THE CAUCHY – SCHWARZ MASTER CLASS - PART 3 pptx
THE CAUCHY – SCHWARZ MASTER CLASS - PART 3 pptx
... Exercise 3. 8. Roots and Branches of Lagrange’s Identity Joseph Louis de Lagrange (1 73 6–1 8 13) developed the case n =3of the identity (3. 4) in 17 73 in the midst of an investigation of the geom- etry ... 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...
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THE CAUCHY – SCHWARZ MASTER CLASS - PART 7 pptx
... 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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THE CAUCHY – SCHWARZ MASTER CLASS - PART 14 pptx
... (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 ... c N c N+1 c N+2 c N +3 ··· and when 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...
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THE CAUCHY – SCHWARZ MASTER CLASS - PART 2 pot
... 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 two sides of the AM-GM ... 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...
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THE CAUCHY – SCHWARZ MASTER CLASS - PART 4 doc
... 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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THE CAUCHY – SCHWARZ MASTER CLASS - PART 5 doc
... 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 o...
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THE CAUCHY – SCHWARZ MASTER CLASS - PART 6 pot
... have the bound f x + y + z 3 = g x + y + z 3 ≤ 1 3 g(x)+g(y)+g(z) ≤ 1 3 f(x)+f(y)+f(z) . The first and last terms of this bound recover the inequality (6.18) so the solution of the ... 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...
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THE CAUCHY – SCHWARZ MASTER CLASS - PART 8 pot
... of the geometric mean, use it to give another proof that the geometric mean is superadditive; that is, show that the formula (8 .33 ) implies the bound (2 .31 ) on page 34 . Exercise 8.6 (More on the ... expression for the sum of the denominators on the right-hand side. 8 The Ladder of Power Means The quantities that provide the upper bound in Cauchy s inequality are s...
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THE CAUCHY – SCHWARZ MASTER CLASS - PART 9 potx
... 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 ... nonnegative x, y, z one has x +(xy) 1 2 +(xyz) 1 3 3 ≤ x · x + y 2 · x + y + z 3 1 /3 . (9 .36 ) Exercise 9. 13 (Rogers’s Inequality — the Proto-H¨older) The inequalit...
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THE CAUCHY – SCHWARZ MASTER CLASS - PART 10 pps
... Compensating Difficulties The two sums on the right-hand side of the naive bound (10 .3) diverge, but the good news is that they diverge for different reasons. In a sense, the first factor diverges ... n (10 .3) but, unfortunately, both of the last two factors turn out to be infinite. The first factor on the right side of the bound (10 .3) diverges like a harmonic series when w...
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