Pub Date : 2023-12-08DOI: 10.1080/0025570x.2023.2284419
J. K. Denny
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Pub Date : 2023-11-10DOI: 10.1080/0025570x.2023.2266324
Zsolt Lengvárszky, Debbie Shepherd
SummaryWe consider a variation of the “overhang” problem where blocks stick together.MSC: 70C20 AcknowledgmentsZsolt Lengvárszky is the recipient of the Miriam Sklar endowed professorship. This research was supported in part by the Louisiana Board of Regents Endowed Professor/Chair Program.Additional informationNotes on contributorsZsolt LengvárszkyZSOLT LENGVÁRSZKY (MR Author ID: 112490) received his degrees from the University of Szeged, Hungary, and the University of South Carolina. He joined the faculty of the Louisiana State University, Shreveport in 2008. His mathematical interests include lattice theory and the mathematics of paper folding.Debbie ShepherdDeborah K. Shepherd received her Ph. D. in Computational Analysis and Modeling from Louisiana Tech University in 2001. Her MS in Mathematics is from Southern Illinois University, Edwardsville. She is currently an associate professor at Louisiana State University, Shreveport. Her primary research interests are in statistical quality control.
我们考虑“悬垂”问题的一种变体，即块粘在一起。szsolt Lengvárszky是Miriam Sklar捐赠教授职位的获得者。这项研究得到了路易斯安那州校董会教授/主席项目的部分支持。szsolt LengvárszkyZSOLT LENGVÁRSZKY (MR作者ID: 112490)获得了匈牙利塞格德大学和南卡罗莱纳大学的学位。他于2008年加入路易斯安那州立大学什里夫波特分校。他的数学兴趣包括格理论和折纸数学。deborah K. Shepherd, 2001年在路易斯安那理工大学获得计算分析和建模博士学位。她在爱德华兹维尔的南伊利诺伊大学获得数学硕士学位。她目前是路易斯安那州立大学什里夫波特分校的副教授。她的主要研究兴趣是统计质量控制。
{"title":"Maximum Overhang of Sticky Stacks","authors":"Zsolt Lengvárszky, Debbie Shepherd","doi":"10.1080/0025570x.2023.2266324","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2266324","url":null,"abstract":"SummaryWe consider a variation of the “overhang” problem where blocks stick together.MSC: 70C20 AcknowledgmentsZsolt Lengvárszky is the recipient of the Miriam Sklar endowed professorship. This research was supported in part by the Louisiana Board of Regents Endowed Professor/Chair Program.Additional informationNotes on contributorsZsolt LengvárszkyZSOLT LENGVÁRSZKY (MR Author ID: 112490) received his degrees from the University of Szeged, Hungary, and the University of South Carolina. He joined the faculty of the Louisiana State University, Shreveport in 2008. His mathematical interests include lattice theory and the mathematics of paper folding.Debbie ShepherdDeborah K. Shepherd received her Ph. D. in Computational Analysis and Modeling from Louisiana Tech University in 2001. Her MS in Mathematics is from Southern Illinois University, Edwardsville. She is currently an associate professor at Louisiana State University, Shreveport. Her primary research interests are in statistical quality control.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":"121 23","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136344","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 : 2023-11-10DOI: 10.1080/0025570x.2023.2266313
Rex H. Wu
SummaryWe provide a visual proof to Lehmer’s infinite sum of the arctangents of the inverse of the odd-indexed Fibonacci numbers. A few corollaries follow from the diagram, including Euler’s Machin-like formula and Strassnitzky’s formula.MSC: 11B39 Additional informationNotes on contributorsRex H. WuREX H. WU (MR Author ID: 1293646, ORCID 0000-0003-0970-3741) would like to thank the anonymous reviewer and the Editor for their many generous suggestions. Rex has been working with Garfield’s trapezoid for many years. He recently found more applications of it on the Fibonacci numbers. This article is one of them. Talking about Garfield’s trapezoid, which is named after president James A. Garfield for his proof on the Pythagorean theorem, Rex unexpectedly met his great-great-grandson Mr. Peter Garfield recently.
