Values Of Elementary Functions Research Articles

Explicit construction of exact unitary designs

In this paper, we first provide an inductive construction of designs on compact groups G from those on subgroups K when (G,K) is a Gelfand pair. We then use the construction to provide explicit constructions of unitary t-designs in the unitary group U(d) for all t and d. To the best of our knowledge, the explicit constructions were so far known only for very special cases. Here, explicit construction means that the entries of the unitary matrices are given by the values of elementary functions at the root of given polynomials. We will compare our constructions with previously known ones in the case of unitary 4-designs in U(4). We finally remark that this method can be applied to the orthogonal groups O(d) and thus provides another explicit construction of spherical t-design on the sphere Sd−1 in the d dimensional Euclidean space by the induction on d.

On some irrationalities

An elementary proof of the irrationality of the number e is well known, while this is not the case with the number π or even, e.g., e 7 . In this note, simple and quite transparent proofs of the irrationality of the number π and rational degrees of e, as well as some other values of elementary functions, are proposed.

Evaluating Elementary Functions with Guaranteed Precision

Because of changes in the computer markets, the problem of efficient evaluation of elementary functions (which seemed to be already solved) becomes important again. In this paper, after a brief review of current approaches to this problem, algorithms for finding guaranteed bounds for values of elementary functions are suggested. The evaluation time is reduced through increasing the amount of memory used.