Pub Date : 2023-10-31DOI: 10.1080/07468342.2023.2263318
Howard Sporn
AbstractWe define golden triples to be triples of integers satisfying a particular quadratic equation. The first two elements of each triple are consecutive terms of a Fibonacci-like sequence. We show that each golden triple can be represented by a rational point on a particular circle, which we call the golden circle. We also generate a related circle called the aureate circle. AcknowledgmentSupport for this project was provided by a PSC-CUNY Award, jointly funded by the Professional Staff Congress and the City University of New York.Additional informationFundingSupport for this project was provided by a PSC-CUNY Award, jointly funded by the Professional Staff Congress and the City University of New York.Notes on contributorsHoward SpornHoward Sporn (hsporn@qcc.cuny.edu) is an associate professor of mathematics at Queensborough Community College in Bayside, NY. He received his Ed.D. in mathematics education from Teachers College, Columbia University. Previously, Sporn earned M.S. degrees in mathematics and physics from Stony Brook University. His research interest is number theory. He also likes philosophy, history, movies, science fiction, and cats. He lives on Long Island, NY, with his wife Sharon.
摘要金三元组是由满足特定二次方程的整数组成的三元组。每个三元组的前两个元素是类斐波那契序列的连续项。我们证明了每个黄金三重都可以用一个特定圆上的有理点来表示，我们称之为黄金圆。我们还生成了一个相关的圆，叫做金色圆。本项目由PSC-CUNY奖提供支持，由专业工作人员大会和纽约城市大学共同资助。本项目的资金支持由PSC-CUNY奖提供，由专业工作人员大会和纽约城市大学联合资助。作者简介howard Sporn howard Sporn (hsporn@qcc.cuny.edu)是纽约贝赛德Queensborough社区学院的数学副教授。他获得了教育学博士学位。哥伦比亚大学师范学院数学教育硕士。此前，他在石溪大学(Stony Brook University)获得数学和物理硕士学位。他的研究兴趣是数论。他还喜欢哲学、历史、电影、科幻小说和猫。他和妻子莎伦住在纽约长岛。
{"title":"The Golden Circle and the Aureate Circle","authors":"Howard Sporn","doi":"10.1080/07468342.2023.2263318","DOIUrl":"https://doi.org/10.1080/07468342.2023.2263318","url":null,"abstract":"AbstractWe define golden triples to be triples of integers satisfying a particular quadratic equation. The first two elements of each triple are consecutive terms of a Fibonacci-like sequence. We show that each golden triple can be represented by a rational point on a particular circle, which we call the golden circle. We also generate a related circle called the aureate circle. AcknowledgmentSupport for this project was provided by a PSC-CUNY Award, jointly funded by the Professional Staff Congress and the City University of New York.Additional informationFundingSupport for this project was provided by a PSC-CUNY Award, jointly funded by the Professional Staff Congress and the City University of New York.Notes on contributorsHoward SpornHoward Sporn (hsporn@qcc.cuny.edu) is an associate professor of mathematics at Queensborough Community College in Bayside, NY. He received his Ed.D. in mathematics education from Teachers College, Columbia University. Previously, Sporn earned M.S. degrees in mathematics and physics from Stony Brook University. His research interest is number theory. He also likes philosophy, history, movies, science fiction, and cats. He lives on Long Island, NY, with his wife Sharon.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135870155","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 : 2023-10-31DOI: 10.1080/07468342.2023.2263109
Sara Barrows, Emily Noye, Sarah Uttormark, Matthew Wright
AbstractA subprime Fibonacci sequence follows the Fibonacci recurrence, where the next term in a sequence is the sum of the two previous terms, except that composite sums are divided by their least prime factor. We extend the recurrence to three terms, investigating subprime tribonacci sequences. It appears that all such sequences eventually enter a repeating cycle. We compute cycles arising from more than one billion sequences, classifying them as trivial, tame, and wild. We further investigate questions of parity and primality in subprime tribonacci sequences. In particular, we show that any nonzero subprime tribonacci sequence eventually contains an odd term. AcknowledgmentThis article grew out of a final project by the first three authors in the course Modern Computational Mathematics at St. Olaf College in spring 2020.Additional informationNotes on contributorsSara BarrowsSara Barrows (sarabarrows18@gmail.com) received