We investigate the structure of the degrees of provability, which measure the proof-theoretic strength of statements asserting the totality of given computable functions. The degrees of provability can also be seen as an extension of the investigation of relative consistency statements for firstorder arithmetic (which can be viewed as Π1-statements, whereas statements of totality of computable ...

We show that if 0′ is c.e. traceable by a, then a is array non-computable. It follows that there is no minimal almost everywhere dominating degree, in the sense of Dobrinen and Simpson [DS04]. This answers a question of Simpson and a question of Nies [Nie09, Problem 8.6.4]. Moreover, it adds a new arrow in [Nie09, Figure 8.1], which is a diagram depicting the relations of various ‘computational...

In this paper we study the network structure in music and attempt to compose music artificially. Networks are constructed with nodes and edges corresponding to musical notes and their co-occurrences. We analyze sample compositions from Bach, Mozart, Chopin, as well as other types of music including Chinese pop music. We observe remarkably similar properties in all networks constructed from the ...

We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...

Cdf(x) = sup deg(p)=d ∣∣∣∣p.v. ∫ f(x− y)e(p(y)) dy y ∣∣∣∣ in which d is an integer, p is a polynomial of degree d, e(u) := e, f is a Schwarz function and the integral is understood in the principal value sense. This definition is motivated principally by the case d = 1. C1f controls the maximal partial Fourier integrals of f and it extends to a bounded map from L into itself for 1 < p < ∞. The ...

We study the classes of hypersimple and semicomputable sets as well as their intersection in the weak truth table degrees. We construct degrees that are not bounded by hypersimple degrees outside any non-trivial upper cone of Turing degrees and show that the hypersimple-free c.e. wtt degrees are downwards dense in the c.e. wtt degrees. Moreover, we consider the sets that are both hypersimple an...

Let us write f(n, ∆; C2k+1) for the maximal number of edges in a graph of order n and maximum degree ∆ that contains no cycles of length 2k + 1. For n 2 ≤ ∆ ≤ n − k − 1 and n sufficiently large we show that f(n, ∆; C2k+1) = ∆(n −∆), with the unique extremal graph a complete