By Loo Keng Hua, Wang Yuan

Owing to the advancements and purposes of computing device technology, ma­ thematicians started to take a major curiosity within the functions of quantity idea to numerical research approximately 20 years in the past. The growth completed has been either very important essentially in addition to passable from the theoretical view aspect. It'or instance, from the 17th century until eventually now, loads of attempt was once made in constructing tools for approximating unmarried integrals and there have been just a couple of works on a number of quadrature till the 1950's. yet some time past 20 years, a few new tools were devised of which the quantity theoretic procedure is a good one. The quantity theoretic process should be defined as follows. We use num­ ber idea to build a series of uniformly dispensed units within the s­ dimensional unit dice G , the place s ~ 2. Then we use the series to s lessen a tough analytic challenge to an mathematics challenge that may be calculated through desktop. for instance, we may well use the mathematics suggest of the values of integrand in a given uniformly allotted set of G to ap­ s proximate the yes quintessential over G such that the critical order of the s blunders time period is proven to be of the very best style, if the integrand satis­ fies yes conditions.

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La? - 1 = (x - '()(a? For s = 3, since + ('( - + ,(-1), L)x '( >L + V'( - 3 and ('( _ L)2 _ 4,(-1 = '(3 - 2L'(2 + L2'( - 4 = - L'(2 '( hence ,(2) and ,(3) < 0, '( are conjugate complex numbers and The lemma is proved. Remark. Minkowski proved that except when Q(a) is a real quadratic field or Q(a) is a cubic field and a(2), a(3) are conjugate complex numbers, the real algebraic number field Q(a) does not contain a unit 7J such that (Of. H. Minhowshi [1], O. Perron [1]). However the above lemma shows 40 2.

4. 4. 9) can be covered by at mogt C(S}'l1+E Mf parallelepipedg of the type p s •M • Proof. Take C (s) = '" 2Z+ 8~ G-(1+8) + 2- 8i ). 9) is an open interval (-lM, lM), hence it can be covered by = 2lM M intervals of the type [c, c + 2l M], where c is a constant. The lemma holds. Suppose that k is a positive integer and the lemma holds for s = 1, 2, .. " k. Now we proceed to prove that the lemma holds for s = k + 1 too. Divide the domain into 2 [IogzM ] and + 3 parts mk+! 2 il < Imk+ll Suppose that tnk+!

1'" and WI where the w;'s are algebraic integers. O. 2. 13) 1 ~ i ~ 8, where W denotes the maximum of the absolute values of the cofactors of A. rs. 4. The roots of the equation F( x) Let 8;;;::: 2. We denote the largest real root of the equation F(x) = by = 0 7]( = 7](0) XS - x s- 1 - and its other roots by x-I = 0 ••• - 7](2), •• " 7](S).

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