摘要给出了奇索引斐波那契数列的倒数的正切无穷和的一个视觉证明。从图中可以得出一些推论，包括欧拉的类机器公式和斯特拉斯尼茨基的公式。作者简介:rex H. WuREX H. WU(作者ID: 1293646, ORCID 0000-0003-0970-3741)在此感谢匿名审稿人和编辑的慷慨建议。雷克斯多年来一直在研究加菲猫的梯形。他最近在斐波那契数列上发现了更多的应用。这篇文章就是其中之一。最近，雷克斯在谈到以证明毕达哥拉斯定理的美国总统詹姆斯·a·加菲尔德命名的加菲猫梯形时，意外地遇到了他的曾曾孙彼得·加菲尔德先生。
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Pub Date : 2023-11-09DOI: 10.1080/0025570x.2023.2266345
Allen Charest, Ben Coté, Ward Heilman
SummaryThe center cubies on the Rubik’s Cube can change orientation when the puzzle is brought from one solved state to another. The set of all possible reorientations of the center cubies creates what we call the invisible solutions group. We investigate the size and structure of the invisible solutions group for the Rubik’s Cube, Rubik’s Revenge, and the Professor’s Cube.MSC: 20-01 AcknowledgmentsThe authors would like to thank the Adrian Tinsley Program for Undergraduate Research and Creative Scholarship for funding and the anonymous referees for the helpful comments.Notes1 Online version of the article contains color diagrams.Additional informationNotes on contributorsAllen CharestALLEN CHAREST received a Bachelors in Mathematics and Secondary Education from Bridgewater State University in 2019. He currently works as STEAM Math Teacher at Greater Lawrence Technical School and is fascinated by group theory and nature.Ben CotéBEN COTÉ (MR Author ID: 951394, ORCID 0000-0003-2844-1935) received a Ph.D. in Mathematics from the University of California, Santa Barbara in 2016. He currently teaches at Western Oregon University. When not investigating recreational mathematics, he enjoys camping and gardening with his wife Brittany and sons Levi and Oliver.Ward HeilmanWARD HEILMAN received a Ph.D. in Mathematics from Northeastern University. He has been at Bridgewater State University (Mass.) since 1996. He is active in social justice, and fascinated by axioms, cryptology, basketball, Huxley, Kerouac and most recently Thomas Paine.
{"title":"The Invisible Solutions of the Rubik’s Cube","authors":"Allen Charest, Ben Coté, Ward Heilman","doi":"10.1080/0025570x.2023.2266345","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2266345","url":null,"abstract":"SummaryThe center cubies on the Rubik’s Cube can change orientation when the puzzle is brought from one solved state to another. The set of all possible reorientations of the center cubies creates what we call the invisible solutions group. We investigate the size and structure of the invisible solutions group for the Rubik’s Cube, Rubik’s Revenge, and the Professor’s Cube.MSC: 20-01 AcknowledgmentsThe authors would like to thank the Adrian Tinsley Program for Undergraduate Research and Creative Scholarship for funding and the anonymous referees for the helpful comments.Notes1 Online version of the article contains color diagrams.Additional informationNotes on contributorsAllen CharestALLEN CHAREST received a Bachelors in Mathematics and Secondary Education from Bridgewater State University in 2019. He currently works as STEAM Math Teacher at Greater Lawrence Technical School and is fascinated by group theory and nature.Ben CotéBEN COTÉ (MR Author ID: 951394, ORCID 0000-0003-2844-1935) received a Ph.D. in Mathematics from the University of California, Santa Barbara in 2016. He currently teaches at Western Oregon University. When not investigating recreational mathematics, he enjoys camping and gardening with his wife Brittany and sons Levi and Oliver.Ward HeilmanWARD HEILMAN received a Ph.D. in Mathematics from Northeastern University. He has been at Bridgewater State University (Mass.) since 1996. He is active in social justice, and fascinated by axioms, cryptology, basketball, Huxley, Kerouac and most recently Thomas Paine.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":" 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192196","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 : 2023-11-09DOI: 10.1080/0025570x.2023.2266329
Pisheng Ding, Sunil K. Chebolu
SummaryGeneralized sine functions, primarily viewed as real-valued functions, are studied herein as complex analytic functions. Their natural domains of analyticity are identified and their mapping properties are detailed. Some implications of this theory for calculus and geometry are explored.MSC:: 30-02 Notes* For z near 1, factor (1−z4)3/4 into g(z)(z−1)3/4 where g is continuous and nonzero at 1. Thus, it suffices to track the phase change of (z−1)3/4 as z traverses an arc in V¯4 from 1−ϵ to 1+ϵ.* The Schwarz reflection principle also guarantees that the extended function is analytic on [O,B4).* At the other half of the vertices, i.e., the translates of the ω3kA3’s, the extended sin 3 is bounded and hence analytic by Riemann’s principle of removable singularity.Additional informationNotes on contributorsPisheng DingPISHENG DING (MR Author ID: 784635) studied physics and mathematics as an undergraduate at the legendary City College of New York, where Joseph Bak kindled in him a keen interest in complex analysis. He earned his doctorate in 2003 from the Courant Institute under the direction of Sylvain Cappell. So that his family could be under one roof, he joined Illinois State University in 2010 as an adjunct and has since remained in this position. While deploring the state of mathematics education in the United States, he currently enjoys teaching his 6th-grade daughter Elaine authentic Euclidean geometry.Sunil K. CheboluSUNIL CHEBOLU (MR Author ID: 781874) received his Ph.D. in mathematics from the University of Washington in 2005. After completing a three-year postdoctoral fellowship at the University of Western Ontario in Canada, he joined the faculty at Illinois State University in 2008. Although his primary research interests lie in algebra and number theory, he embraces all areas of mathematics. During his spare time, he enjoys playing his guitar or observing deep-sky objects through his telescope.
