<p style='text-indent:20px;'>Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the of corresponding points uniformly bounded. Bounded equivalence an relation. We show hull repetitive set with finite local complexity has either one class or uncountably many.</p>

Let K be the kernel of an epimorphism G→ Z, where G is a finitely presented group. If K has infinitely many subgroups of index 2, 3 or 4, then it has uncountably many. Moreover, if K is the commutator subgroup of a classical knot group G, then any homomorphism from K onto the symmetric group S2 (resp. Z3) lifts to a homomorphism onto S3 (resp. alternating group A4).

We examine the quantum structure of continuum entanglement and in the context of short-pulse down-conversion we answer the open question of how many of the uncountably many frequency modes contribute effectively to the entanglement. We derive a set of two-photon mode functions that provide an exact, discrete, and effectively finite basis for characterizing pairwise entanglement. Our analysis pr...

We prove the existence of nowhere continuous bijections that satisfy the Costas property, as well as (countably and uncountably) infinite Golomb rulers. We define and prove the existence of real and rational Costas clouds, namely nowhere continuous Costas injections whose graphs are everywhere dense in a region of the real plane, based on nonlinear solutions of Cauchy’s functional equation. We ...

We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CM-trivial.

We investigate the behavior of a complete flat metric on surface near puncture. call puncture regular if it has neighborhood which is isometric to that point at infinity cone. prove there are punctures not and only curvature $4\pi$. classify irregular up modification equivalence, where two called modification-equivalent they have neighborhoods. show uncountably many modification-equivalence cla...

Abstract We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on $\omega $ which are not generalized density ideals. In addition, they nonpathological. This answers a question posed by Borodulin-Nadzieja et al. in [this Journal , vol. 80 (2015), pp. 1268–1289]. Lastly, we provide sufficient conditions for ideal to be necessarily ideal.