We define a metric on the class of spectral triples, which is null exactly between unitarily equivalent triples. This dominates propinquity, and thus implies convergence quantum compact spaces induced by In process our construction, we also introduce covariant modular as key component for definition propinquity.

We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose points enjoy several unexpected properties. In particular, they have zero lower box dimension and infinite upper box dimension.

This paper is mainly concerned with Hausdorff dimensions of Besicovitch-Eggleston subsets in countable symbolic space. A notable point is that, the dimension values posses a universal lower bound depending only on the underlying metric. As a consequence of the main results, we obtain Hausdorff dimension formulas for sets of real numbers with prescribed digit frequencies in their Lüroth expansions.

In this paper we establish an alternative characterization of the completion theory for metric spaces which makes fundamental use of a special type of real valued maps, and we derive alternative descriptions for the completions of both Hausdorff uniform and Hausdorff uniform approach spaces. Mathematics Subject Classifications (2000): 54B30, 54D35, 54E15, 54E35, 54E99.

We discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence (σn) of integers such that σn/ √ 2n tends to some σ ∈ [0,∞]. For every n ≥ 1, we call qn a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having n faces and 2σn half-edges on the...