a BA in Physics and a concentration in Engineering Studies from St. Olaf College in 2022. She currently works in R&D for Saint-Gobain, a materials company. As a Research Engineer, Sara investigates innovative solutions within the building materials industry.Emily NoyeEmily Noye (emily.noye@yahoo.com) has her BA in Mathematics with a concentration in Statistics and Data Science from St. Olaf College. She now works as a Retirement Actuary for Aon Consulting while working toward her ASA certification.Sarah UttormarkSarah Uttormark (smu32@cornell.edu) earned a BA in Physics, Mathematics, Norwegian, and Nordic Studies with a concentration in Engineering Studies from St. Olaf College in 2022. She is now an Applied Physics graduate student at Cornell University, where she works under Professor Lois Pollack on instrumentation and methods for studying the structures and dynamics of biomolecules.Matthew WrightMatthew Wright (wright5@stolaf.edu) is an Associate Professor at St. Olaf College (Northfield, MN), where he teaches applied and computational math courses. He is an author of the RIVET software for topological data analysis. Matthew lives in Minnesota with his wife and two children, and also enjoys juggling. Find him online at mlwright.org.
{"title":"Three’s A Crowd: An Exploration of Subprime Tribonacci Sequences","authors":"Sara Barrows, Emily Noye, Sarah Uttormark, Matthew Wright","doi":"10.1080/07468342.2023.2263109","DOIUrl":"https://doi.org/10.1080/07468342.2023.2263109","url":null,"abstract":"AbstractA subprime Fibonacci sequence follows the Fibonacci recurrence, where the next term in a sequence is the sum of the two previous terms, except that composite sums are divided by their least prime factor. We extend the recurrence to three terms, investigating subprime tribonacci sequences. It appears that all such sequences eventually enter a repeating cycle. We compute cycles arising from more than one billion sequences, classifying them as trivial, tame, and wild. We further investigate questions of parity and primality in subprime tribonacci sequences. In particular, we show that any nonzero subprime tribonacci sequence eventually contains an odd term. AcknowledgmentThis article grew out of a final project by the first three authors in the course Modern Computational Mathematics at St. Olaf College in spring 2020.Additional informationNotes on contributorsSara BarrowsSara Barrows (sarabarrows18@gmail.com) received a BA in Physics and a concentration in Engineering Studies from St. Olaf College in 2022. She currently works in R&D for Saint-Gobain, a materials company. As a Research Engineer, Sara investigates innovative solutions within the building materials industry.Emily NoyeEmily Noye (emily.noye@yahoo.com) has her BA in Mathematics with a concentration in Statistics and Data Science from St. Olaf College. She now works as a Retirement Actuary for Aon Consulting while working toward her ASA certification.Sarah UttormarkSarah Uttormark (smu32@cornell.edu) earned a BA in Physics, Mathematics, Norwegian, and Nordic Studies with a concentration in Engineering Studies from St. Olaf College in 2022. She is now an Applied Physics graduate student at Cornell University, where she works under Professor Lois Pollack on instrumentation and methods for studying the structures and dynamics of biomolecules.Matthew WrightMatthew Wright (wright5@stolaf.edu) is an Associate Professor at St. Olaf College (Northfield, MN), where he teaches applied and computational math courses. He is an author of the RIVET software for topological data analysis. Matthew lives in Minnesota with his wife and two children, and also enjoys juggling. Find him online at mlwright.org.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135813426","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 : 2023-10-31DOI: 10.1080/07468342.2023.2266315
Skip Garibaldi
{"title":"The Luckiest Strategy on Earth? A Better Way to Buy Lottery Scratch-off Tickets","authors":"Skip Garibaldi","doi":"10.1080/07468342.2023.2266315","DOIUrl":"https://doi.org/10.1080/07468342.2023.2266315","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135869309","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 : 2023-10-26DOI: 10.1080/07468342.2023.2266314
