October 13 2019
Finding Ellipses by Ulrich Daepp, Pamela Gorkin, Andrew Shaffer, Karl Voss Mathematicians delight in finding surprising connections between seemingly disparate areas of mathematics. Whole domains of modern mathematics have arisen from exploration of such connections--consider analytic number theory or algebraic topology. Finding Ellipses is a delight-filled romp across a three-way unexpected connection between complex analysis, linear algebra, and projective geometry. The book begins with Blaschke products, complex-analytic functions that are generalizations of disk automorphisms. In the analysis of Blaschke products, we encounter, in a quite natural way, an ellipse inside the unit disk. The story continues by introducing the reader to Poncelet's theorem--a beautiful result in projective geometry that ties together two conics and, in particular, two ellipses, one circumscribed by a polygon that is inscribed in the second. The Blaschke ellipse and the Poncelet ellipse turn out to be the same ellipse, and the connection is illuminated by considering the numerical range of a $2 imes 2$ matrix. The numerical range is a convex subset of the complex plane that contains information about the geometry of the transformation represented by a matrix. Through the numerical range of $n imes n$ matrices, we learn more about the interplay between Poncelet's theorem and Blaschke products. The story ranges widely over analysis, algebra, and geometry, and the exposition of the deep and surprising connections is lucid and compelling. Written for advanced undergraduates or beginning graduate students, this book would be the perfect vehicle for an invigorating and enlightening capstone exploration. The exercises and collection of extensive projects could be used as an embarkation point for a satisfying and rich research project. You are invited to read actively using the accompanying interactive website, which allows you to visualize the concepts in the book, experiment, and develop original conjectures.
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Sentence repetition tasks are widely used in the diagnosis and assessment of children with language difficulties. This paper seeks to clarify the nature of sentence repetition tasks and their relationship to other language skills. We present the results from a 2-year longitudinal study of 216 MLA Handbook 8th Edition.pdf | Plagiarism | Citation
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Is there any way in InDesign to type prime/double prime symbols (feet and inches symbols) without having to go into the Glyphs panel? There are a host of characters I wish I could type, but which only appear in the Glyphs panel — accents, dingbats, symbols, ornaments, and more. In an earlier post Exponential And Logistic Functions, Their Graphs And ...
Writing exponential models¶. Source: Made it up.. Explanation:. This artifact demonstrates writing exponential models. I used the exponential formula (y = ab^x) to solve this problem. (a) stands for the initial amount. (b) stands for the rate. Since the rate is a percentage that is growing, I need to express it as 1 + 0.031, or else my function would be decaying and inaccurate. Garrison's NCLEX Tutoring - YouTube
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