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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Vector & tensor analysis
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Finding Ellipses - What Blaschke Products, Poncelet's Theorem, and the Numerical Range Know about Each Other (Hardcover)
Loot Price: R1,872
Discovery Miles 18 720
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Finding Ellipses - What Blaschke Products, Poncelet's Theorem, and the Numerical Range Know about Each Other (Hardcover)
Series: Carus Mathematical Monographs
Expected to ship within 10 - 15 working days
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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
$2 \times 2$ 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 $n \times n$ 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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