Synthesis and structures of poly(perfluoroethyl)[60]fullerenes: 1,7,16,36,46,49-C60(C2F5)6 and 1,6,11,18,24,27,32,35-C60(C2F5)8

The high-temperature reaction of C60 and C2F5I produced poly(perfluoroethyl)fullerenes with unprecedented addition patterns.     

A Relation Between Closure Operators on a Small Category and Its Category of Presheaves

In this paper we show that each factorization structure on a small category , satisfying certain conditions, yields a presheaf on and a morphism of presheaves . We then give connections, and set up one to one correspondences, between subclasses of the following classes: (a) closure operators on (b) subobjects of (c) morphisms from to (d) weak Lawvere–Tierney topologies (e) weak Grothendieck topologies (f) closure operators on .     

Density and invariant means in left amenable semigroups

A left cancellative and left amenable semigroup S satisfies the Strong Følner Condition. That is, given any finite subset H of S and any >0, there is a finite nonempty subset F of S such that for each sH, |sFF|<|F|. This condition is useful in defining a very well behaved notion of density, which we call Følner density, via the notion of a left Følner net, that is a net F_ααD of finite nonempty subsets of S such that for each sS, (|sF_αF_α|)/|F_α| converges to 0. Motivated by a desire to show that this density behaves as it should on cartesian products, we were led to consider the set LIM₀(S) which is the set of left invariant means which are weak^* limits in l_∞(S)^* of left Følner nets. We show that the set of all left invariant means is the weak^* closure of the convex hull of LIM₀(S). (If S is a left amenable group, this is a relatively old result of C. Chou.) We obtain our desired density result as a corollary. We also show that the set of left invariant means on is actually equal to . We also derive some properties of the extreme points of the set of left invariant means on S, regarded as measures on βS, and investigate the algebraic implications of the assumption that there is a left invariant mean on S which is non-zero on some singleton subset of βS.     

Proof of Modulational Instability of Stokes Waves in Deep Water

It is proven that small-amplitude steady periodic water waves with infinite depth are unstable with respect to long-wave perturbations. This modulational instability was first observed more than half a century ago by Benjamin and Feir. It has been proven rigorously only in the case of finite depth. We provide a completely different and self-contained approach to prove the spectral modulational instability for water waves in both the finite and infinite depth cases. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.     

Eutectic Media Open a Synthetic Route to Oligocitrazinic Acid Fluorophores of Purple Hue

Under isochoric and solvent-free conditions, the reaction between ammonium formate and citric acid results in a deeply purple reaction product with fluorescent properties. This brings this reaction in the realm of bio-based fluorophores and bottom-up carbon nanodots from citric acid. The reaction conditions are optimized in terms of UV–vis spectroscopic properties and, subsequently, the main reaction product is separated. While the structural analysis does not give any indication for carbon nanodots in a general sense, it points towards the formation of molecular fluorophores that consist of oligomerized citrazinic acid derivatives. Furthermore, EPR spectroscopy reveals the presence of stable free radicals in the product. We hypothesize that such open-shell structures may play a general role in molecular fluorophores from citric acid and are not yet sufficiently explored. Therefore, we believe that analysis of these newly discovered fluorophores may contribute to a better understanding of the properties of fluorophores and CND from citric acid in general.