Pub Date : 2002-06-01DOI: 10.5642/HMNJ.200201.26.09
D. Joyner, P. Lejarraga
Our aim is to sketch some ideas related to how we (as in, we two) think we (as in, we humans) think. "That theory is useless. It isn't even wrong." Wolfgang Pauli. Our hope in this paper is to provide a theory, admittedly somewhat vague, of how we think about mathematics. We also hope our ideas do not cause the reader to be reminded of Pauli's quote above. These notes were motivated by the interesting book by Changeaux and Connes [CC]. REALISM VS CONSTRUCTIVISM Realism: Mathematical objects exist independently of experience (or "physical reality") which we process using our senses (smell, touch, sight, ... ) and interpret using our brain. For example, Descartes speaks of a triangle as an "immutable and eternal" figure whose existence is independent of the mind which imagines it. Similar statements are made regarding God by many religious experts. Constructivism: Mathematical objects exist solely in the mind as a certain electro-chemo-biological pattern of neurons, synapses, chemicals, ... in the brain. For an extreme example, Hume believed that ideas are merely copies of sense impressions. Examples: Alain Cannes (and probably most mathematicians) are realists. For example, the famous quote of Kronecker's, "The integers are made by God, all else is made by man," indicates a realist point-of-view. On the other hand, the biologist Jean Pierre Changeux and philosopher David Hume are constructivists (though Hume is the more extreme). Poincare was possibly a constructivist in this sense (see [D], chapter 9). The realist position might be roughly summarized as 28 follows: The physical world is modeled as much as possible by mathematics. Mathematicians merely discover what is already in existence. The constructivist position might be summarized as follows: Models for the physical world are constructions of the mind (only) and all such mental constructs exist solely as electro-chemo-biological patterns of neurons, ... in the brain. To the question, "Why is mathematics so well-suited to the description of physics?", the constructivist might counter that physicists tend to examine reproducible phenomena which tend to have "universal" characteristics. Hence mathematics, which is also universal, is admirably suited for physical description. POINTS OF AGREEMENT • Mathematics provides a "universal language", i.e., a grammar and set of terms which can be understood by anyone (sufficiently trained), independently of their cultural background. • There is a "physical world" independent of our mind (which, however, we sense using our brain and sensory organs). • Mathematical objects can be represented as a certain electro-chemo-biological pattern of neurons, synapses, chemicals, ... in the brain. • A given mathematical construction can be represented as a program in a "Turing machine". (Using an over-simplification, these representations are modeled using neural networks, which are related to Turing machines [M].) FORMAL CONSTRUCTIVISM AssoRTED THOUGHTS OF OuR OwN •
我们的目的是概述一些与我们(也就是我们两人)如何思考我们(也就是我们人类)如何思考有关的想法。“这种理论是没用的。这根本不算错。”沃尔夫冈·泡利不相容。我们在这篇论文中的希望是提供一个理论，诚然有些模糊，关于我们如何思考数学。我们也希望我们的想法不会让读者想起泡利上面的话。这些笔记的灵感来自Changeaux和Connes的一本有趣的书[CC]。现实主义与建构主义现实主义:数学对象独立于经验(或“物理现实”)而存在，我们用感官(嗅觉、触觉、视觉等)处理经验。用我们的大脑来解释。例如，笛卡尔说三角形是一个“不变的和永恒的”图形，它的存在是独立于想象它的心灵的。许多宗教专家也对上帝作了类似的陈述。建构主义:数学对象仅以神经元、突触、化学物质等组成的某种电化学生物模式存在于大脑中。在大脑里。举个极端的例子，休谟认为观念只是感觉印象的复制。例子:阿兰·坎纳(可能还有大多数数学家)是现实主义者。例如，克罗内克的名言“整数是上帝创造的，其他都是人创造的”，表明了一种现实主义的观点。另一方面，生物学家让·皮埃尔·昌格(Jean Pierre Changeux)和哲学家大卫·休谟(David Hume)则是建构主义者(尽管休谟更为极端)。在这个意义上，庞加莱可能是一个建构主义者(见[D]，第9章)。现实主义的立场可以大致概括如下:物理世界是尽可能用数学来建模的。数学家只是发现已经存在的东西。建构主义的立场可以概括如下:物理世界的模型(仅仅)是心灵的构造，所有这些心理构造都仅以神经元的电化学-生物模式存在……在大脑里。对于“为什么数学如此适合描述物理?”这个问题，建构主义者可能会反驳说，物理学家倾向于研究具有“普遍”特征的可再现现象。因此，同样具有普遍性的数学非常适合于物理描述。•数学提供了一种“通用语言”，即任何人(经过充分训练)都能理解的语法和一套术语，而不受其文化背景的影响。•有一个“物质世界”独立于我们的思想(然而，我们用我们的大脑和感觉器官来感知)。•数学对象可以表示为神经元、突触、化学物质等的某种电化学生物学模式。在大脑里。•给定的数学结构可以表示为图灵机中的程序。(通过过度简化，这些表示使用与图灵机相关的神经网络建模[M]。)虽然数学可能确实是普遍的，但它的发展和现状受到文化和人类经验的启发和影响。人文数学网络学报第26期
