An uncountable family of finitely generated residually finite groups

We study a family of finitely generated residually finite groups. These groups are doubles F 2 * H F 2 F_{2}*_{H}F_{2} of a rank-2 free group F 2 F_{2} along an infinitely generated subgroup 𝐻. Varying 𝐻 yields uncountably many groups up to isomorphism.

Mallows permutations as stable matchings

We show that the Mallows measure on permutations of $1,\dots ,n$ arises as the law of the unique Gale–Shapley stable matching of the random bipartite graph with vertex set conditioned to be perfect, where preferences arise from the natural total ordering of the vertices of each gender but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely, every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{{\mathbb Z},{\mathbb Z}}$ falls into one of two classes: a countable family $(\sigma _n)_{n\in {\mathbb Z}}$ of tame stable matchings, in which the length of the longest edge crossing k is $O(\log |k|)$ as $k\to \pm \infty $ , and an uncountable family of wild stable matchings, in which this length is $\exp \Omega (k)$ as $k\to +\infty $ . The tame stable matching $\sigma _n$ has the law of the Mallows permutation of ${\mathbb Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$ . The permutation $\sigma _{n+1}$ dominates $\sigma _{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

Generic point equivalence and Pisot numbers

Let $\unicode[STIX]{x1D6FD}>1$ be an integer or, generally, a Pisot number. Put $T(x)=\{\unicode[STIX]{x1D6FD}x\}$ on $[0,1]$ and let $S:[0,1]\rightarrow [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm \unicode[STIX]{x1D6FD}^{m}$ with positive integers $m$. We give a sufficient condition for $T$ and $S$ to have the same generic points. We also give an uncountable family of maps which share the same set of generic points.

ON CONSTRAINTS AND DIVIDING IN TERNARY HOMOGENEOUS STRUCTURES

Let ${\cal M}$ be ternary, homogeneous and simple. We prove that if ${\cal M}$ is finitely constrained, then it is supersimple with finite SU-rank and dependence is k-trivial for some k < ω and for finite sets of real elements. Now suppose that, in addition, ${\cal M}$ is supersimple with SU-rank 1. If ${\cal M}$ is finitely constrained then algebraic closure in ${\cal M}$ is trivial. We also find connections between the nature of the constraints of ${\cal M}$, the nature of the amalgamations allowed by the age of ${\cal M}$, and the nature of definable equivalence relations. A key method of proof is to “extract” constraints (of ${\cal M}$) from instances of dividing and from definable equivalence relations. Finally, we give new examples, including an uncountable family, of ternary homogeneous supersimple structures of SU-rank 1.

Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets

The main objective of the present work is to provide a procedure to construct Gabor orthonormal bases generated by indicator functions of parallelepiped-shaped sets. Given two full-rank lattices of the same volume, we investigate conditions under which there exists a common fundamental domain which is the image of a unit cube under an invertible linear operator. More precisely, we provide a characterization of pairs of full-rank lattices in{\mathbb{R}^{d}}admitting common connected fundamental domains of the type{N[0,1)^{d}}, whereNis an invertible matrix. As a byproduct of our results, we are able to construct a large class of Gabor windows which are indicator functions of sets of the type{N[0,1)^{d}}. We also apply our results to construct multivariate Gabor frames generated by smooth windows of compact support. Finally, we prove in the two-dimensional case that there exists an uncountable family of pairs of lattices of the same volume which do not admit a common connected fundamental domain of the type{N[0,1)^{2}}, whereNis an invertible matrix.

On conjugacy of maximal abelian subalgebras and the outer automorphism group of the Cuntz algebra

We investigate the structure of the outer automorphism group of the Cuntz algebra and the closely related problem of conjugacy of maximal abelian subalgebras in . In particular, we exhibit an uncountable family of maximal abelian subalgebras, conjugate to the standard maximal abelian subalgebra via Bogolubov automorphisms, that are not inner conjugate to .