New PDF release: Symmetry results for solutions of semilinear elliptic

By Pacella F.

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Extra info for Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities

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13 of [AB], £ n(a) 9(a)<7V £ [n7V O qN(k-l+^ 9(2-*-" (1 fc)r E 9 ~ )}+ r 2. In an obvious sense yet again, it follows that the stated "asymptotic density" D exists. 14) COROLLARY. As N —> oo; i/ie total number A/-(JV) of coprime ordered k-tuples cti, . . 1. 1 but, since our treatment of the latter category uses facts about it which may be less familiar to some readers, we begin with a direct discussion of J-q alone.

1 and the equation d2*(y) = {Z (y)]4 / Z ( y 2 ) . 5) PROPOSITION. of degree N is The average value of d2(a) for elements a G Q }]-lNz + O(N2) as N -> oo. 6) PROPOSITION. The average value of (3 for dements of degree N in Q is if V = |, ifv>\. 5: Error estimates for ... PROOF. We have /3#(y) = Z ( y ) g # ( y } , where g # ( y ) = Z(y^Z(y3}/Z(y6}. Then g*(y] = Z(y2)h*(y), where h*(y) = Z ( y 3 ) / Z ( y 6 ) is absolutely convergent for \y\ < q~z. Hence g(N) = since h f ( q ~ ? ) is absolutely convergent.

1 below, for example), attention may and normally will be confined to specifying the asymptotic behaviour of the relevant numbers f ( N ) as N —> oo, since the same denominator G(N] appears throughout and is given asymptotically by Axiom A*. -l implies that = £ /(«)- £ 7H n oo; thus m(/) is also the asymptotic mean-value of / in the present sense. 1) THEOREM. Let /, g be arithmetical functions such that4 f * ( y ) = [Z(y)}kg#(y), where k is a positive integer, and g^(y) is absolutely convergent for \y\ < qT , T > — 1.