{"title":"Notes on Formal Constructivism","authors":"D. Joyner, P. Lejarraga","doi":"10.5642/HMNJ.200201.26.09","DOIUrl":"https://doi.org/10.5642/HMNJ.200201.26.09","url":null,"abstract":"Our aim is to sketch some ideas related to how we (as in, we two) think we (as in, we humans) think. \"That theory is useless. It isn't even wrong.\" Wolfgang Pauli. Our hope in this paper is to provide a theory, admittedly somewhat vague, of how we think about mathematics. We also hope our ideas do not cause the reader to be reminded of Pauli's quote above. These notes were motivated by the interesting book by Changeaux and Connes [CC]. REALISM VS CONSTRUCTIVISM Realism: Mathematical objects exist independently of experience (or \"physical reality\") which we process using our senses (smell, touch, sight, ... ) and interpret using our brain. For example, Descartes speaks of a triangle as an \"immutable and eternal\" figure whose existence is independent of the mind which imagines it. Similar statements are made regarding God by many religious experts. Constructivism: Mathematical objects exist solely in the mind as a certain electro-chemo-biological pattern of neurons, synapses, chemicals, ... in the brain. For an extreme example, Hume believed that ideas are merely copies of sense impressions. Examples: Alain Cannes (and probably most mathematicians) are realists. For example, the famous quote of Kronecker's, \"The integers are made by God, all else is made by man,\" indicates a realist point-of-view. On the other hand, the biologist Jean Pierre Changeux and philosopher David Hume are constructivists (though Hume is the more extreme). Poincare was possibly a constructivist in this sense (see [D], chapter 9). The realist position might be roughly summarized as 28 follows: The physical world is modeled as much as possible by mathematics. Mathematicians merely discover what is already in existence. The constructivist position might be summarized as follows: Models for the physical world are constructions of the mind (only) and all such mental constructs exist solely as electro-chemo-biological patterns of neurons, ... in the brain. To the question, \"Why is mathematics so well-suited to the description of physics?\", the constructivist might counter that physicists tend to examine reproducible phenomena which tend to have \"universal\" characteristics. Hence mathematics, which is also universal, is admirably suited for physical description. POINTS OF AGREEMENT • Mathematics provides a \"universal language\", i.e., a grammar and set of terms which can be understood by anyone (sufficiently trained), independently of their cultural background. • There is a \"physical world\" independent of our mind (which, however, we sense using our brain and sensory organs). • Mathematical objects can be represented as a certain electro-chemo-biological pattern of neurons, synapses, chemicals, ... in the brain. • A given mathematical construction can be represented as a program in a \"Turing machine\". (Using an over-simplification, these representations are modeled using neural networks, which are related to Turing machines [M].) FORMAL CONSTRUCTIVISM AssoRTED THOUGHTS OF OuR OwN •","PeriodicalId":176215,"journal":{"name":"Humanistic Mathematics Network Journal","volume":" 1226","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120826632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2002-06-01DOI: 10.5642/HMNJ.200201.26.22
A. White, Sandra Z. Keith
{"title":"Teaching As Though Students Mattered: A Biography of Alvin White As Told to Sandra Keith","authors":"A. White, Sandra Z. Keith","doi":"10.5642/HMNJ.200201.26.22","DOIUrl":"https://doi.org/10.5642/HMNJ.200201.26.22","url":null,"abstract":"","PeriodicalId":176215,"journal":{"name":"Humanistic Mathematics Network Journal","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121232402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2002-06-01DOI: 10.5642/HMNJ.200201.26.08
P. C. Kenschaft
