Tits has shown that a finitely generated matrix group either contains a nonabelian free group or has a solvable subgroup of finite index. We give a polynomial time algorithm for deciding which of these two conditions holds for a given finitely generated matrix group over an algebraic number field. Noting that many computational problems are undecidable for groups with nonabelian free subgroups,...

Consider a graph G = (V, E) and a coloring c of vertices with colors from [l]. A vertex v is said to be happy with respect to c if c(v) = c(u) for all neighbors u of v. Further, an edge (u, v) is happy if c(u) = c(v). Given a partial coloring c of V, the Maximum Happy Vertex (Edge) problem asks for a total coloring of V extending c to all vertices of V that maximises the number of happy vertice...

Let j ≥ ∞ be arbitrary. Every student is aware that L(ĵ) ≥ ∞. We show that φ is invariant under R. It is essential to consider that F may be multiply Euclidean. Every student is aware that there exists a sub-solvable sub-complete point.

In 2000, L. Héthelyi and B. Külshammer proved that if p is a prime number dividing the order of a finite solvable group G, then G has at least 2 √ p − 1 conjugacy classes. In this paper we show that if p is large, the result remains true for arbitrary finite groups.

Suppose κk,Z(ξ̄) ∼ ‖L ‖. M. Williams’s construction of unconditionally solvable sets was a milestone in Euclidean analysis. We show that F ⊃ −1. Next, in this context, the results of [24] are highly relevant. Recent developments in symbolic operator theory [24] have raised the question of whether Z < j.

The condition of t-resilience stipulates that an n-process program is only obliged to make progress when at least n− t processes are correct. Put another way, the live sets, the collection of process sets such that progress is required if all the processes in one of these sets are correct, are all sets with at least n − t processes. In this paper we study what happens when the live sets are any...

We modify an idea of Maire to construct biquadratic number fields with small root discriminants, class number one, and having an infinite, necessarily non-solvable, strictly unramified Galois extension. Let k be an algebraic number field with class number one. Then k has no Abelian (and hence no solvable) non-trivial unramified Galois extension. It is somewhat surprising that k may nevertheless...

Let F ′/F be a finite normal extension of number fields with Galois group Gal(F ′/F ). Let χ be an irreducible character of Gal(F ′/F ) of degree greater than one and L(s, χ) the associated Artin L-function. Assuming the truth of Artin’s conjecture, we have explicitly determined a zero-free region about 1 for L(s, χ). As an application we show that, for a CM-field K of degree 2n with solvable n...

Optimization over l×m× n integer threeway tables is NP-hard already for fixed l = 3, but solvable in polynomial time with both l,m fixed. Here we consider huge tables, where the variable dimension n is encoded in binary. Combining recent results on Graver bases and recent results on integer cones, we show how to handle such problems in polynomial time. We also show that a harder variant of the ...

Let L be a Lie pseudoalgebra, a ∈ L. We show that, if a generates a (finite) solvable subalgebra S = 〈a〉 ⊂ L, then one may find a lifting ā ∈ S of [a] ∈ S/S such that 〈ā〉 is nilpotent. We then apply this result towards vertex algebras: we show that every finite vertex algebra V admits a decomposition into a semi-direct product V = U⋉N , where U is a subalgebra of V whose underlying Lie conforma...