Pub Date : 2023-08-08DOI: 10.1080/07468342.2023.2237851
Melissa Holly
SummaryLarge cliques with cardinality |ω|>⌈23n⌉ cannot exist with large independent sets where |α|>⌈23n⌉ in the same simple connected graph G of order n≥2. Additional informationNotes on contributorsMelissa Holly Melissa Holly (mholly@vcu.edu) received her bachelor’s degree from University of Illinois at Chicago Circle in 1974. In 2017, she became a graduate student in the Department of Mathematics and Applied Mathematics at Virginia Commonwealth University. Her favorite cycling trail is one with tall trees that reach for the sky where they hold hands.
{"title":"Large Independent Set vs. Large Clique","authors":"Melissa Holly","doi":"10.1080/07468342.2023.2237851","DOIUrl":"https://doi.org/10.1080/07468342.2023.2237851","url":null,"abstract":"SummaryLarge cliques with cardinality |ω|>⌈23n⌉ cannot exist with large independent sets where |α|>⌈23n⌉ in the same simple connected graph G of order n≥2. Additional informationNotes on contributorsMelissa Holly Melissa Holly (mholly@vcu.edu) received her bachelor’s degree from University of Illinois at Chicago Circle in 1974. In 2017, she became a graduate student in the Department of Mathematics and Applied Mathematics at Virginia Commonwealth University. Her favorite cycling trail is one with tall trees that reach for the sky where they hold hands.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840440","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 : 2023-08-08DOI: 10.1080/07468342.2023.2237850
Jathan Austin, Brian G. Kronenthal, Jonathon Miller, Susanna Molitoris-Miller
SummaryCatan is a dynamic property-building and trading board game in which players build a new board every time they play by arranging tiles, number tokens, and port markers. In the interest of creating an equitable board, the official Catan rules restrict how the number tokens are placed. In this paper, we use three techniques to count the number of nonequivalent boards satisfying this restriction, as well as determine the probability that a randomly generated board will be legal. AcknowledgmentWe thank the anonymous referees whose time and attention supported the publication of this work.Additional informationNotes on contributorsJathan Austin Jathan Austin (jwaustin@salisbury.edu) is an associate professor of mathematics at Salisbury University in Maryland. He earned a B.S. in mathematics from Salisbury, and both an M.S. in mathematics and a Ph.D. in mathematics education from the University of Delaware. In his spare time he enjoys doing jigsaw puzzles and watching classic sci-fi.Brian G. Kronenthal Brian G. Kronenthal (kronenthal@kutztown.edu) is a professor of mathematics at Kutztown University of Pennsylvania. He earned his B.S. in mathematics from Lafayette College (Easton, Pennsylvania), as well as his M.S. and Ph.D. in mathematics from the University of Delaware. His favorite research problems are combinatorial, often with an algebraic flair. In addition to teaching and research, he enjoys playing ping pong, watching movies, and rooting for Philadelphia sports teams.Jonathon Miller Jonathon Miller (jonathonamiller@gmail.com) is a software developer for Amazon. He earned his degree in mathematics and a minor in computer science from Salisbury University in Maryland. He also earned his MS in mathematics from The University of Delaware. In his free time Jonathon enjoys being a husband and father, playing board games, and serving as an excellent DM for Dungeons and Dragons.Susanna Molitoris-Miller Susanna Molitoris-Miller (susannamolitorismiller@gmail.com) earned her B.S. in mathematics from The University of Scranton, and M.S. in mathematics and Ph.D. in mathematics education from The University of Delaware. Susanna’s research focuses on how students learn mathematical concepts in creative ways. In her free time she enjoys spending time with her family, fiber arts, tea and, of course, game night with friends.
