College Mathematics Journal最新文献

{"title":"A Curious Property of Tangents","authors":"David Treeby, Marty Ross","doi":"10.1080/07468342.2022.2160602","DOIUrl":"https://doi.org/10.1080/07468342.2022.2160602","url":null,"abstract":"Summary We give two proofs of a surprising elementary result concerning tangents drawn to the graphs of convex functions.","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":"46295058","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 Note on Teaching the Riemann Integral","authors":"B. S. Thomson","doi":"10.1080/07468342.2022.2160588","DOIUrl":"https://doi.org/10.1080/07468342.2022.2160588","url":null,"abstract":"Summary There are alternative definitions for the Riemann integral, many of which avoid some of the unpleasant computations that arise when using Riemann sums. In this version a simple distance function for step functions is used and the Riemann integral is defined and developed by employing exclusively “convergent” sequences of step functions. While only modestly different from the standard presentation it might have some extra intuitive appeal. This is a common device in advanced theories of integration and can be introduced at this elementary level.","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":"41730534","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":"When Triangles Are Similar","authors":"Ricardo E. Rojas","doi":"10.1080/07468342.2022.2150506","DOIUrl":"https://doi.org/10.1080/07468342.2022.2150506","url":null,"abstract":"Summary For any given triangle, we define its index of similarity. We then prove that two triangles are similar if and only if they share both an index of similarity and a pair of congruent angles.","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":"48624815","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":"Infinite Families of Infinite Series With Integer Sums","authors":"Damiano Fulghesu, James A. Sellers, C. K. Taylor","doi":"10.1080/07468342.2022.2160611","DOIUrl":"https://doi.org/10.1080/07468342.2022.2160611","url":null,"abstract":"Summary Motivated by a desire to show first-year calculus students examples of infinite series whose sums are relatively straightforward to find, we demonstrate a technique to calculate such sums using tools often demonstrated to calculus students. We then use these results, which include an explicit formula for the sums in question, to prove the existence of an infinite family of infinite series whose sums are integers.","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":"47340039","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: A Geometric Series through the Dissection of Regular Hexagons","authors":"Rex H. Wu","doi":"10.1080/07468342.2022.2153563","DOIUrl":"https://doi.org/10.1080/07468342.2022.2153563","url":null,"abstract":"Summary We provide a visual proof to the geometric series with the common ratio 1/4 through the dissection of regular hexagons.","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":"48499146","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":"10.1080/07468342.2023.2165374","DOIUrl":"https://doi.org/10.1080/07468342.2023.2165374","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":"44074718","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":"Multiplicity of Hexagon Numbers","authors":"Cameron G. Hale, Jonathan R. Kelleher, J. Mayer","doi":"10.1080/07468342.2022.2120326","DOIUrl":"https://doi.org/10.1080/07468342.2022.2120326","url":null,"abstract":"Summary Imagine you have an unlimited supply of congruent equilateral triangles. Polygon numbers are the number of these triangles used to tile a convex polygon. For example, triangle numbers are square integers n 2, where the positive integer n is the side length of a tiled equilateral triangle. Hexagon numbers are the number of triangles used to tile a convex hexagon, and can be realized by removing corners from a tiled equilateral triangle. The number of ways (multiplicity up to congruence) that a given hexagon number can be constructed geometrically from such tiles is the subject of our paper.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48268557","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":"George Pólya Awards for 2022","authors":"Sarah Ann Stewart Fleming, J. Previte, Michelle Previte","doi":"10.1080/07468342.2022.2122670","DOIUrl":"https://doi.org/10.1080/07468342.2022.2122670","url":null,"abstract":"In “The Beautiful Chaotic Dynamics f i,” Joseph and Michelle Previte guide their readers on an engaging exploration of the principal branch of the complex map f (z) = i. While Brouwer’s Fixed-Point Theorem guarantees that this function has at least one fixed point, the authors establish that there are, in fact, an infinite number of fixed points—all but one of which are unstable. Of course, with this initial groundwork in place, exotic and ever-enchanting fractal images cannot be far behind! By iterating i numerically for a large collection of initial points, the Prevites create a graph to identify those points in the plane which lie in the basin of attraction of the stable fixed point and those initial points which escape to infinite. Earlier in the paper, technology was used to explore the locations of the sought after fixed points and to follow up with careful mathematical analysis to verify the information alluded to in the resulting graphs. This helpful side of technology is counter-balanced as the Prevites use mathematical analysis to carefully