We prove the conjecture formulated in Litvak and Ejov (2009), namely, that the trace of the fundamental matrix of a singularly perturbed Markov chain that corresponds to a stochastic policy feasible for a given graph is minimised at policies corresponding to Hamiltonian cycles.

Let G be a subgroup of the symmetric group Sn, and let δG=∣Sn/G∣−1 where ∣Sn/G∣ is the index of G in Sn. Then there are at most On,ϵ(Hn−1+δG+ϵ) monic integer polynomials of degree n that have Galois group G and height not exceeding H, so there are only a “few” polynomials having a “small” Galois group.

We refine a result of Dubickas on the maximal multiplicity of the roots of a complex polynomial, and obtain several separability criteria for complex polynomials with large leading coefficient. We also give p-adic analogous results for polynomials with integer coefficients.

Let α be a totally positive algebraic integer of degree d≥2 and α1=α,α2,…,αd be all its conjugates. We use explicit auxiliary functions to improve the known lower bounds of Sk/d, where Sk=∑ di=1αki and k=1,2,3. These improvements have consequences for the search of Salem numbers with negative traces.

We consider logarithmic averages, over friable integers, of non-negative multiplicative functions. Under logarithmic, one-sided or two-sided hypotheses, we obtain sharp estimates that improve upon known results in the literature regarding both the quality of the error term and the range of validity. The one-sided hypotheses correspond to classical sieve assumptions. They are applied to provide an effective form of the Johnsen–Selberg prime power sieve.

A new lower bound is established for the distance between two roots of an integer polynomial, and a new upper bound for the distance between a given real number and the set of zeros of an integer polynomial. The latter result is applied to improve a metrical result in Diophantine approximation.