《卡坦岛》是一款动态的建造和交易棋盘游戏，玩家每次玩游戏时都会通过排列瓷砖、数字标记和港口标记来建造一个新的棋盘。为了创建一个公平的董事会，官方的卡坦规则限制了数字代币的放置方式。在本文中，我们使用三种技术来计算满足此限制的非等效棋盘的数量，以及确定随机生成的棋盘合法的概率。我们感谢匿名审稿人，他们的时间和精力支持了本文的出版。附加信息贡献者说明贾森·奥斯汀贾森·奥斯汀(jwaustin@salisbury.edu)是马里兰州索尔兹伯里大学的数学副教授。他在索尔兹伯里大学获得数学学士学位，在特拉华大学获得数学硕士学位和数学教育博士学位。在业余时间，他喜欢玩拼图游戏和看经典科幻电影。Brian G. Kronenthal (kronenthal@kutztown.edu)是宾夕法尼亚州库茨敦大学的数学教授。他在Lafayette College (Easton, Pennsylvania)获得数学学士学位，并在特拉华大学(University of Delaware)获得数学硕士和博士学位。他最喜欢的研究问题是组合问题，通常带有代数天赋。除了教学和研究之外，他还喜欢打乒乓球、看电影和为费城运动队加油。乔纳森·米勒(jonathonamiller@gmail.com)是亚马逊的软件开发人员。他在马里兰州索尔兹伯里大学(Salisbury University)获得了数学学位，辅修了计算机科学。他还获得了特拉华大学数学硕士学位。在业余时间，Jonathon喜欢做一个丈夫和父亲，玩棋盘游戏，并担任《龙与地下城》的优秀DM。苏珊娜·莫里托里斯-米勒(susannamolitorismiller@gmail.com)，毕业于斯克兰顿大学数学学士学位，特拉华大学数学硕士和数学教育博士学位。苏珊娜的研究重点是学生如何以创造性的方式学习数学概念。在空闲时间，她喜欢与家人共度时光，纤维艺术，喝茶，当然还有与朋友们的游戏之夜。
{"title":"Caught “Red”-Handed? The Probability of Randomly Constructing a Legal <i>Catan</i> Board","authors":"Jathan Austin, Brian G. Kronenthal, Jonathon Miller, Susanna Molitoris-Miller","doi":"10.1080/07468342.2023.2237850","DOIUrl":"https://doi.org/10.1080/07468342.2023.2237850","url":null,"abstract":"SummaryCatan is a dynamic property-building and trading board game in which players build a new board every time they play by arranging tiles, number tokens, and port markers. In the interest of creating an equitable board, the official Catan rules restrict how the number tokens are placed. In this paper, we use three techniques to count the number of nonequivalent boards satisfying this restriction, as well as determine the probability that a randomly generated board will be legal. AcknowledgmentWe thank the anonymous referees whose time and attention supported the publication of this work.Additional informationNotes on contributorsJathan Austin Jathan Austin (jwaustin@salisbury.edu) is an associate professor of mathematics at Salisbury University in Maryland. He earned a B.S. in mathematics from Salisbury, and both an M.S. in mathematics and a Ph.D. in mathematics education from the University of Delaware. In his spare time he enjoys doing jigsaw puzzles and watching classic sci-fi.Brian G. Kronenthal Brian G. Kronenthal (kronenthal@kutztown.edu) is a professor of mathematics at Kutztown University of Pennsylvania. He earned his B.S. in mathematics from Lafayette College (Easton, Pennsylvania), as well as his M.S. and Ph.D. in mathematics from the University of Delaware. His favorite research problems are combinatorial, often with an algebraic flair. In addition to teaching and research, he enjoys playing ping pong, watching movies, and rooting for Philadelphia sports teams.Jonathon Miller Jonathon Miller (jonathonamiller@gmail.com) is a software developer for Amazon. He earned his degree in mathematics and a minor in computer science from Salisbury University in Maryland. He also earned his MS in mathematics from The University of Delaware. In his free time Jonathon enjoys being a husband and father, playing board games, and serving as an excellent DM for Dungeons and Dragons.Susanna Molitoris-Miller Susanna Molitoris-Miller (susannamolitorismiller@gmail.com) earned her B.S. in mathematics from The University of Scranton, and M.S. in mathematics and Ph.D. in mathematics education from The University of Delaware. Susanna’s research focuses on how students learn mathematical concepts in creative ways. In her free time she enjoys spending time with her family, fiber arts, tea and, of course, game night with friends.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840443","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 : 2023-08-08DOI: 10.1080/07468342.2023.2237854
Peter M. Higgins
AbstractFor any polynomial p(x) with real coefficients and of degree n≥1, for sufficiently large positive x there is a unique y, distinct from x, such that p(x) and p(y) are equal in absolute value. We show that, in the limit, the mean of x and y is equal to the mean of the roots of p(x).
