In this paper we introduce and investigate the notions of diagrams and discrete extensions in the study of finitary 2-representations of finitary 2-categories.

In this paper, we study families of singular surfaces in ℂ3 parametrised by $\mathcal {A}$-finitely determined map germs. We consider the topological triviality and Whitney equisingularity of an unfolding F of a finitely determined map germ f : (ℂ2, 0) → (ℂ3, 0). We investigate the following question: topological triviality implies Whitney equisingularity of the unfolding F? We provide a complete answer to this question, by giving counterexamples showing how the conjecture can be false.

We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato–Tate density. Examples of such sequences come from coefficients of several L-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ⩽ n11/2(logn)−1/2+o(1) for a set of n of asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations of n by a binary quadratic form one has slightly more than square-root cancellations for almost all integers n.

In addition, we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato–Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally and might be within reach unconditionally using the currently established potential automorphy.

For Γ = ℤp, Iwasawa was the first to construct Γ-extensions over number fields with arbitrarily large μ-invariants. In this work, we investigate other uniform pro-p groups which are realisable as Galois groups of towers of number fields with arbitrarily large μ-invariant. For instance, we prove that this is the case if p is a regular prime and Γ is a uniform pro-p group admitting a fixed-point-free automorphism of odd order dividing p−1. Both in Iwasawa's work, and in the present one, the size of the μ-invariant appears to be intimately related to the existence of primes that split completely in the tower.

Bal and DeBiasio [Partitioning random graphs into monochromatic components, Electron. J. Combin. 24 (2017), Paper 1.18] put forward a conjecture concerning the threshold for the following Ramsey-type property for graphs G: every k-colouring of the edge set of G yields k pairwise vertex disjoint monochromatic trees that partition the whole vertex set of G. We determine the threshold for this property for two colours.

Let 𝕂n be the intermediate steps in the cyclotomic ℤp-extension of a CM number field 𝕂. For p ⧧ 2 the minus part of the p-class groups is given by An− = $\frac{1}{2}$(1 − j)An. We will give a new definition of the minus part for p = 2 and will prove that there is no finite submodule in lim∞←nAn−. Furthermore we will show that μ = 0 if and only if μ− = 0 in this new definition.

We are concerned with sets of generic points for shift-invariant measures in the countable symbolic space. We measure the sizes of the sets by the Billingsley-Hausdorff dimensions defined by Gibbs measures. It is shown that the dimension of such a set is given by a variational principle involving the convergence exponent of the Gibbs measure and the relative entropy dimension of the Gibbs measure with respect to the invariant measure. This variational principle is different from that of the case of finite symbols, where the convergent exponent is zero and is not involved. An application is given to a class of expanding interval dynamical systems.