Pub Date : 2021-01-19DOI: 10.1080/17459737.2021.1871788
B. Limkar, G. Chandekar
Operational Modal Analysis (OMA) of Tibetan singing bowl is performed to extract natural frequencies and mode shapes without measuring excitation data. It is kept free on a rigid surface, which is a common way of playing this musical instrument. OMA results are validated using Experimental Modal Analysis (EMA) and Numerical Methods using FEA. Numerical simulations using ANSYS® software establishes a benchmark for EMA results. The input and response data for 144 response points are collected using instrumented hammer and accelerometer, connected to a four-channel FFT analyser. A self-generated MATLAB® code processes the response signals for EMA and OMA. For natural frequencies, the absolute error lies within 6%, except for the first mode. For mode shapes, the Modal Assurance Criteria (MAC) value is more than 70%, except for the fourth mode. Thus, OMA is the best available method compared to the EMA and Numerical method using FEA for structural analysis under actual performance conditions.
{"title":"Structural dynamic analysis of a musical instrument: Tibetan singing bowl","authors":"B. Limkar, G. Chandekar","doi":"10.1080/17459737.2021.1871788","DOIUrl":"https://doi.org/10.1080/17459737.2021.1871788","url":null,"abstract":"Operational Modal Analysis (OMA) of Tibetan singing bowl is performed to extract natural frequencies and mode shapes without measuring excitation data. It is kept free on a rigid surface, which is a common way of playing this musical instrument. OMA results are validated using Experimental Modal Analysis (EMA) and Numerical Methods using FEA. Numerical simulations using ANSYS® software establishes a benchmark for EMA results. The input and response data for 144 response points are collected using instrumented hammer and accelerometer, connected to a four-channel FFT analyser. A self-generated MATLAB® code processes the response signals for EMA and OMA. For natural frequencies, the absolute error lies within 6%, except for the first mode. For mode shapes, the Modal Assurance Criteria (MAC) value is more than 70%, except for the fourth mode. Thus, OMA is the best available method compared to the EMA and Numerical method using FEA for structural analysis under actual performance conditions.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77768155","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 : 2021-01-02DOI: 10.1080/17459737.2020.1716404
Reinhard Blutner, Peter beim Graben
Metaphors involving motion and forces are a source of inspiration for understanding tonal music and tonal harmonies since ancient times. Starting with the rise of quantum cognition, the modern interactional conception of forces as developed in gauge theory has recently entered the field of theoretical musicology. We develop a gauge model of tonal attraction based on SU(2) symmetry. This model comprises two earlier attempts, the phase model grounded on U(1) gauge symmetry, and the spatial deformation model derived from SO(2) gauge symmetry. In the neutral, force-free case both submodels agree and generate the same predictions as a simple qubit approach. However, there are several differences in the force-driven case. It is claimed that the deformation model gives a proper description of static tonal attraction. The full model combines the deformation model with the phase model through SU(2) gauge symmetry and unifies static and dynamic tonal attraction.
{"title":"Gauge models of musical forces","authors":"Reinhard Blutner, Peter beim Graben","doi":"10.1080/17459737.2020.1716404","DOIUrl":"https://doi.org/10.1080/17459737.2020.1716404","url":null,"abstract":"Metaphors involving motion and forces are a source of inspiration for understanding tonal music and tonal harmonies since ancient times. Starting with the rise of quantum cognition, the modern interactional conception of forces as developed in gauge theory has recently entered the field of theoretical musicology. We develop a gauge model of tonal attraction based on SU(2) symmetry. This model comprises two earlier attempts, the phase model grounded on U(1) gauge symmetry, and the spatial deformation model derived from SO(2) gauge symmetry. In the neutral, force-free case both submodels agree and generate the same predictions as a simple qubit approach. However, there are several differences in the force-driven case. It is claimed that the deformation model gives a proper description of static tonal attraction. The full model combines the deformation model with the phase model through SU(2) gauge symmetry and unifies static and dynamic tonal attraction.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88653624","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 : 2020-12-08DOI: 10.1080/17459737.2021.1979116
Tom Goodman, Karoline van Gemst, P. Tiňo
This paper presents a geometric approach to pitch estimation (PE) – an important problem in music information retrieval (MIR), and a precursor to a variety of other problems in the field. Though there exist a number of highly accurate methods, both mono-pitch estimation and multi-pitch estimation (particularly with unspecified polyphonic timbre) prove computationally and conceptually challenging. A number of current techniques, while incredibly effective, are not targeted towards eliciting the underlying mathematical structures that underpin the complex musical patterns exhibited by acoustic musical signals. Tackling the approach from both theoretical and experimental perspectives, we present a novel framework, a basis for further work in the area, and results that (while not state of the art) demonstrate relative efficacy. The framework presented in this paper opens up a completely new way to tackle PE problems and may have uses both in traditional analytical approaches as well as in the emerging machine learning (ML) methods that currently dominate the literature.