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Pub Date : 2023-08-08DOI: 10.1080/07468342.2023.2237849
Hayden Ruff, Kenneth Schilling
AbstractWe explore a family of closed curves in the plane. Each member of the family is the limit of self-intersecting polygonal curves. Nonetheless, most members of the family are simple closed curves. One, the “attractive attractor,” surrounds an infinite set of disjoint regions. AcknowledgmentsWe are grateful to the referee for a careful reading and many helpful suggestions.Additional informationNotes on contributorsHayden Ruff Hayden Ruff (hr442@drexel.edu) is a pre-candidacy mathematics Ph.D. student enrolled at Drexel University in Philadelphia, PA. He completed a B.S. in mathematics and physics at the University of Michigan – Flint, where he collaborated with and received instruction from Dr. Schilling. His research interests include signal processing, psychoacoustics, and applications of mathematics to the study of musicality.Kenneth Schilling Kenneth Schilling (ksch@umich.edu) received his Ph.D. from the University of California, Berkeley in 1981. He is professor emeritus of mathematics from the University of Michigan-Flint. During his years on the faculty, he loved working with students and interactiing with colleagues and pretty much everything about university life except grading exams.
摘要研究平面上的闭曲线族。族中的每一个成员都是自交多边形曲线的极限。尽管如此，这个家族的大多数成员都是简单的封闭曲线。一种是“吸引吸引子”，它围绕着一组无限不相交的区域。我们非常感谢推荐人的仔细阅读和许多有益的建议。Hayden Ruff (hr442@drexel.edu)是宾夕法尼亚州费城德雷塞尔大学的数学博士生。他在密歇根大学弗林特分校(University of Michigan - Flint)完成了数学和物理学士学位，在那里他与Schilling博士合作并接受他的指导。他的研究兴趣包括信号处理、心理声学和数学在音乐性研究中的应用。Kenneth Schilling (ksch@umich.edu) 1981年获得加州大学伯克利分校博士学位。他是密歇根大学弗林特分校的数学名誉教授。在教职期间，他喜欢和学生一起工作，喜欢和同事交流，喜欢大学生活的一切，除了批改考试。
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Pub Date : 2023-08-08DOI: 10.1080/07468342.2023.2229224
Greg Oman, Charles N. Curtis
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Pub Date : 2023-08-08DOI: 10.1080/07468342.2023.2240204
James S. Wolper
AbstractOne can learn a lot about a differential equation without solving it. Using the relationship between the sign of the derivative of a function and its behavior, given by the Mean Value Theorem, makes many important properties of solutions clear, and allows one to consider variations that are difficult or ugly to solve exactly. Additional informationNotes on contributorsJames S. WolperJames Wolper (wolpjame@isu.edu) received his Ph.D. in Mathematics from Brown University. He is an Emeritus Professor at Idaho State University. Most of his research has been in Algebraic Geometry, most recently into the statistical properties of periods of Riemann Surfaces.