Brian D. Jones
{"title":"Taxicab Metric of a Different Kind","authors":"Brian D. Jones","doi":"10.1080/07468342.2023.2266314","DOIUrl":"https://doi.org/10.1080/07468342.2023.2266314","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134909812","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 : 2023-10-24DOI: 10.1080/07468342.2023.2263029
Daniel Solow
SummaryIn a straightforward proof by induction of the Collatz conjecture, it is easy to show that the statement is true for n=1 (the base case) but proving the induction step is more challenging—in fact, no one has done so to date. In this work, a different induction proof of the conjecture, based on a binary representation of the starting integer, is given in which it is possible to prove the induction step but there is no known proof for the base case. In fact, if one could prove the base case, then the Collatz conjecture is true. Additional informationNotes on contributorsDaniel SolowDaniel Solow (daniel.solow@case.edu) received a BS in math from Carnegie-Mellon University; an MS in Operations Research from Berkeley; and a Ph.D. in Operations Research from Stanford University. He has been a professor in the Weatherhead School of Management at Case Western Reserve University since 1978. In addition to specializing in optimization and mathematical modeling, he is best known for his seminal book titled How to Read and Do Proofs that, when published in 1982, was the first systematic approach for teaching students how to read, understand, think about and do mathematical proofs. He has also established an annual award through the Mathematical Association of America to recognize outstanding contributions of undergraduate educational materials.
在Collatz猜想的一个简单的归纳法证明中，很容易证明该命题对n=1(基本情况)是正确的，但证明归纳法步骤更具挑战性——事实上，迄今为止还没有人这样做过。在这项工作中，基于开始整数的二进制表示，给出了猜想的另一种归纳证明，其中可以证明归纳步骤，但没有已知的基本情况的证明。事实上，如果可以证明基本情况，那么Collatz猜想就是正确的。daniel Solow (daniel.solow@case.edu)获得卡内基梅隆大学数学学士学位;伯克利大学运筹学硕士学位;获得斯坦福大学运筹学博士学位。自1978年以来，他一直担任凯斯西储大学韦瑟黑德管理学院的教授。除了专注于优化和数学建模之外，他最著名的著作是1982年出版的《如何阅读和做证明》(How to Read and Do Proofs)，这是教学生如何阅读、理解、思考和做数学证明的第一个系统方法。他还通过美国数学协会设立了一个年度奖项，以表彰在本科教育材料方面的杰出贡献。
{"title":"Half an Induction Proof of the Collatz Conjecture","authors":"Daniel Solow","doi":"10.1080/07468342.2023.2263029","DOIUrl":"https://doi.org/10.1080/07468342.2023.2263029","url":null,"abstract":"SummaryIn a straightforward proof by induction of the Collatz conjecture, it is easy to show that the statement is true for n=1 (the base case) but proving the induction step is more challenging—in fact, no one has done so to date. In this work, a different induction proof of the conjecture, based on a binary representation of the starting integer, is given in which it is possible to prove the induction step but there is no known proof for the base case. In fact, if one could prove the base case, then the Collatz conjecture is true. Additional informationNotes on contributorsDaniel SolowDaniel Solow (daniel.solow@case.edu) received a BS in math from Carnegie-Mellon University; an MS in Operations Research from Berkeley; and a Ph.D. in Operations Research from Stanford University. He has been a professor in the Weatherhead School of Management at Case Western Reserve University since 1978. In addition to specializing in optimization and mathematical modeling, he is best known for his seminal book titled How to Read and Do Proofs that, when published in 1982, was the first systematic approach for teaching students how to read, understand, think about and do mathematical proofs. He has also established an annual award through the Mathematical Association of America to recognize outstanding contributions of undergraduate educational materials.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135266620","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 : 2023-10-24DOI: 10.1080/07468342.2023.2266319
Alexander Holley, Anastasiia Minenkova
