We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ω-c.e. iff every set in it is weak truth-table reducible to a hypersimple, or ranked, set. We also show that ...

Let r be a commutative field (finite or infinite) and let P = P(n, r) be the n-dimensional projective space over ZY Then every point x E P can be expressed by n + 1 homogene coordinates x = (x,,..., x,), not all zero and (x0,..., x,) = @x0,..., Ax,) for OflET. By a hypersurface of degree d we simply mean the set of all points x E P with p(x) = 0, where p(x) is a homogenous polynomial of degree ...

We present an explicit expression for the number of decompositions of an n−cycle as a product of any two permutations of cycle types given by partitions λ and μ. The same expression is also counting the number of unicellular rooted bicolored maps on an orientable surface of genus g with vertex degree distribution given by λ and μ. The relation between the genus and the partitions λ and μ is giv...

Let A be a recursive structure, and let R be a recursive relation on A. Harizanov ( 1991) isolated a syntactical condition which (with additional effectiveness conditions) is necessary and sufficient for A to have recursive copies in which the image of R is r.e. of arbitrary te. degree. We had conjectured that a certain extension of Harizanov’s syntactical condition would (with some effectivene...