The Cauchy-Schwarz Master Class: An Introduction to the Art by J. Michael Steele PDF

By J. Michael Steele

Michael Steele describes the basic subject matters in mathematical inequalities and their makes use of. utilizing the Cauchy-Schwarz inequality as a consultant, Steele offers a desirable selection of difficulties regarding inequalities and coaches readers via ideas, in a mode corresponding to George Polya, by means of educating easy options and sprucing challenge fixing abilities even as. Undergraduate and starting graduate scholars in arithmetic, theoretical computing device technology, statistics, engineering, and economics will locate the booklet acceptable for self-study.

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Perhaps no other discipline can contribute more to one’s effectiveness as a solver of mathematical problems. 1 (More from Leap-forward Fall-back Induction) Cauchy’s leap-forward, fall-back induction can be used to prove more than just the AM-GM inequality; in particular, it can be used to show that Cauchy’s inequality for n = 2 implies the general result. For example, by Cauchy’s inequality for n = 2 applied twice, one has a1 b1 + a2 b2 + a3 b3 + a4 b4 = {a1 b1 + a2 b2 } + {a3 b3 + a4 b4 } 1 1 1 1 ≤ (a21 + a22 ) 2 (b21 + b22 ) 2 + (a23 + a24 ) 2 (b23 + b24 ) 2 1 1 ≤ (a21 + a22 + a23 + a24 ) 2 (b21 + b22 + b23 + b24 ) 2 , which is Cauchy’s inequality for n = 4.

These are not isolated examples. In fact, they are pointers to one of the most prevalent themes in the theory of inequalities. Whenever we hope to apply some underlying inequality to a new problem, the success or failure of the application will often depend on our ability to recast the problem so that the inequality is applied in one of those pleasing circumstances where the inequality is sharp, or nearly sharp. In the cases we have seen so far, normalization helped us reframe our problems so that an underlying inequality could be applied more efficiently, but sometimes one must go to greater lengths.

29) is known as the Cram´er–Rao lower bound, and it has extensive applications in mathematical statistics. 1) and one may rightly wonder how so much value can be drawn from a bound which comes from the trivial observation that (x − y)2 ≥ 0. 1) has a deeper physical or geometric interpretation that might reveal the reason for its effectiveness? 1) simply says that the area of the rectangle with sides x and y is never greater than the average of the areas of the two squares with sides x and y, and although this interpretation is modestly interesting, one can do much better with just a small change.

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The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities (Maa Problem Books Series.) by J. Michael Steele

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