一个人不求解微分方程也能学到很多东西。利用中值定理给出的函数导数的符号与其行为之间的关系，使解的许多重要性质变得清晰，并允许人们考虑难以或难看地精确求解的变化。james S. WolperJames Wolper (wolpjame@isu.edu)获得布朗大学数学博士学位。他是爱达荷州立大学的名誉教授。他的大部分研究都集中在代数几何上，最近研究的是黎曼曲面周期的统计性质。
{"title":"Not Solving Differential Equations","authors":"James S. Wolper","doi":"10.1080/07468342.2023.2240204","DOIUrl":"https://doi.org/10.1080/07468342.2023.2240204","url":null,"abstract":"AbstractOne can learn a lot about a differential equation without solving it. Using the relationship between the sign of the derivative of a function and its behavior, given by the Mean Value Theorem, makes many important properties of solutions clear, and allows one to consider variations that are difficult or ugly to solve exactly. Additional informationNotes on contributorsJames S. WolperJames Wolper (wolpjame@isu.edu) received his Ph.D. in Mathematics from Brown University. He is an Emeritus Professor at Idaho State University. Most of his research has been in Algebraic Geometry, most recently into the statistical properties of periods of Riemann Surfaces.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840437","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 : 2023-08-08DOI: 10.1080/07468342.2023.2238581
Armengol Gasull
"Proof Without Words: Tangent of π/8.." The College Mathematics Journal, ahead-of-print(ahead-of-print), p. 1
“不用语言证明:π/8的正切…”《大学数学学报》，第1页
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Pub Date : 2023-08-08DOI: 10.1080/07468342.2023.2223509
Don Chakerian, Stephen Erfle
Click to increase image sizeClick to decrease image size AcknowledgmentsWe are thankful for the LaTeX wizardry of our colleagues Xiaozhou Ding and David Richeson.Additional informationNotes on contributorsDon Chakerian Don Chakerian (dmandgd@aol.com) received his Ph.D. from UC Berkeley and is Emeritus Professor of Mathematics at the University of California, Davis. His area of research is the theory of convex sets and geometric inequalities. He earned the George Pólya Award from the MAA in 1981 for his paper Circles and Spheres, College Mathematics Journal, vol.11, pp.26–41. In it, he mentions an elegant proof of a theorem of N.A. Court, found by one of his favorite undergraduate students, by name of Steve Erfle.Stephen Erfle Stephen Erfle (erfle@dickinson.edu) took as many classes as he could from Don Chakerian while attending UC Davis as an undergraduate. He received his Ph.D. in Economics from Harvard and is Professor of International Business and Management at Dickinson College. He is currently working on a recreational mathematics book entitled Playing with Polygons. Some of the Excel files that form the backbone of this book allow users to produce electronic string art on a polygonal vertex frame. Some images were sufficiently interesting that he enticed his mentor and now coauthor into jointly working on this paper.
{"title":"Up the Hill and Down Again","authors":"Don Chakerian, Stephen Erfle","doi":"10.1080/07468342.2023.2223509","DOIUrl":"https://doi.org/10.1080/07468342.2023.2223509","url":null,"abstract":"Click to increase image sizeClick to decrease image size AcknowledgmentsWe are thankful for the LaTeX wizardry of our colleagues Xiaozhou Ding and David Richeson.Additional informationNotes on contributorsDon Chakerian Don Chakerian (dmandgd@aol.com) received his Ph.D. from UC Berkeley and is Emeritus Professor of Mathematics at the University of California, Davis. His area of research is the theory of convex sets and geometric inequalities. He earned the George Pólya Award from the MAA in 1981 for his paper Circles and Spheres, College Mathematics Journal, vol.11, pp.26–41. In it, he mentions an elegant proof of a theorem of N.A. Court, found by one of his favorite undergraduate students, by name of Steve Erfle.Stephen Erfle Stephen Erfle (erfle@dickinson.edu) took as many classes as he could from Don Chakerian while attending UC Davis as an undergraduate. He received his Ph.D. in Economics from Harvard and is Professor of International Business and Management at Dickinson College. He is currently working on a recreational mathematics book entitled Playing with Polygons. Some of the Excel files that form the backbone of this book allow users to produce electronic string art on a polygonal vertex frame. Some images were sufficiently interesting that he enticed his mentor and now coauthor into jointly working on this paper.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840442","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 : 2023-08-08DOI: 10.1080/07468342.2023.2237853
Ron Lycan
SummaryA quasigroup is a set with a binary operation in which both left and right division are unique. Equivalently, every row and column in a quasigroup table is a permutation of its elements. The commuting probability of a quasigroup is the probability that two of its elements, chosen at random, will commute. In this paper, we show that a quasigroup may have any rational number in (0,1] as a commuting probability. AcknowledgmentsThe author would like to thank their advisor, Vadim Ponomarenko, for helping and supporting them throughout the process of writing this article.
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