Humanistic Mathematics Network Journal #26 25 stone in Basle. Today, the double helix carries both a biological meaning as well as an intimation of human destiny. In my childhood, the circle persisted as a potent magic figure in the playtime doggerel "Make a magic circle and sign it with a dot." The interested reader will find thousands of allusions to the phrase "magic circle" on the Web. Magic ellipses or rectangles are less frequent. The Buddhist mandalas which are objects of spiritual contemplation, embody highly stylized geometrical arrangements. The amulets and talismans that are worn on the body, placed on walls, displayed in cars; the ankhs, the crosses, the hexagrams, the outlined fish, the horseshoes, the triangular abracadabra arrangements and magical squares, the sigils (magical signs or images) of which whole dictionaries were compiled in the 17th century, the hex signs placed on house exteriors, all point to geometry in the service of religious or quasi-religious practice. There is a multitude of geometrical figures signs employed in kabbalistic practices, each associated with stars, planets, metals, stones, spirits, demons, and whose mode of production and use is specified rigorously. Wallis Budge, student of Near Eastern antiquities wrote: According to Cornelius Agrippa [physician and magician, 1486-1535], it is necessary to be careful when using a magical square as an amulet, that it is drawn when the sun or moon or the planet is exhibiting a benevolent aspect, for otherwise the amulet will bring misfortune and calamity upon the wearer instead of prosperity and happiness. Let semanticists and semioticists explain the relationship between our geometrical symbols and our psyches for it lies deeper than simple designation (e.g., crescent = Islam). The geometrical swastika, which over the millennia and cultures has carried different interpretations, is now held in abhorrence by most Americans. The memory of World War II is certainly at work here, but the geometry can go "abstract" and its meaning become detached from an original historic context. Why has Salvador Dali (1904-1989) in his large painting Corpus Hypercubus in the Metropolitan Museum in New York, placed a crucifixion against a representation of a four dimensional cube? Art historian Mar-tin Kemp has commented: Dali's painting does stand effectively for an age-old striving in art, theology, mathematics, and cosmology for access to those dimensions that lie beyond the visual and tactile scope of the finite spaces of up-and-down, left and right, and in-and-out that …
{"title":"Pat's Prologues: Introductions to the First Two Airings of Math Medley, A Radio Talk Show.","authors":"P. C. Kenschaft","doi":"10.5642/HMNJ.200201.26.08","DOIUrl":"https://doi.org/10.5642/HMNJ.200201.26.08","url":null,"abstract":"Humanistic Mathematics Network Journal #26 25 stone in Basle. Today, the double helix carries both a biological meaning as well as an intimation of human destiny. In my childhood, the circle persisted as a potent magic figure in the playtime doggerel \"Make a magic circle and sign it with a dot.\" The interested reader will find thousands of allusions to the phrase \"magic circle\" on the Web. Magic ellipses or rectangles are less frequent. The Buddhist mandalas which are objects of spiritual contemplation, embody highly stylized geometrical arrangements. The amulets and talismans that are worn on the body, placed on walls, displayed in cars; the ankhs, the crosses, the hexagrams, the outlined fish, the horseshoes, the triangular abracadabra arrangements and magical squares, the sigils (magical signs or images) of which whole dictionaries were compiled in the 17th century, the hex signs placed on house exteriors, all point to geometry in the service of religious or quasi-religious practice. There is a multitude of geometrical figures signs employed in kabbalistic practices, each associated with stars, planets, metals, stones, spirits, demons, and whose mode of production and use is specified rigorously. Wallis Budge, student of Near Eastern antiquities wrote: According to Cornelius Agrippa [physician and magician, 1486-1535], it is necessary to be careful when using a magical square as an amulet, that it is drawn when the sun or moon or the planet is exhibiting a benevolent aspect, for otherwise the amulet will bring misfortune and calamity upon the wearer instead of prosperity and happiness. Let semanticists and semioticists explain the relationship between our geometrical symbols and our psyches for it lies deeper than simple designation (e.g., crescent = Islam). The geometrical swastika, which over the millennia and cultures has carried different interpretations, is now held in abhorrence by most Americans. The memory of World War II is certainly at work here, but the geometry can go \"abstract\" and its meaning become detached from an original historic context. Why has Salvador Dali (1904-1989) in