In 1958 E. Lehmer found an explicit description of those primes p for which a given prime q is a cubic residue. Her result essentially states that if one writes 4p = L + 27M, then q is a cubic residue if and only if M L ≡ t 2 − 1 t3 − 9t mod q for some integer t. In this paper we demonstrate that a similar result may be obtained for cubic nonresidues, yielding a cubic character for fixed p that...

An explicit representation of a C1 piecewise rational cubic spline has been developed, which can produce a monotonic interpolant to given monotonic data. The explicit representation is easily constructed, and numerical experiments indicate that the method produces visually pleasing curves. Furthermore, an error analysis of the interpolant is given.

R0<b162R0 gcd(b1, N )1/2 R 0 (HP) . The second line is false and in fact one has M1 = 1 in Proposition 3. The author is very grateful to Professor Hongze Li for drawing his attention to this flaw. The error can be fixed by introducing an average over b1 into the statement of Proposition 3. This allows us to recover the main theorem in [1], and also [2, Lemma 11], via the following modification....

In this paper we analyze the convergence properties of an iterative algorithm based on cubic interpolation which was originally proposed for the calculation of the real stability radius 5]. When applied to the calculation of complex stability radii of generalized eigenvalue problems and of the H 1-norm of an arbitrary rational transfer function , this new algorithm has global linear convergence...

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description...