SummaryBy telling this story, we discuss how to blend research problems into the classrooms to enhance the curriculum starting already with the linear algebra course. In particular, we present an elementary approach to the reconstruction of persymmetric Jacobi matrices from their eigenvalues. This work is done in collaboration with and for undergraduate students taking linear algebra. We wrote it with a thought in mind to fill the gaps and to show the depth of results still keeping it within the boundaries of the undergraduate underclassman level. So if a student reads it and the web supplements, it will be completely understandable and by comparing to the original paper [Citation2] the student would realize how to break down the results and proofs to fully comprehend them. As for educators, we hope this would show an example of how to present an establish knowledge and modern theories by making a problem feasible for the broader undergraduate audience. We also briefly discuss the motivation of studying the problem in the prism of modern theories. Namely, we link this algorithm to the perfect quantum transfer problem. AcknowledgmentThe second author wishes to thank their linear algebra students who kept working on their projects in the midst of the COVID-19 pandemic and survived the switch to the distance learning. They were an incredible source of inspiration.Additional informationNotes on contributorsAlexander HolleyAlexander Holley (alexander.holley@uconn.edu) is pursuing a B.S. in Mechanical Engineering at the University of Connecticut. He plans on working in the aerospace field after graduation, or returning to school for a M. Sc. in a related engineering field. His interests include being in nature, cooking, and watching or playing sports.Anastasiia MinenkovaAnastasiia Minenkova (aminenkova@mtholyoke.edu) is a visiting lecturer at Mount Holyoke College. She received her M.Sc. in Mathematics from the University of Mississippi and her doctorate from the University of Connecticut. Her research interests are in numerical linear algebra. She is very passionate about teaching. Anastasiia’s interests outside of academia include cross-stitching and culinary tourism.
{"title":"A Linear Algebra Story: How We Reconstructed a Matrix from its Eigenvalues","authors":"Alexander Holley, Anastasiia Minenkova","doi":"10.1080/07468342.2023.2266319","DOIUrl":"https://doi.org/10.1080/07468342.2023.2266319","url":null,"abstract":"SummaryBy telling this story, we discuss how to blend research problems into the classrooms to enhance the curriculum starting already with the linear algebra course. In particular, we present an elementary approach to the reconstruction of persymmetric Jacobi matrices from their eigenvalues. This work is done in collaboration with and for undergraduate students taking linear algebra. We wrote it with a thought in mind to fill the gaps and to show the depth of results still keeping it within the boundaries of the undergraduate underclassman level. So if a student reads it and the web supplements, it will be completely understandable and by comparing to the original paper [Citation2] the student would realize how to break down the results and proofs to fully comprehend them. As for educators, we hope this would show an example of how to present an establish knowledge and modern theories by making a problem feasible for the broader undergraduate audience. We also briefly discuss the motivation of studying the problem in the prism of modern theories. Namely, we link this algorithm to the perfect quantum transfer problem. AcknowledgmentThe second author wishes to thank their linear algebra students who kept working on their projects in the midst of the COVID-19 pandemic and survived the switch to the distance learning. They were an incredible source of inspiration.Additional informationNotes on contributorsAlexander HolleyAlexander Holley (alexander.holley@uconn.edu) is pursuing a B.S. in Mechanical Engineering at the University of Connecticut. He plans on working in the aerospace field after graduation, or returning to school for a M. Sc. in a related engineering field. His interests include being in nature, cooking, and watching or playing sports.Anastasiia MinenkovaAnastasiia Minenkova (aminenkova@mtholyoke.edu) is a visiting lecturer at Mount Holyoke College. She received her M.Sc. in Mathematics from the University of Mississippi and her doctorate from the University of Connecticut. Her research interests are in numerical linear algebra. She is very passionate about teaching. Anastasiia’s interests outside of academia include cross-stitching and culinary tourism.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135315710","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 : 2023-10-23DOI: 10.1080/07468342.2023.2263074
