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By Hans G. Kaper, Marc Garbey

ISBN-10: 0585319677

ISBN-13: 9780585319674

ISBN-10: 082478538X

ISBN-13: 9780824785383

Integrates fields typically held to be incompatible, if now not downright antithetical, in sixteen lectures from a February 1990 workshop on the Argonne nationwide Laboratory, Illinois. the subjects, of curiosity to commercial and utilized mathematicians, analysts, and desktop scientists, comprise singular according to

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Example text

P rove that the set o f all polynom ials with rational coefficients is countable. P rove or disprove: L et A be a countable set o f real numbers, and let P be the set o f all polynom ials with coefficients in A. T hen P is countable. 8 cannot be used to prove th at the repeating decim als (rational num bers) in the interval 0 < x < 1 are n ot countable. Show that all num bers o f the form 0 . 8 in w hich every elem ent on the m ain diagonal is 1. 8 th at all infinite sequences o f members o f 91 constitute an uncountable set.

3 / FUNCTIONS 19 are true. The first of these is clear. The second is verified as follows: If, on the contrary, {a} = {c, d), then a = c and a = d, so that c = d. But this violates the assumption that c 9^ d. Now, the equality (a, b) = (c, d) can be written in the form { {« } , {a, 6}} = {{c}, {c, d}}, from which we find that {a} € {{c}, [c, d } }, so that {0 } = {c},and consequently a = c. But we also see that {c, d\ £ {{a}, {a, 6}}. This, together with the inequality {0 } {c, d}, shows that {c, d} = { 0, 6}.

12 / Example Consider the differentiation operator D{ f) = f . Clearly, D is not a function on 6 because of functions such as |x — \ |which are not differentiable. On the other hand, D is a function on the set of polynomials. The function D, which is linear, is called a linear operator. The reader should discover for himself the major differences between the functions D and I. 13 / Example A very important function on 6 into (R is the so called supremum function which relates to each / € 6 the number II/11 = max |f(x) | 0^*<1 (see Figure 7 ).

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Asymptotic analysis and numerical solution of partial differential equations by Hans G. Kaper, Marc Garbey

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