeitnetze on common cyclic rhythms or timelines. Their different topologies – whether orientable, bounded, manifold, etc. – reveal some of the topological character of musical concepts.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76161633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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