广义正弦函数最初被视为实值函数，本文将其作为复解析函数进行研究。确定了它们的自然分析域，并详细说明了它们的映射属性。探讨了这一理论对微积分和几何的一些启示。注*对于z接近1时，因子(1−z4)3/4化为g(z)(z−1)3/4，其中g在1处连续且非零。因此，它足以跟踪(z−1)3/4的相位变化，因为z在V¯4中从1−λ到1+ λ。* Schwarz反射原理也保证了扩展函数在[O,B4)上是解析的。*在另一半的顶点，即ω3kA3的平移，扩展的sin3是有界的，因此可以用黎曼可移动奇点原理解析。作者ID: 784635)在传奇的纽约城市学院(City College of New York)读本科时学习物理和数学，在那里，约瑟夫·巴克点燃了他对复杂分析的浓厚兴趣。2003年，他在Sylvain Cappell的指导下从Courant Institute获得博士学位。为了让家人住在一个屋檐下，他于2010年加入伊利诺伊州立大学(Illinois State University)，成为一名兼职教师，此后一直担任这一职位。虽然他对美国的数学教育状况感到遗憾，但他目前很喜欢教他六年级的女儿伊莱恩正宗的欧几里得几何。Sunil K. CHEBOLU(作者ID: 781874)， 2005年在华盛顿大学获得数学博士学位。在加拿大西安大略大学(University of Western Ontario)完成三年博士后研究后，他于2008年加入伊利诺伊州立大学(Illinois State University)任教。虽然他的主要研究兴趣是代数和数论，但他也涉猎数学的各个领域。在业余时间，他喜欢弹吉他或通过望远镜观察深空物体。
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SummaryAccording to Buffon’s needle problem, you can approximate the value of π by throwing needles onto arrays of parallel lines and counting the number of needles crossing the lines. However, it is time-consuming to do the experiment in reality (in 1901, Lazzarini threw 3408 needles to approximate the value of π). In this paper, we tried to approximate the value of π from easily-accesible images of fallen leaves. We extended Buffon’s needle problem in two ways: instead of a simple needle we considered a complex shape, and instead of a uniform size we considered various sizes. After we confirmed the extension by a computer simulation, we approximated the value of π from the images we took experimentally and we while walking. From the experimentally-taken images (50 leaves/image × 20 images), the value of π was approximated as 3.1569. The approximated value of π using the images we took while walking was 3.5961 (the accuracy was not good because it was calculated from one image).MSC: 60-01 Notes1 Note that the online version of this article has color diagramsAdditional informationNotes on contributorsRuka IguchiRUKA IGUCHI is a 3rd-year high school student at Hiroshima University High School. She is a runner and has often participated in a marathon.Hideaki KidaHIDEAKI KIDA is a mathematics teacher at Hiroshima University High School. After receiving a Master of Education degree from Hiroshima University, he has been a high school teacher for 22 years.Keisuke MatsudaKEISUKE MATSUDA is a graduate student at Osaka University, studying beetle horn formation using mathematical modeling. He seeks the rules underlying the natural phenomena around us and the shapes of living things.Miki OnoMIKI ONO is a 3rd-year high school student at Hiroshima University High School and a singer who has an attractive voice.Miu ShibataMIU SHIBATA is a 3rd-year high school student at Hiroshima University High School and a violinist who is good at computer programming.Haruka TakanoHARUKA TAKANO is a 3rd-year high school student at Hiroshima University High School and a contrabassist who loves grilled meat and ice cream as much as mathematics.
