This article examines lower bounds for the representation growth of finitely generated (particularly profinite and pro-p) groups. It also considers the related question of understanding the maximal multiplicities of character degrees in finite groups, and in particular simple groups.

Write Θ for the stable character associated to a finite dimensional representation E of a connected real reductive group G. Let M be the centralizer of a maximal torus T , and denote by ΦM (γ,Θ ) Arthur’s extension of |D M (γ)| 1 2Θ(γ) to T (R). In this paper we give a simple explicit expression for ΦM (γ,Θ ), when γ is elliptic in G.

The number of triangles in arrangements of lines and pseudolines has been object of some research Most results however concern arrangements in the projective plane In this article we add results for the number of triangles in Euclidean arrange ments of pseudolines Though the change in the embedding space from projective to Euclidean may seem small there are interesting changes both in the resul...

We provide here the first steps toward a Classification Theory of Abstract Elementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some λ greater than its Löwenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non μsplitting types. Most importantly, the existence of saturated models in...

For a given set M of positive integers, a well-known problem of Motzkin asks for determining the maximal density μ(M) among sets of nonnegative integers in which no two elements differ by M . The problem is completely settled when |M | ≤ 2, and some partial results are known for several families of M for |M | ≥ 3. In this paper, we consider the case M = {a, b, c}, with c a multiple of a or b. I...

We study the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semi-simple Lie group using an extension of the method due to Borel and Prasad.

We prove that the sizes of the maximal dissociated subsets of a given finite subset of an abelian group differ by a logarithmic factor at most. On the other hand, we show that the set {0, 1}n ⊆ Zn possesses a dissociated subset of size Ω(n log n); since the standard basis of Zn is a maximal dissociated subset of {0, 1}n of size n, the result just mentioned is essentially sharp.

We establish necessary and sufficient conditions for the existence of and the expressions for the general real and complex Hermitian solutions to the classical system of quaternion matrix equationsA1X C1, XB1 C2, and A3XA3 C3. Moreover, formulas of the maximal andminimal ranks of four real matrices X1, X2, X3, and X4 in solution X X1 X2i X3j X4k to the system mentioned above are derived. As app...

We prove R-sectoriality or, equivalently, Lp-maximal regularity for a class of operators on cylindrical domains of the form Rn−k × V , where V ⊂ Rk is a domain with compact boundary, Rk, or a half-space. Instead of extensive localization procedures, we present an elegant approach via operator valued multiplier theory, which takes advantage of the cylindrical shape of both, the domain and the op...

Bigrassmannian permutations are known as permutations which have precisely one left descent and one right descent. They play an important role in the study of Bruhat order. Fulton introduced the essential set of a permutation and studied its combinatorics. As a consequence of his work, it turns out that the essential set of bigrassmannian permutations consists of precisely one element. In this ...