By Charles J.(Charles J. Stone) Stone

ISBN-10: 0534233287

ISBN-13: 9780534233280

This author's smooth method is meant essentially for honors undergraduates or undergraduates with an excellent math heritage taking a mathematical facts or statistical inference direction. the writer takes a finite-dimensional sensible modeling point of view (in distinction to the normal parametric method) to reinforce the relationship among statistical thought and statistical method.

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**Example text**

1/x + h/2 Fig. 4. Interpretation of Plimpton 322 in terms of a “cut–and–paste” geometry problem. Left: a rectangle of unit area with reciprocal sides, x and 1/x, the former exceeding the latter by an integer amount h. Right: cutting oﬀ width 12 h and placing it on top produces an L shape within a square of side 1/x + 12 h. The area of this square minus that of the smaller shaded square, of side 12 h, must be equal to 1. 2 Theorem of Pythagoras 17 The interpretation of Plimpton 322 as a compilation of “cut–and–paste” geometry exercises involving regular reciprocal pairs is perhaps more mundane (but more credible) than “number theory” or “trigonometry” interpretations.

Let Πk denote the set of all polynomials p(t) of degree ≤k. Suppose we seek the polynomial pˆ(t) ∈ Πk that exhibits the least possible absolute diﬀerence k = min max p(t)∈Πk t∈[ −1,+1 ] | p(t) − tk+1 | from the monomial tk+1 over the interval [ −1, +1 ]. The error of this best “min–max” polynomial approximant pˆ(t) of degree ≤ k to tk+1 proves to be a multiple of the Chebyshev polynomial Tk+1 (t) — namely, tk+1 − pˆ(t) = 2−k Tk+1 (t) , and since |Tk+1 (t)| ≤ 1 for t ∈ [ −1, +1 ], the value of the minimum possible error is k = 2−k .

Thus, for a simple root rj with mj = 1, we have Cj1 = (t − rj ) p(t) q(t) . t=rj Partial fraction expansions of rational functions play an important role in algorithms for their systematic integration. In this context, the coeﬃcients C11 , . . , CN 1 have a special signiﬁcance — they are called the residues of the rational function p(t)/q(t) at its distinct poles r1 , . . , rN . We will encounter them again in Chaps. 4 and 16. 4 Complex Numbers Wessel’s development [of complex numbers] proceeded rather directly from geometric problems, through geometric–intuitive reasoning, to an algebraic formula.

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