It is shown that certain commonly occurring conditions may be factored out of sums of multiplicative arithmetic functions.
A function is arithmetic if it is defined on the positive integers. Those complex-valued arithmetic functions g which satisfy the relation g(ab) = g(a)g(b) for all coprime pairs of positive integers a, b are here called multiplicative. In this paper g will be a multiplicative function which satisfies |g(n)| ≤ 1 for all positive integers n.
Every Galois extension of odd degree has a self-dual normal basis.
For all positive a the point spectrum of the (C, α) matrix is determined, where the matrix is regarded as an operator on certain Banach sequence spaces. In particular the point spectrum is obtained in the spaces Ip(X), with 1<p≤∞, where X is a Banach space.
We show that through g + 5 general points of g−1 passes a canonical curve of genus g. For low, odd values of g one may assign more points (g+5+[6/g−2]). The proof is based on a study of the normal bundle of g-cuspidal rational curves.
These facts have consequences for the smoothability of the curve singularity consisting of r lines through the origin of kn, n<r≤(2n+1), in generic position. We strengthen some results of Pinkham and Greuel on such curves.
It is shown that the minimal resolution of a determinantal ring has the structure of an associative differential graded algebra.
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