THE CAUCHY – SCHWARZ MASTER CLASS - PART 16 ppt

... of the few articles to advocate the inductive approach to Cauchy s inequality that is favored in this chapter. The Cram´er–Rao inequality of Exercise 1.15 illustrates one way that the Cauchy Schwarz ... 101 Cauchy, Augustin-Louis, 10 Cauchy Binet identity, 49 Cauchy Schwarz inequality, 8 as accidental corollary, 57 cross term defects, 83 geometric proof, 58 self-generalization,...

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... characterization of the case of equality for the n-dimensional Cauchy inequality. Problem 3.1 (On Equality in Cauchy s Bound) Show that if (b 1 ,b 2 , ,b n ) =0then equality holds in Cauchy s in- equality ... mathematical address of all time. In his lecture, Hilbert de- scribed 23 problems which he believed to be worth the attention of the world’s mathematicians at the dawn of...

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... may suspect that the fundamental theorem of calculus will somehow help. This is The Cauchy- Schwarz Master Class, so here one may not need long to think of applying the 1-trick and Schwarz s inequality ... replaced by 1. The essence of the challenge is therefore to beat the naive immediate application of Schwarz s inequality. Taking the Hint If we want to apply the pat...

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... (14.23) This is known as the Rademacher–Menchoff inequality, and it is surely among the most important results in the theory of orthogonal series. For us, much of the charm of the Rademacher–Menchoff inequality ... we sum along the “down-left” diagonals we see that the ex- tended sequence satisfies the identity 3 N  n=1 c n = N+2  n=1 2  h=0 c n−h . In the exactly same way,...

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... focus the appli- cation of an inequality on the point (or the region) where the inequality is most effective. For example, in the derivation of the AM-GM inequal- ity from the bound 1 + x ≤ e x , the ... The AM-GM Inequality 35 Fig. 2.4. The curve y = x/e x−1 helps us measure the extent to which the individual terms of the averages must be squeezed together when the...

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... bonus. The numer- ator on the right-hand side of the identity (4.7) must also be positive, and this observation gives us yet another proof of the Cauchy Schwarz inequality. There are even two further ... a Plan If the Cauchy Schwarz Master Class were to have a final exam, then the light cone inequality would provide fertile ground for the develop- ment of good problems. O...

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... A}. Exercise 5.3 (Cauchy Schwarz and the Cross-Term Defect) If u and v are elements of the real inner product space V for which on has the upper bounds u, u≤A 2 and v, v≤B 2 , then Cauchy s inequality ... connec- tion to probability theory, and it has many applications in other areas of mathematics. Nevertheless, the probabilistic interpretation of the bound 5 Consequences of...

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... inspired by the proof given by Cauchy s for the AM-GM inequality, and, in an effort to get to the heart of Cauchy s argument, Jensen introduced the class of functions that satisfy the inequality f  x ... near the end of his 1906 article, Jensen ex- pressed the bold view that perhaps someday the class of convex function might seen to be as fundamental as the class of...

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... relationships between the power means. Perhaps the most productive of these is simply the fact that M t is a monotone increasing function of the power t, but all of the elements of the diagram have their day. The ... (189 6–1 981) observed in his lectures on the geome- try of numbers that the limit representation (8.2) for the geometric mean can be used to prove an elega...

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... central core of the classical theory of inequal- ities, and we have already seen three of these: the Cauchy Schwarz inequality, the AM-GM inequality, and Jensen’s inequality. The quartet is completed ... proofs of his bound (9.37). In the first of these he called on the Cauchy Binet formula [see (3.7), page 49], and the second he used the AM-GM inequality which he wrote in...

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