his large painting Corpus Hypercubus in the Metropolitan Museum in New York, placed a crucifixion against a representation of a four dimensional cube? Art historian Mar-tin Kemp has commented: Dali's painting does stand effectively for an age-old striving in art, theology, mathematics, and cosmology for access to those dimensions that lie beyond the visual and tactile scope of the finite spaces of up-and-down, left and right, and in-and-out that …","PeriodicalId":176215,"journal":{"name":"Humanistic Mathematics Network Journal","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117003916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2002-06-01DOI: 10.5642/HMNJ.200201.26.13
M. E. Goldberg
{"title":"When Is a Math Problem Really \"Real\"?","authors":"M. E. Goldberg","doi":"10.5642/HMNJ.200201.26.13","DOIUrl":"https://doi.org/10.5642/HMNJ.200201.26.13","url":null,"abstract":"","PeriodicalId":176215,"journal":{"name":"Humanistic Mathematics Network Journal","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117110661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2002-06-01DOI: 10.5642/HMNJ.200201.26.17
Lacie Juris
{"title":"Magic in a Box","authors":"Lacie Juris","doi":"10.5642/HMNJ.200201.26.17","DOIUrl":"https://doi.org/10.5642/HMNJ.200201.26.17","url":null,"abstract":"","PeriodicalId":176215,"journal":{"name":"Humanistic Mathematics Network Journal","volume":"118 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125399942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2002-06-01DOI: 10.5642/hmnj.200201.26.11
Patricia Baggett, A. Ehrenfeucht
{"title":"Marcy's Dots: A Problem on National Test Revisited","authors":"Patricia Baggett, A. Ehrenfeucht","doi":"10.5642/hmnj.200201.26.11","DOIUrl":"https://doi.org/10.5642/hmnj.200201.26.11","url":null,"abstract":"","PeriodicalId":176215,"journal":{"name":"Humanistic Mathematics Network Journal","volume":"491 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127030824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2002-06-01DOI: 10.5642/HMNJ.200201.26.21
Sandra Z. Keith
{"title":"Special Section: On the Publication of the 26th Issue of the Humanistic Mathematics Network Journal","authors":"Sandra Z. Keith","doi":"10.5642/HMNJ.200201.26.21","DOIUrl":"https://doi.org/10.5642/HMNJ.200201.26.21","url":null,"abstract":"","PeriodicalId":176215,"journal":{"name":"Humanistic Mathematics Network Journal","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130987227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2002-06-01DOI: 10.5642/HMNJ.200201.26.06
A. Katsap
{"title":"Humanizing Mathematics: The Humanistic Impression in the Course for Mathematics Teaching.","authors":"A. Katsap","doi":"10.5642/HMNJ.200201.26.06","DOIUrl":"https://doi.org/10.5642/HMNJ.200201.26.06","url":null,"abstract":"","PeriodicalId":176215,"journal":{"name":"Humanistic Mathematics Network Journal","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123956208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2002-06-01DOI: 10.5642/HMNJ.200201.26.05
A. Garciadiego
The main goal of this essay is to discuss, informally, an intuitive approach to the history of mathematics as an academic discipline. The initial point of departure includes the analysis of some traditional definitions of the concept of 'history' taken from standard dictionaries. This concise dissection attempts to suggest the complexity of the discipline. The term 'history' is familiar to almost everyone, and most people believe they know its meaning intuitively. The lay person sometimes thinks of history in terms of dates, names, places and of colorful anecdotes on interesting characters. History provides a record of where someone lived and what he did. In short, history is the repository of the past. But what exactly is the history of mathematics as an academic discipline? The events that took place yesterday are now part of history. Photographs are history: they reflect the way we once were. History is studied in elementary school, especially that of student's native countries. Teachers, who often seem old enough to have taken part in some of the historical events, teach-over and over again-anecdotes, names, places, dates, and so on. Historical movies, TV programs and books are popular. The media affects the way people understand history as a