point out the limits of technology by identifying points within the basin of attraction that the computer-generated plot clearly mis-identified. While some of the points the computer identified as being outside the basin of attraction actually approach the stable fixed point, the authors provide a compactness argument to show that there are indeed points in the plane with orbits whose moduli tend to infinity. The paper continues by examining the composite maps f 2(z) and f 3(z) to identify period two and period three points of f (z). Thus, one concludes that f (z) is a chaotic map having periods of all orders. The authors conclude by giving readers six open problems to investigate on their own. The Prevites’ clear exposition makes it easy for a reader to interact with this paper at a variety of levels. There are five exercises sprinkled throughout the paper that allows one to, at first, skip some of the technical details and more quickly get to the “good stuff”—that is, the beautiful chaotic dynamics of i. However, these exercises contain some very nice analysis for students to grapple with and help to reiterate the usefulness of one-sided limits, monotonicity, and notions of convergence that students have likely seen in their mathematics courses. Students who already have some familiarity with complex numbers could use this paper as a nice introduction to the ideas of fractals and chaos. Taking time to fill in some of the details, to reproduce some of the lovely plots, and to explore the open problems would make for a truly engaging and worthwhile project for students and instructors alike.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43525081","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 Investigation of Metropolitan Crime Distribution","authors":"L. Bell, P. Hill","doi":"10.1080/07468342.2022.2125263","DOIUrl":"https://doi.org/10.1080/07468342.2022.2125263","url":null,"abstract":"Have you ever researched crime data for your neighborhood? There are a variety of websites that offer crime map filters that display the distribution of various crimes by location. The distribution of crimes by location is of importance to many. From the inquiry of a potential homeowner to the deployment of resources for law enforcement, the dispersion or spatial arrangement of crimes can provide insights into a community. Several factors such as the degree of urbanization, unemployment and other socioeconomic demographics may influence the distribution of crimes in a particular location. In the following discussion, we examine the distribution of crimes in the District of Columbia (DC). In particular, we evaluate whether the spatial arrangement of crimes in the third district of DC is random, uniformly dispersed or clustered. We will consider three methods that may be applied to study the allocation of crimes.","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47973176","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":"10.1080/07468342.2022.2127286","DOIUrl":"https://doi.org/10.1080/07468342.2022.2127286","url":null,"abstract":"During the fall of 2021, Josh Wardle, a software engineer in Brooklyn, created a word guessing game “WORDLE” for his partner. After Wardle released the game to the public, it rapidly became a sensation, with players posting progress on social media sites. WORDLE asks you to guess a five letter word in six attempts. Each attempt involves entering an entire five letter word, after which colors indicate whether each letter appears in the word but is incorrectly placed (yellow) or whether it appears and is in the correct location (green). Information theory, pioneered by mathematician Claude Shannon in the 1940’s gives insight about optimal guessing strategies. Rather than trying to build one’s guess around letters one has successfully identified and placed, sometimes substantially more information about the remaining letters can be recovered by guessing words one knows are wrong. Honner illustrates this concretely with an example in which the word is known after several attempts to be ATCH. Because prior guesses have ruled out certain letters, the best next guess is the incorrect word CHIMP, because this guess guarantees to determine the last unknown letter. The key formula is I = log2( 1 p ), where I denotes bits of information obtained from the guessed letter and p is the probability of the event. The article gives an application using a toy WORDLE game with 16 two letter words. Because log2( 16 1 ) = 4, 4 bits of information are needed to know the 2 letter word from the list of 16. Because only 4 of the 16 words contain an A, the information contained in the knowledge of the presence of the letter A is log2( 16 4 ) = 2, which is like halving the possible list of words twice. Similarly the information in knowing the word contains a T is 2, and together, because probability is additive, knowing both A and T are in the word suffice to identify the word at from the list. Of course, 5 letter WORDLE is harder: there are 2315 possible solution words, and an additional 10,657 words that are accepted as guesses. Calculating the actual information yielded by a particular guess would require more information than the average player has at hand, but bearing the information theory concepts in mind may improve your game. KW","PeriodicalId":38710,"journal":{"name":"College Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46095947","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}