{"title":"Intuitive Interpretations of SVD Vectors","authors":"Robert D. Graff, K. Jabbusch, David James","doi":"10.1080/07468342.2023.2201567","DOIUrl":"https://doi.org/10.1080/07468342.2023.2201567","url":null,"abstract":"Summary The SVD is currently becoming an increasingly important central pillar in linear algebra. A voice as authoritative and influential as Gil Strang’s remarked that although a few years ago the SVD was not even part of an introductory linear algebra course, “now it has to be.” To balance the already firmly established theoretical basis, we survey several intuitive conceptualizations for singular vectors and values, expand on them, and provide some original contributions.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"54 1","pages":"200 - 211"},"PeriodicalIF":0.0,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48219062","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}
{"title":"Exploring Endless Space","authors":"D. Aldous","doi":"10.1080/07468342.2023.2201150","DOIUrl":"https://doi.org/10.1080/07468342.2023.2201150","url":null,"abstract":"Summary 4X games are a genre of modern computer games, exemplified by the Civilization series, whose initial stages implicitly involve exploring an unknown graph. One exploration scheme is “move to nearest unvisited vertex” and this is in fact forced in the game Endless Space. We revisit the small body of old “worst case” results about the efficiency of this scheme, and comment on the unstudied “average case” problem.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"54 1","pages":"179 - 185"},"PeriodicalIF":0.0,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44252725","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}
{"title":"Problems and Solutions","authors":"Greg Oman, Charles N. Curtis","doi":"10.1080/07468342.2023.2186094","DOIUrl":"https://doi.org/10.1080/07468342.2023.2186094","url":null,"abstract":"This section contains problems intended to challenge students and teachers of college mathematics. We urge you to participate actively both by submitting solutions and by proposing problems that are new and interesting. To promote variety, the editors welcome problem proposals that span the entire undergraduate curriculum. Proposed problems should be uploaded to the submission management system Submittable by visiting the web address https://cmj.submittable.com/submit (instructions are provided at this site). Alternatively, problem proposals may be sent to Greg Oman, either by email (preferred) as a pdf, TEX, or Word attachment or by mail to the address provided above. Whenever possible, a proposed problem should be accompanied by a solution, appropriate references, and any other material that would be helpful to the editors. Proposers should submit problems only if the proposed problem is not under consideration by another journal. Solutions to the problems in this issue should be uploaded to the submission management system Submittable by visiting the web address https://cmj.submittable.com/submit (instructions are provided at this site). Alternatively, solutions may be sent to Chip Curtis, either by email as a pdf, TEX, or Word attachment (preferred) or by mail to the address provided above, no later than September 15, 2023. Sending both pdf and TEX files is ideal.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"54 1","pages":"147 - 160"},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45609336","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}
{"title":"On that most over Skinned of Improper Integrals","authors":"S. Stewart","doi":"10.1080/07468342.2023.2186085","DOIUrl":"https://doi.org/10.1080/07468342.2023.2186085","url":null,"abstract":"Summary Continuing a much discussed topic of the various ways a particular improper integral can be evaluated, we give three further ways its generalization can be evaluated. Using techniques typically encountered immediately after the calculus sequence of courses we show how the improper integral can be evaluated using the beta and gamma functions, by first converting it to a double integral, and using a property of the Laplace transform.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"54 1","pages":"123 - 129"},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43073526","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}
{"title":"Bryn Mawr College Matriculation Exams from a Century Ago","authors":"William Dunham","doi":"10.1080/07468342.2023.2189820","DOIUrl":"https://doi.org/10.1080/07468342.2023.2189820","url":null,"abstract":"William Dunham (bdunham@brynmawr.edu) is a historian of mathematics who has written four books on the subject: Journey Through Genius, The Mathematical Universe, Euler: The Master of Us All, and The Calculus Gallery. He most recently co-edited (with Don Albers and Jerry Alexanderson) The G. H. Hardy Reader and is featured in “Great Thinkers, Great Theorems,” a DVD course from the Teaching Company. Since his retirement from Muhlenberg College (emeritus, 2014), Dunham has held visiting positions at Harvard, Princeton, Penn, Cornell, and Bryn Mawr, where he is currently a Research Associate in Mathematics.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"54 1","pages":"83 - 89"},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43988239","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}
{"title":"An Arithmetic Triangle Arising From a One-Way Street Grid","authors":"T. Richmond, Mia Sword","doi":"10.1080/07468342.2023.2186083","DOIUrl":"https://doi.org/10.1080/07468342.2023.2186083","url":null,"abstract":"Summary Pascal’s triangle arises by counting the number of shortest paths from (0, 0) to (n, k) on a square street grid. The length of the shortest path is the Manhattan distance from (0, 0) to (n, k). We consider the case of a square street grid of one-way streets, with successive parallel streets being oppositely directed. We investigate the associated distance function q (which is only a quasi-metric, since q(A, B) may differ from q(B, A)) and the arithmetic triangle obtained by counting shortest routes on the one-way grid from (0, 0) to (n, k).","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"54 1","pages":"113 - 122"},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44390011","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}
{"title":"Solving Linear Systems with Column Reduction","authors":"Romain Boulet","doi":"10.1080/07468342.2023.2186082","DOIUrl":"https://doi.org/10.1080/07468342.2023.2186082","url":null,"abstract":"Summary In order to solve a system of linear equations, students or teachers are used to performing a row reduction with the Gauss method. In this paper we propose to adopt a column point of view through two fundamental subspaces of a matrix—its kernel and its image—linked to the homogeneous system and to a particular solution of the system. This paper provides a method to find the set of solutions of a system by performing a column reduction of a double augmented matrix of the system.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"54 1","pages":"104 - 112"},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42060287","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}
{"title":"Media Highlights","authors":"T. Leise, P. Straffin","doi":"10.1080/07468342.2023.2186092","DOIUrl":"https://doi.org/10.1080/07468342.2023.2186092","url":null,"abstract":"Frank Farris is a well-known expositor on symmetry and art, and author of Creating Symmetry, The Artful Mathematics of Wallpaper Patterns (Princeton University Press, 2015). Farris describes his journey that began with an Illustrating Mathematics semester at the Institute for Computational and Experimental Research in Mathematics, where he learned to use Rhino, a program designed for architects, and Grasshopper, a plug-in to create shapes from formulas. He first designed the planar profile of a knot in the complex plane, then used Grasshopper to create a two-dimensional array of identical knots with their centers at the Eisenstein integers. These are complex numbers of the form a + bω, where ω = −1/2 + √3i/2 and a and b are integers. To design the 7-Color Torus, a digital print shown on the cover of the Math Horizons issue, Farris exploited the property that every Eisenstein integer yields one of 7 remainders when divided by 2 − ω. This allowed him to color the knots with 7 colors, according to the congruence classes of their centers. Four of the nearest centers with a specified color form a parallelogram, and a torus is the quotient of the plane by vectors that span the parallelogram. As each knot is surrounded by 6 knots of different colors, this construction shows that at least 7 colors must be used to color a map on a torus. The final figure shows an attractive scarf designed by mathematical artist Ellie Baker, based on the 7-color pattern in the article. PR","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"54 1","pages":"161 - 168"},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47606780","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}
{"title":"Proof Without Words: The Sum of Even Powers of Three","authors":"Rex H. Wu","doi":"10.1080/07468342.2023.2189821","DOIUrl":"https://doi.org/10.1080/07468342.2023.2189821","url":null,"abstract":"Summary We provide a visual proof to the sum of the even powers of 3, which can be generalized to for .","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":"54 1","pages":"139 - 139"},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47431893","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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