‎let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $f$ of characteristic $pgeq 0$‎. ‎we survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2‎, ‎3‎, ‎7)$-generated and of those which are $(2,3)$-generated‎.

‎let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $mathbb{f}$ of characteristic $pgeq 0$‎. ‎we survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2,3,7)$-generated and of those which are $(2,3)$-generated‎.

Let A be any nonrecursive set. For each k 0, we show that there are functions (k + 1)-truth-table reducible to A that are not k-weak-truth-table reducible to A; and we show that there are functions (k + 1)-Turing reducible to A that are not k-Turing reducible to A. Intuitively, this means that k + 1 queries to a nonrecursive oracle are strictly more useful than k queries to the same oracle (for...

Let ` be a prime number and let k be an algebraically closed field with char(k) 6= `. Let K/k be a function field of transcendence degree td(K/k) > 1 with absolute Galois group GK . Take G K = GK and for all i ≥ 1, let G (i+1) K = [G , G](G) ∞ . In other words, the groups {G K }i≥1 form the descending central `∞-series of GK . Then ΠK = G /G = Gal(K ′/K) is the Galois group of the maximal pro-`...

The main topic of the present work is the relation that a set X is strongly hyperimmune-free (shif) relative to Y . Here X is shif-below Y if and only if for every partial X-recursive function p there is a partial Y -recursive function q such that every a in the domain of p is also in the domain of q and satisfies p(a) < p(q). It is shown that between degrees not above the halting problem this ...

This paper extends Post’s programme to finite levels of the Ershov hierarchy of ∆2 sets. Our initial characterisation, in the spirit of Post [27], of the degrees of the immune and hyperimmune n-enumerable sets leads to a number of results setting other immunity properties in the context of the Turing and wtt-degrees derived from the Ershov hierarchy. For instance, we show that any n-enumerable ...

Introduction. These notes are based on E. L. Post's paper Recursively enumerable sets of positive integers and their decision problems1 to which we shall refer as RES. The reader is assumed to be familiar with §§1-5 and 9 of this paper. In the first note we shall discuss some algebraic properties of simple and hypersimple sets. In the second note we shall prove the existence of a recursively en...

A set S of vertices in a graph G is a connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by S is connected. The connected domination number γc(G) is the minimum size of such a set. Let δ(G) = min{δ(G), δ(G)}, where G is the complement of G and δ(G) is the minimum vertex degree. We prove that when G and G are both connected, γc(G) + γc(G) ≤...