{"title":"A geometric framework for pitch estimation on acoustic musical signals","authors":"Tom Goodman, Karoline van Gemst, P. Tiňo","doi":"10.1080/17459737.2021.1979116","DOIUrl":"https://doi.org/10.1080/17459737.2021.1979116","url":null,"abstract":"This paper presents a geometric approach to pitch estimation (PE) – an important problem in music information retrieval (MIR), and a precursor to a variety of other problems in the field. Though there exist a number of highly accurate methods, both mono-pitch estimation and multi-pitch estimation (particularly with unspecified polyphonic timbre) prove computationally and conceptually challenging. A number of current techniques, while incredibly effective, are not targeted towards eliciting the underlying mathematical structures that underpin the complex musical patterns exhibited by acoustic musical signals. Tackling the approach from both theoretical and experimental perspectives, we present a novel framework, a basis for further work in the area, and results that (while not state of the art) demonstrate relative efficacy. The framework presented in this paper opens up a completely new way to tackle PE problems and may have uses both in traditional analytical approaches as well as in the emerging machine learning (ML) methods that currently dominate the literature.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90140050","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 : 2020-11-22DOI: 10.1080/17459737.2020.1836687
A. Popoff, Jason Yust
This paper develops a framework based on category theory which unifies the simultaneous consideration of timepoints, metrical relations, and meter inclusion founded on the category of sets and binary relations. Metrical relations are defined as binary relations on the set of timepoints, and the subsequent use of the monoid they generate and of the corresponding functor to allows us to define meter networks, i.e. networks of timepoints (or sets of timepoints) related by metrical relations. We compare this to existing theories of metrical conflict, such as those of Harald Krebs and Richard Cohn, and illustrate that these tools help to more effectively combine displacement and grouping dissonance and reflect analytical claims concerning nineteenth-century examples of complex hemiola and twentieth-century polymeter. We show that meter networks can be transformed into each other through meter network morphisms, which allows us to describe both meter displacements and meter inclusions. These networks are applied to various examples from the nineteenth and twentieth century.
{"title":"Meter networks: a categorical framework for metrical analysis","authors":"A. Popoff, Jason Yust","doi":"10.1080/17459737.2020.1836687","DOIUrl":"https://doi.org/10.1080/17459737.2020.1836687","url":null,"abstract":"This paper develops a framework based on category theory which unifies the simultaneous consideration of timepoints, metrical relations, and meter inclusion founded on the category of sets and binary relations. Metrical relations are defined as binary relations on the set of timepoints, and the subsequent use of the monoid they generate and of the corresponding functor to allows us to define meter networks, i.e. networks of timepoints (or sets of timepoints) related by metrical relations. We compare this to existing theories of metrical conflict, such as those of Harald Krebs and Richard Cohn, and illustrate that these tools help to more effectively combine displacement and grouping dissonance and reflect analytical claims concerning nineteenth-century examples of complex hemiola and twentieth-century polymeter. We show that meter networks can be transformed into each other through meter network morphisms, which allows us to describe both meter displacements and meter inclusions. These networks are applied to various examples from the nineteenth and twentieth century.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75569943","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 : 2020-11-12DOI: 10.1080/17459737.2020.1825844
Ben Grant, F. Knights, P. Padilla, Dan Tidhar
Mathematical methods, specifically Network Theory, are used here to investigate musical complexity as a marker of stylistic development. Proceeding from the premise that an 18th century classical composer's musical language becomes more complex over time, we suggest that this method, insofar as it quantifies and graphically represents complexity, could be a useful tool for exploring musical style, compositional maturity, and also issues of authorship or chronology. As a preliminary study of this concept, we chose a sample of six minuet movements (and one scherzo) from Haydn's string quartets from throughout his career, and analysed the melodic content of the first violin part. This intentional limitation to a small sample of works in a single genre whose authorship and chronology are beyond question allows us to focus on fundamental issues of musical content, and how that might develop and change during the period in which the works were composed.