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Pub Date : 2023-11-07DOI: 10.1080/0025570x.2023.2266334
Pentti Haukkanen, Timo Tossavainen
SummaryWe introduce a set of axioms that define two relations: planar perpendicularity and parallelism. We show how these relations can be explored informally through play with secondary school students. Our aim is to demonstrate that axiomatic thinking can be made accessible and fun even for young learners.MSC: Primary 97-01Secondary 51M04 Additional informationNotes on contributorsPentti HaukkanenPentti Haukkanen (MR Author ID: 82550) received his PhD in mathematics from the University of Tampere, Finland, under the supervision of Seppo Hyyrö. Currently, he is a university lecturer of mathematics at Tampere University in Finland. In his spare time, he enjoys various sports and culture.Timo TossavainenTimo Tossavainen (MR Author ID: 664009) received his PhD in mathematics from Jyväskylä University, Finland, under the supervision of Pekka Koskela. Currently, he is a professor of mathematics education at the Lulea University of Technology in Sweden. Among his proudest accomplishments is having learnt to play the guitar solo to the smash hit “The Final Countdown” by the Swedish rock band Europe.
我们引入了一组公理来定义平面垂直和平行两种关系。我们展示了如何通过与中学生的游戏来非正式地探索这些关系。我们的目的是证明，即使是年轻的学习者，公理思维也可以变得容易理解和有趣。spentti Haukkanen(作者ID: 82550)获得芬兰坦佩雷大学数学博士学位，在Seppo Hyyrö的指导下。目前，他是芬兰坦佩雷大学的数学讲师。在业余时间，他喜欢各种体育和文化。Timo Tossavainen(作者ID: 664009)，芬兰Jyväskylä大学数学博士，导师为Pekka Koskela。目前，他是瑞典吕勒奥理工大学(Lulea University of Technology)数学教育学教授。他最自豪的成就之一是学会了为瑞典摇滚乐队“欧洲”(Europe)的热门歌曲《最后的倒计时》(the Final Countdown)演奏吉他独奏。
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Pub Date : 2023-11-06DOI: 10.1080/0025570x.2023.2266389
Matthew C. King, Noah A. Rosenberg
How many ways are there to arrange the sequence of games in a single-elimination sports tournament? We consider the connection between this enumeration problem and the enumeration of “labeled histories,” or sequences of asynchronous branching events, in mathematical phylogenetics. The possibility of playing multiple games simultaneously in different arenas suggests an extension of the enumeration of labeled histories to scenarios in which multiple branching events occur simultaneously. We provide a recursive result enumerating game sequences and labeled histories in which simultaneity is allowed. For a March Madness basketball tournament of 68 labeled teams, the number of possible sequences of games is ∼1.91×1078 if arbitrarily many arenas are available, but only ∼3.60×1068 if all games must be played sequentially in the same arena.
{"title":"A Mathematical Connection Between Single-Elimination Sports Tournaments and Evolutionary Trees","authors":"Matthew C. King, Noah A. Rosenberg","doi":"10.1080/0025570x.2023.2266389","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2266389","url":null,"abstract":"How many ways are there to arrange the sequence of games in a single-elimination sports tournament? We consider the connection between this enumeration problem and the enumeration of “labeled histories,” or sequences of asynchronous branching events, in mathematical phylogenetics. The possibility of playing multiple games simultaneously in different arenas suggests an extension of the enumeration of labeled histories to scenarios in which multiple branching events occur simultaneously. We provide a recursive result enumerating game sequences and labeled histories in which simultaneity is allowed. For a March Madness basketball tournament of 68 labeled teams, the number of possible sequences of games is ∼1.91×1078 if arbitrarily many arenas are available, but only ∼3.60×1068 if all games must be played sequentially in the same arena.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135589104","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 : 2023-11-02DOI: 10.1080/0025570x.2023.2266346
Matthew Glomski, K. Peter Krog, Mason Nakamura, Elizabeth M. Reid
SummaryInspired by a problem presented in a New York Times essay, we consider the number of triangles formed by a finite collection of lines in the plane, under the restrictions that no more than two lines intersect at any point and that no two are parallel. We explore the ramifications of relaxing each of these requirements, and we derive a ‘unified triangle counting formula’ for any arrangement of finitely many lines in the plane.MSC: 05B30 Additional informationNotes on contributorsMatthew GlomskiMATTHEW GLOMSKI earned his Ph.D. at the University at Buffalo and joined the mathematics faculty of Marist College. In his spare time he enjoys hiking in the nearby Catskill Mountains.K. Peter KrogK. PETER KROG earned his Ph.D. at the University of Connecticut and joined the mathematics faculty at Marist College in 1996. His mathematical interests include group theory, statistics, and combinatorics.Mason NakamuraMASON NAKAMURA is an applied mathematics student at Marist College and plans to pursue his doctorate. He enjoys golfing, hiking, and snorkeling during his time away from mathematics.Elizabeth M. ReidELIZABETH M. REID is a member of the mathematics faculty at Marist College. She earned her Ph.D. at the University at Buffalo and enjoys hiking in her spare time.
