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We show that a complete [Formula: see text]-dimensional Riemannian manifold [Formula: see text] with finitely generated first homology has macroscopic dimension [Formula: see text] if it satisfies the following “macroscopic curvature” assumptions: every ball of radius [Formula: see text] in [Formula: see text] has volume at most [Formula: see text], and every loop in every ball of radius [Formula: see text] in [Formula: see text] is null-homologous in the concentric ball of radius [Formula: see text]. 

Abstract How many chess rooks or queens does it take to guard all squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can “see” whichever squares the rook or queen attacks. We show that $$\lfloor {\frac{n}{2}} \rfloor $$ ⌊ n 2 ⌋ rooks or $$\lfloor {\frac{n}{3}} \rfloor $$ ⌊ n 3 ⌋ queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We then prove that finding the minimum number of rooks or queens needed to guard a polyomino is NP-hard. These results also apply to d -dimensional rooks and queens on d -dimensional polycubes. Finally, we use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes. 

ABSTRACT The formation and evolution of local brightest cluster galaxies (BCGs) is investigated by determining the stellar populations and dynamics from the galaxy core, through the outskirts and into the intracluster light (ICL). Integral spectroscopy of 23 BCGs observed out to $$4\, r_{e}$$ is collected and high signal-to-noise regions are identified. Stellar population synthesis codes are used to determine the age, metallicity, velocity, and velocity dispersion of stars within each region. The ICL spectra are best modelled with populations that are younger and less metal-rich than those of the BCG cores. The average BCG core age of the sample is $$\rm 13.3\pm 2.8\, Gyr$$ and the average metallicity is $$\rm [Fe/H] = 0.30\pm 0.09$$, whereas for the ICL the average age is $$\rm 9.2\pm 3.5\, Gyr$$ and the average metallicity is $$\rm [Fe/H] = 0.18\pm 0.16$$. The velocity dispersion profile is seen to be rising or flat in most of the sample (17/23), and those with rising values reach the value of the host cluster’s velocity dispersion in several cases. The most extended BCGs are closest to the peak of the cluster’s X-ray luminosity. The results are consistent with the idea that the BCG cores and inner regions formed quickly and long ago, with the outer regions and ICL forming more recently, and continuing to assemble through minor merging. Any recent star formation in the BCGs is a minor component, and is associated with the cluster cool core status.