{"title":"Network-theoretic analysis and the exploration of stylistic development in Haydn's string quartets","authors":"Ben Grant, F. Knights, P. Padilla, Dan Tidhar","doi":"10.1080/17459737.2020.1825844","DOIUrl":"https://doi.org/10.1080/17459737.2020.1825844","url":null,"abstract":"Mathematical methods, specifically Network Theory, are used here to investigate musical complexity as a marker of stylistic development. Proceeding from the premise that an 18th century classical composer's musical language becomes more complex over time, we suggest that this method, insofar as it quantifies and graphically represents complexity, could be a useful tool for exploring musical style, compositional maturity, and also issues of authorship or chronology. As a preliminary study of this concept, we chose a sample of six minuet movements (and one scherzo) from Haydn's string quartets from throughout his career, and analysed the melodic content of the first violin part. This intentional limitation to a small sample of works in a single genre whose authorship and chronology are beyond question allows us to focus on fundamental issues of musical content, and how that might develop and change during the period in which the works were composed.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76095946","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 : 2020-10-14DOI: 10.1080/17459737.2020.1825845
A. Popoff
Traditional transformational music theory describes transformations between musical elements as functions between sets and studies their subsequent algebraic properties and their use for music analysis. This is formalized from a categorical point of view by the use of functors where is a category, often a group or a monoid. At the same time, binary relations have also been used in mathematical music theory to describe relations between musical elements, one of the most compelling examples being Douthett's and Steinbach's parsimonious relations on pitch-class sets. Such relations are often used in a geometrical setting, for example through the use of so-called parsimonious graphs to describe how musical elements relate to each other. This article examines a generalization of transformational approaches based on functors , called relational presheaves, which focuses on the algebraic properties of binary relations defined over sets of musical elements. While binary relations include the particular case of functions, they provide additional flexibility as they also describe partial functions and allow the definition of multiple images for a given musical element. Our motivation to expand the toolbox of transformational music theory is illustrated in this paper by practical examples of monoids and categories generated by parsimonious and common-tone cross-type relations. At the same time, we describe the interplay between the algebraic properties of such objects and the geometrical properties of graph-based approaches.
{"title":"On the use of relational presheaves in transformational music theory","authors":"A. Popoff","doi":"10.1080/17459737.2020.1825845","DOIUrl":"https://doi.org/10.1080/17459737.2020.1825845","url":null,"abstract":"Traditional transformational music theory describes transformations between musical elements as functions between sets and studies their subsequent algebraic properties and their use for music analysis. This is formalized from a categorical point of view by the use of functors where is a category, often a group or a monoid. At the same time, binary relations have also been used in mathematical music theory to describe relations between musical elements, one of the most compelling examples being Douthett's and Steinbach's parsimonious relations on pitch-class sets. Such relations are often used in a geometrical setting, for example through the use of so-called parsimonious graphs to describe how musical elements relate to each other. This article examines a generalization of transformational approaches based on functors , called relational presheaves, which focuses on the algebraic properties of binary relations defined over sets of musical elements. While binary relations include the particular case of functions, they provide additional flexibility as they also describe partial functions and allow the definition of multiple images for a given musical element. Our motivation to expand the toolbox of transformational music theory is illustrated in this paper by practical examples of monoids and categories generated by parsimonious and common-tone cross-type relations. At the same time, we describe the interplay between the algebraic properties of such objects and the geometrical properties of graph-based approaches.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77695475","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 : 2020-10-05DOI: 10.1080/17459737.2020.1836686
J. Elliott
We provide an application of the theory of group actions to the study of musical scales. For any group G, finite G-set S, and real number t, we define the t-power diameter to be the size of any maximal orbit of S divided by the t-power mean orbit size of the elements of S. The symmetric group acts on the set of all tonic scales, where a tonic scale is a subset of containing 0. We show that for all , among all the subgroups G of , the t-power diameter of the G-set of all heptatonic scales is the largest for the subgroup Γ, and its conjugate subgroups, generated by . The unique maximal Γ-orbit consists of the 32 thāts of Hindustani classical music popularized by Bhatkhande. This analysis provides a reason why these 32 scales, among all 462 heptatonic scales, are of mathematical interest. We also apply our analysis, to a lesser degree, to hexatonic and pentatonic scales.
{"title":"Group actions, power mean orbit size, and musical scales","authors":"J. Elliott","doi":"10.1080/17459737.2020.1836686","DOIUrl":"https://doi.org/10.1080/17459737.2020.1836686","url":null,"abstract":"We provide an application of the theory of group actions to the study of musical scales. For any group G, finite G-set S, and real number t, we define the t-power diameter to be the size of any maximal orbit of S divided by the t-power mean orbit size of the elements of S. The symmetric group acts on the set of all tonic scales, where a tonic scale is a subset of containing 0. We show that for all , among all the subgroups G of , the t-power diameter of the G-set of all heptatonic scales is the largest for the subgroup Γ, and its conjugate subgroups, generated by . The unique maximal Γ-orbit consists of the 32 thāts of Hindustani classical music popularized by Bhatkhande. This analysis provides a reason why these 32 scales, among all 462 heptatonic scales, are of mathematical interest. We also apply our analysis, to a lesser degree, to hexatonic and pentatonic scales.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75931397","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 : 2020-09-20DOI: 10.1080/17459737.2020.1812128
W. Sethares, Wayne Vitale
A primary esthetic in the performance practice of Balinese gamelan is the ombak (Indonesian for wave), which is manifest in musical form, performance, and tuning. The ombak arises in a paired tuning system in which corresponding unisons of two instruments (or instrumental groups) are tuned to slightly different frequencies, one higher and one lower, to produce beats. Pitch classes are not necessarily tuned to octaves in an exact 2:1 frequency ratio; instead, octaves are often stretched or compressed. This paper discusses the relationship between the ombak rate and octave tempering, and demonstrates that the beating rate, combined with the octave tuning strategy chosen, can be modeled using a tempering parameter that determines the amount of stretching or compression. This model is then used to analyze tuning data of nine complete gamelan.