受《纽约时报》一篇文章中提出的问题的启发，我们考虑平面上有限的直线集合在不超过两条直线相交且不平行的条件下形成的三角形的数量。我们探索了放宽这些要求的后果，并推导出一个“统一三角形计数公式”，适用于平面上有限多条线的任何排列。matthew GLOMSKI在布法罗大学(University at Buffalo)获得博士学位，并加入圣母学院(Marist College)数学系。在业余时间，他喜欢在附近的卡茨基尔山脉徒步旅行。彼得KrogK。PETER KROG在康涅狄格大学获得博士学位，并于1996年加入圣母学院数学系。他的数学兴趣包括群论、统计学和组合学。Mason NAKAMURA是圣母学院应用数学专业的学生，他计划继续攻读博士学位。在不学习数学的时间里，他喜欢打高尔夫球、远足和浮潜。Elizabeth M. REID是圣母学院数学系的一员。她在布法罗大学获得博士学位，业余时间喜欢徒步旅行。
{"title":"Counting Triangles with Combinatorics","authors":"Matthew Glomski, K. Peter Krog, Mason Nakamura, Elizabeth M. Reid","doi":"10.1080/0025570x.2023.2266346","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2266346","url":null,"abstract":"SummaryInspired by a problem presented in a New York Times essay, we consider the number of triangles formed by a finite collection of lines in the plane, under the restrictions that no more than two lines intersect at any point and that no two are parallel. We explore the ramifications of relaxing each of these requirements, and we derive a ‘unified triangle counting formula’ for any arrangement of finitely many lines in the plane.MSC: 05B30 Additional informationNotes on contributorsMatthew GlomskiMATTHEW GLOMSKI earned his Ph.D. at the University at Buffalo and joined the mathematics faculty of Marist College. In his spare time he enjoys hiking in the nearby Catskill Mountains.K. Peter KrogK. PETER KROG earned his Ph.D. at the University of Connecticut and joined the mathematics faculty at Marist College in 1996. His mathematical interests include group theory, statistics, and combinatorics.Mason NakamuraMASON NAKAMURA is an applied mathematics student at Marist College and plans to pursue his doctorate. He enjoys golfing, hiking, and snorkeling during his time away from mathematics.Elizabeth M. ReidELIZABETH M. REID is a member of the mathematics faculty at Marist College. She earned her Ph.D. at the University at Buffalo and enjoys hiking in her spare time.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135933836","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 : 2023-11-01DOI: 10.1080/0025570x.2023.2266415
Tom Edgar
SummaryWe use a square inside another to provide two geometric series dissection proofs. Notes1 Note that the online version of this article has color diagramsAdditional informationNotes on contributorsTom EdgarTOM EDGAR (MR Author ID: 821633) is a professor of mathematics at Pacific Lutheran University and the editor of Math Horizons. He has recently been animating his favorite visual proofs on his YouTube channel www.youtube.com/@MathVisualProofs/.
{"title":"Two Geometric Series","authors":"Tom Edgar","doi":"10.1080/0025570x.2023.2266415","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2266415","url":null,"abstract":"SummaryWe use a square inside another to provide two geometric series dissection proofs. Notes1 Note that the online version of this article has color diagramsAdditional informationNotes on contributorsTom EdgarTOM EDGAR (MR Author ID: 821633) is a professor of mathematics at Pacific Lutheran University and the editor of Math Horizons. He has recently been animating his favorite visual proofs on his YouTube channel www.youtube.com/@MathVisualProofs/.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":"59 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135270601","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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