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The congruence x1x2≡x3x4 (mod p), the equation x1x2=x3x4, and mean values of character sums

✍ Scribed by Ayyad, Anwar (author);Cochrane, Todd (author);Zheng, Zhiyong (author)

Publisher
Academic Press Inc.
Year
1996
Tongue
English
Weight
748 KB
Volume
59
Category
Article
ISSN
0022-314X

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We obtain the asymptotic formulae |B & V| =( |B|Âp)+O(-|B| log 2 p) for the number of solutions of the congruence x 1 x 2 #x 3 x 4 (mod p) in a box B of arbitrary size and position, and N(B)=(12Â? 2 ) B 2 log B+CB 2 +O(B 19Â13 log 7Â13 B), with C given explicitly, for the number of solutions of the

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It is shown that if k is a field of characteristic zero, then the kernel of any w x triangular k-derivation of k X , X , X , X is finitely generated as a k-algebra. This is obtained as a corollary of a more general result concerning triangular w x R-derivations of R X, Y, Z for certain rings R.

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## Ä Ž . 4 Experimental results of the high-pressure phase behaviour for xCHF q 1 y x H O and 3 2 Ä Ž . 4 Ž . x CHF q x NaCl q 1 y x y x H O are presented. Gas q gas equilibria for five 1 3 2 1 2 2 Ž .Ž . Ž . compositions with x CHF 0.0334 " 0.0002x -0.1389 " 0.0002 as well as for six 3 Ž . Ž . Ž