{"title":"Ombak and octave stretching in Balinese gamelan","authors":"W. Sethares, Wayne Vitale","doi":"10.1080/17459737.2020.1812128","DOIUrl":"https://doi.org/10.1080/17459737.2020.1812128","url":null,"abstract":"A primary esthetic in the performance practice of Balinese gamelan is the ombak (Indonesian for wave), which is manifest in musical form, performance, and tuning. The ombak arises in a paired tuning system in which corresponding unisons of two instruments (or instrumental groups) are tuned to slightly different frequencies, one higher and one lower, to produce beats. Pitch classes are not necessarily tuned to octaves in an exact 2:1 frequency ratio; instead, octaves are often stretched or compressed. This paper discusses the relationship between the ombak rate and octave tempering, and demonstrates that the beating rate, combined with the octave tuning strategy chosen, can be modeled using a tempering parameter that determines the amount of stretching or compression. This model is then used to analyze tuning data of nine complete gamelan.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73664694","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 : 2020-09-06DOI: 10.1080/17459737.2021.1953169
Ricardo Gómez Aíza
We study classes of musical scales obtained from shift spaces in symbolic dynamics through the first symbol rule, which yields scales in any n-TET tuning system. The modes are thought as elements of orbit equivalence classes of cyclic shift actions on languages, and we study their orbitals and transversals. We present explicit formulations of the generating functions that allow us to deduce the orbital and transversal dimensions of classes of musical scales generated by vertex shifts, for all n, in particular for the 12-TET tuning system.
{"title":"Symbolic dynamical scales: modes, orbitals, and transversals","authors":"Ricardo Gómez Aíza","doi":"10.1080/17459737.2021.1953169","DOIUrl":"https://doi.org/10.1080/17459737.2021.1953169","url":null,"abstract":"We study classes of musical scales obtained from shift spaces in symbolic dynamics through the first symbol rule, which yields scales in any n-TET tuning system. The modes are thought as elements of orbit equivalence classes of cyclic shift actions on languages, and we study their orbitals and transversals. We present explicit formulations of the generating functions that allow us to deduce the orbital and transversal dimensions of classes of musical scales generated by vertex shifts, for all n, in particular for the 12-TET tuning system.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78906037","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 : 2020-07-17DOI: 10.1080/17459737.2020.1775902
L. Nuño
In this paper, pitch-class sets are analyzed in terms of their intervallic structures and those related by transposition are called a set type. Then, non-inversionally-symmetrical set classes are split into two set types related by inversion. As a higher version of the interval-class vector, I introduce the trichord-type vector, whose elements are the number of times each trichord type is contained in a set type, as well as a trichord-class vector for set classes. By using the interval-class, trichord-class, and trichord-type vectors, a list of set classes and types is developed, including, apart from the usual information, the intervallic structures and the trichord-type vectors. The inclusion of this last characteristic is the most significant difference with respect to previously published lists of set classes. Finally, a compact periodic table containing all set classes is given, showing their main characteristics and relationships at a glance.
{"title":"A detailed list and a periodic table of set classes","authors":"L. Nuño","doi":"10.1080/17459737.2020.1775902","DOIUrl":"https://doi.org/10.1080/17459737.2020.1775902","url":null,"abstract":"In this paper, pitch-class sets are analyzed in terms of their intervallic structures and those related by transposition are called a set type. Then, non-inversionally-symmetrical set classes are split into two set types related by inversion. As a higher version of the interval-class vector, I introduce the trichord-type vector, whose elements are the number of times each trichord type is contained in a set type, as well as a trichord-class vector for set classes. By using the interval-class, trichord-class, and trichord-type vectors, a list of set classes and types is developed, including, apart from the usual information, the intervallic structures and the trichord-type vectors. The inclusion of this last characteristic is the most significant difference with respect to previously published lists of set classes. Finally, a compact periodic table containing all set classes is given, showing their main characteristics and relationships at a glance.","PeriodicalId":50138,"journal":{"name":"Journal of Mathematics and Music","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85547245","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：