Michelle Cordier, Meaghan Wheeler
SummarySuppose there are n points that we wish to locate on a plane. Instead of the locations of the points, we are given all the lines of k distinct slopes that contain the points. We show that the minimum number of slopes needed, in general, to find all the point locations is n + 1 and we provide an algorithm to do so. Additional informationNotes on contributorsMichelle Cordier Michelle Cordier (M.Doyle@chatham.edu) is a professor at Chatham University where she teaches mathematics and physics. She received her Ph.D. in mathematics from Kent State University. She enjoys being a member of the Mathematical Association of America Project New Experiences in Teaching (NExT) where she continually is changing her teaching style to incorporate her students.Meaghan Wheeler Meaghan Wheeler (meaghanwheeler99@gmail.com) is a microbiologist in the medical device industry. She received her bachelors in Biomedical Engineering from the University of Miami. She enjoys working as a microbiologist where she assists in the development of biocompatibility, cleaning, disinfection, and sterilization strategies for product launches.
{"title":"Searching for Point Locations Using Lines","authors":"Michelle Cordier, Meaghan Wheeler","doi":"10.1080/07468342.2023.2263074","DOIUrl":"https://doi.org/10.1080/07468342.2023.2263074","url":null,"abstract":"SummarySuppose there are n points that we wish to locate on a plane. Instead of the locations of the points, we are given all the lines of k distinct slopes that contain the points. We show that the minimum number of slopes needed, in general, to find all the point locations is n + 1 and we provide an algorithm to do so. Additional informationNotes on contributorsMichelle Cordier Michelle Cordier (M.Doyle@chatham.edu) is a professor at Chatham University where she teaches mathematics and physics. She received her Ph.D. in mathematics from Kent State University. She enjoys being a member of the Mathematical Association of America Project New Experiences in Teaching (NExT) where she continually is changing her teaching style to incorporate her students.Meaghan Wheeler Meaghan Wheeler (meaghanwheeler99@gmail.com) is a microbiologist in the medical device industry. She received her bachelors in Biomedical Engineering from the University of Miami. She enjoys working as a microbiologist where she assists in the development of biocompatibility, cleaning, disinfection, and sterilization strategies for product launches.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135322755","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 : 2023-10-23DOI: 10.1080/07468342.2023.2264719
Kenichi Hirose
SummaryWe present simple ways to compute finite sums of k-th powers of integers with the hockey-stick identity and the Stirling numbers.
本文给出了用曲棍球棍恒等式和斯特林数计算整数k次幂有限和的简单方法。
{"title":"Computing Sums of Powers of Integers with the Hockey-Stick Identity and the Stirling Numbers","authors":"Kenichi Hirose","doi":"10.1080/07468342.2023.2264719","DOIUrl":"https://doi.org/10.1080/07468342.2023.2264719","url":null,"abstract":"SummaryWe present simple ways to compute finite sums of k-th powers of integers with the hockey-stick identity and the Stirling numbers.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135413492","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 : 2023-10-23DOI: 10.1080/07468342.2023.2265478
Marc Frantz
SummaryWe show how a natural mistake in integration leads to insights about the natural logarithm.
我们展示了积分中的一个自然错误如何导致对自然对数的认识。
{"title":"The Natural Natural Logarithm","authors":"Marc Frantz","doi":"10.1080/07468342.2023.2265478","DOIUrl":"https://doi.org/10.1080/07468342.2023.2265478","url":null,"abstract":"SummaryWe show how a natural mistake in integration leads to insights about the natural logarithm.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135413657","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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