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Abstract Algebra with Applications: Rings and Fields by Karlheinz Spindler

By Karlheinz Spindler

A finished presentation of summary algebra and an in-depth therapy of the functions of algebraic recommendations and the connection of algebra to different disciplines, akin to quantity concept, combinatorics, geometry, topology, differential equations, and Markov chains.

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Then p and q are points on the boundary of ∆, and the line segment [p, q] is contained in ∆. Let f (t) = (1 − t)p + tq, for 0 ≤ t ≤ 1, so that f maps [0, 1] onto the line segment [p, q]. f (ci /(ci + cj )) = c, so that c is an interior point of [p, q]. Thus the function g(t) = π(f (t)) has a maximum at ci /(ci + cj ). Now g(t) = t(1 − t)(ci + cj )2 ck , k=i,j so that dg = (1 − 2t)(ci + cj )2 dt ck , k=i,j and dg dt ci ci + cj = (cj − ci )(ci + cj )2 ck = 0. k=i,j Thus ci = cj . Since this holds for all pairs of indices i, j, the maximum is attained at (a, .

Then a is an affine function on R, a(x) = f (x) and a(y) ≤ f (y) for y ∈ I. Thus f is the supremum of the affine functions which it dominates. We now return to Jensen’s inequality. Suppose that µ is a probability measure on the Borel sets of a (possibly unbounded) open interval I = (a, b). In analogy with the discrete case, we wish to define the barycentre µ ¯ to be x dµ(x). There is no problem if I is bounded; if I is unbounded, we require I 1 that the identity function i(x) = x is in L (µ): that is, I |x| dµ(x) < ∞.

A), and at no other point. We shall refer to the arithmetic mean–geometric mean inequality as the AM–GM inequality. 2 Applications We give two applications of the AM–GM inequality. In elementary analysis, it can be used to provide polynomial approximations to the exponential function. 1 (i) If nt > −1, then (1 − t)n ≥ 1 − nt. (ii) If −x < n < m then (1 + x/n)n ≤ (1 + x/m)m . (iii) If x > 0 and α > 1 then (1 − x/nα )n → 1. (iv) (1 + x/n)n converges as n → ∞, for all real x. Proof (i) Take a1 = 1 − nt and a2 = · · · = an = 1.

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