{"title":"An Elementary Treatment of the Lambert-W Relation","authors":"M. Moore","doi":"10.1080/07468342.2023.2188040","DOIUrl":"https://doi.org/10.1080/07468342.2023.2188040","url":null,"abstract":"Summary In this discussion, we introduced the Lambert-W relation, how to solve equations that are solvable via this relation and not solvable otherwise, discussed how to numerically calculate values, how to generate the real and imaginary parts of W for which have sufficiently negative or complex arguments. We then gave a brief insight into one of the physical applications of the Lambert-W relation that is used in fluid mechanics. Although one could introduce several new functions into the algebraic toolbox for those exploring mathematics, we have demonstrated that the Lambert-W relation is one that can easily be adapted into the toolbox of elementary-level mathematics students.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49238773","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":"Solutions of the Van der Pol Equation","authors":"S. D’Alessio","doi":"10.1080/07468342.2023.2191376","DOIUrl":"https://doi.org/10.1080/07468342.2023.2191376","url":null,"abstract":"Summary Presented in this paper are various solutions to the Van der Pol equation. Numerical solutions are utilized as an independent means of validating the various solutions discussed. A new solution in the form of a power series has been found. Although this solution is exact, its interval of convergence can only be estimated for a special case. Numerical experiments reveal that the power series solution can provide an exact solution over intervals where other approximate solutions are not valid. Thus, the new solution represents an additional solution that can complement other existing solutions. This work also emphasizes the importance and the role of computation. Although the power series solution along with the other approximate solutions mentioned are of theoretical interest, their restrictions limit their usefulness in real applications, and therefore numerical methods should also be considered.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43662373","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":"Two Comments on the Binomial Theorem","authors":"S. Litvinov, F. Marko","doi":"10.1080/07468342.2023.2186084","DOIUrl":"https://doi.org/10.1080/07468342.2023.2186084","url":null,"abstract":"Summary We present a calculus and a probabilistic proofs of the binomial theorem. We believe that the material presented in this article would be most suitable to students with some background in calculus and discrete mathematics. It can be used by instructors looking for interesting applications in a calculus or a discrete mathematics/probability course.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46559942","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":"A Generalization of Euler’s Limit","authors":"B. Chakraborty, Sagar Chakraborty","doi":"10.1080/07468342.2023.2183543","DOIUrl":"https://doi.org/10.1080/07468342.2023.2183543","url":null,"abstract":"Summary The famous Euler’s limit is In this note, we observe yet another generalization of Euler’s limit as follows: Let and be two sequence of real numbers such that an > 1 and and bn is satisfying the asymptotic formula , where k > 0, then","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49519134","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 Elementary Proof that Reduced Row Echelon form of a Matrix is Unique","authors":"B. Lotto","doi":"10.1080/07468342.2023.2184168","DOIUrl":"https://doi.org/10.1080/07468342.2023.2184168","url":null,"abstract":"Many results in a first course in linear algebra rely on the uniqueness of reduced echelon form of a given matrix. Most textbooks either omit the proof of this important result or use ideas and results in a proof that are not typically available when the result is introduced. For example, the proof [1] is relegated to an appendix and relies on the concept of linear dependence relations among columns, while the proof in [3] uses elementary matrices, permutation matrices, and block matrix calculations. Since the first course in linear algebra is often a first introduction to higher level conceptual thinking in mathematics and the careful use of definitions, theorems, and proofs, it is desirable to have an elementary proof of the uniqueness of reduced echelon form that is accessible to students at the time the fact is presented using only the fact that the solution sets of linear systems represented by row equivalent augmented matrices are the same. Consequently, this proof can be offered as a supplement to the main narrative of the class and might spark some interest among potential future majors. The ideas in this proof are not new (see [4] and [2], for example) but the idea of using augmented matrices in the argument is novel. The proof here is also written specifically for undergraduate students in their first linear algebra class.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45201744","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":"Report on the 13th Annual USA Junior Mathematical Olympiad","authors":"B. Bajnok","doi":"10.1080/07468342.2023.2165373","DOIUrl":"https://doi.org/10.1080/07468342.2023.2165373","url":null,"abstract":"Summary We present the problems and solutions to the 13th Annual United States of America Junior Mathematical Olympiad.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48233193","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":"A Simple Algorithm for Drawing Apportionment Diagrams","authors":"David McCune","doi":"10.1080/07468342.2022.2160618","DOIUrl":"https://doi.org/10.1080/07468342.2022.2160618","url":null,"abstract":"Abstract This article gives an algorithm that will build an apportionment diagram for any apportionment method when there are three states. We then provide several images that are produced by the algorithm and discuss what these images can reveal about the given apportionment method.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44080577","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":"Do Dogs Know Calculus With Early Transcendentals?","authors":"G. Weyenberg","doi":"10.1080/07468342.2022.2160615","DOIUrl":"https://doi.org/10.1080/07468342.2022.2160615","url":null,"abstract":"Summary A standard applications problem in differential calculus involves minimizing the travel time needed for a dog on a beach to reach a toy floating in the water, taking into account the different velocities the dog can achieve when running versus swimming. The solution usually presented to such problems are formulated in terms of a Cartesian coordinate. Here, a solution that is based in trigonometry is presented that, while quite natural, appears to be relatively unknown. The solution to the optimization problem in the trigonometric setting has a clear interpretation which is lacking in the Cartesian solution.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46641142","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":"Covering a Star Number with Pentagonal Numbers","authors":"Günhan Caglayan","doi":"10.1080/07468342.2022.2153554","DOIUrl":"https://doi.org/10.1080/07468342.2022.2153554","url":null,"abstract":"","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59407587","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":"Disjoint Placement Probability of Line Segments via Geometry","authors":"C. Ennis, J. Shier","doi":"10.1080/07468342.2022.2160619","DOIUrl":"https://doi.org/10.1080/07468342.2022.2160619","url":null,"abstract":"Abstract We have shown that when any finite number n, of line segments with total combined length less than one, have their centers placed randomly inside the unit interval , the probability of obtaining a mutually disjoint placement of the segments within , is given by the expression where , and denotes the length of the k-th segment, Lk . The result is established by a careful analysis of the geometry of the event, “all segments disjoint and contained within [0,1],” considered as a subset of the uniform probability space of n centers, each of which is in ; that is to say, the unit n-cube of . This event has an interesting geometric structure consisting of disjoint, congruent, (up to a mirror image) polytopes within the unit n-cube. It is shown these event polytopes fit together perfectly to form, except for a set of measure zero, a partition of an n-dimensional cube with common edge length , and hence an n-volume given by the formula. In the case of n = 3 segments, the polytopes form one of the known tetrahedral partitions of the cube as discussed, for example in [4]. In fact for all n > 0, the polytopes comprise a partition of the n-dimensional hypercube, and are therefore n-dimensional space filling.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45049989","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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