Let I(n) denote the number of isomorphism classes of subgroups of (Z/nZ)×, and let G(n) denote the number of subgroups of (Z/nZ)× counted as sets (not up to isomorphism). We prove that both logG(n) and log I(n) satisfy Erdős–Kac laws, in that suitable normalizations of them are normally distributed in the limit. Of note is that logG(n) is not an additive function but is closely related to the s...

We deene the VC-dimension of a set of permutations A S n to be the maximal k such that there exist distinct i 1 ; :::; i k 2 f1; :::; ng that appear in A in all possible linear orders, that is, every linear order of fi 1 ; :::; i k g is equivalent to the standard order of f(i 1); :::; (i k)g for at least one permutation 2 A. In other words, the VC-dimension of A is the maximal k such that for s...

The reactions of Zn-, Zn,Cd-, and Cd-thioneins with EDTA and apo-carbonic anhydrase have been studied. The ligand substitution reaction of zinc with EDTA is multiphasic, having both associative and dissociative components in the rate expression. The cadmium sites are about 2 orders of magnitude less reactive. In contrast, apo-carbonic anhydrase abstracts zinc from Zn-thionein and Zn,Cd-thionein...

Over the course of a few billion years, nature has developed extraordinary nanomaterials for the efficient conversion of solar energy into chemical energy. One of these materials, photosystem I (PSI), functions as a photodiode capable of generating a charge separation with nearly perfect quantum efficiency. Because of the favorable properties and natural abundance of PSI, researchers around the...

Understanding how mammalian genomes have been reshuffled through structural changes is fundamental to the dynamics of its composition, evolutionary relationships between species and, in the long run, speciation. In this work, we reveal the evolutionary genomic landscape in Rodentia, the most diverse and speciose mammalian order, by whole-genome comparisons of six rodent species and six represen...

We describe a construction of topological orders from coupled lower-dimensional symmetry-protected orders, which is closely related to gauging subsystem symmetry. Our yields both conventional and exotic fracton type-I type-II. In particular, we find spin-chain Haah's cubic code, coupled-layer Yoshida's fractal spin liquids.

The study of orders over surfaces is an integral aspect of noncommutative algebraic geometry. Although there is a substantial amount known about orders, relatively few concrete examples have been constructed explicitly. Of those already constructed, most are del Pezzo orders, noncommutative analogues of del Pezzo surfaces, the simplest case. We reintroduce a noncommutative analogue of the well-...