discipline. Unfortunately, sometimes the public's knowledge of important historical issues is derived from the popular media (especially movies), and not from professional sources. Thus, their understanding of these events can be distorted. Unlike the word mathematics, the term history is used on a daily basis by the news media. Some reporters may believe that history is actually being made when they type or read the news. On occasions, reporters and anchormen have trouble attempting to disassoci-ate themselves from the event they are reporting on. Of course, in most cases, the term history is misused by professional reporters. Often, for example, TV sports commentators discuss player's individual records while narrating a baseball game. These statistics include the player's batting average, number of times at bat in the game, number of stolen bases, etc. Sometimes, the commentators also narrates the player's background. They mention the college the player attended; where he played in the minor leagues, his previous professional teams and so on. The commentator may also discuss some of the player's qualities as a human being (e.g., generosity, sportsmanship). Then, the sportscaster may try to explain why the player was motivated to become a professional. After the commentator has finished …
{"title":"History of Mathematics, an Intuitive Approach","authors":"A. Garciadiego","doi":"10.5642/HMNJ.200201.26.05","DOIUrl":"https://doi.org/10.5642/HMNJ.200201.26.05","url":null,"abstract":"The main goal of this essay is to discuss, informally, an intuitive approach to the history of mathematics as an academic discipline. The initial point of departure includes the analysis of some traditional definitions of the concept of 'history' taken from standard dictionaries. This concise dissection attempts to suggest the complexity of the discipline. The term 'history' is familiar to almost everyone, and most people believe they know its meaning intuitively. The lay person sometimes thinks of history in terms of dates, names, places and of colorful anecdotes on interesting characters. History provides a record of where someone lived and what he did. In short, history is the repository of the past. But what exactly is the history of mathematics as an academic discipline? The events that took place yesterday are now part of history. Photographs are history: they reflect the way we once were. History is studied in elementary school, especially that of student's native countries. Teachers, who often seem old enough to have taken part in some of the historical events, teach-over and over again-anecdotes, names, places, dates, and so on. Historical movies, TV programs and books are popular. The media affects the way people understand history as a discipline. Unfortunately, sometimes the public's knowledge of important historical issues is derived from the popular media (especially movies), and not from professional sources. Thus, their understanding of these events can be distorted. Unlike the word mathematics, the term history is used on a daily basis by the news media. Some reporters may believe that history is actually being made when they type or read the news. On occasions, reporters and anchormen have trouble attempting to disassoci-ate themselves from the event they are reporting on. Of course, in most cases, the term history is misused by professional reporters. Often, for example, TV sports commentators discuss player's individual records while narrating a baseball game. These statistics include the player's batting average, number of times at bat in the game, number of stolen bases, etc. Sometimes, the commentators also narrates the player's background. They mention the college the player attended; where he played in the minor leagues, his previous professional teams and so on. The commentator may also discuss some of the player's qualities as a human being (e.g., generosity, sportsmanship). Then, the sportscaster may try to explain why the player was motivated to become a professional. After the commentator has finished …","PeriodicalId":176215,"journal":{"name":"Humanistic Mathematics Network Journal","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128689134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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