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Azidoalkyation is an efficient strategy for the conversion of unsaturated precursors into nitrogen-containing structural motifs. Herein, we describe a convenient and highly regioselective iron-catalyzed 1,2-azidoalkylation of 1,3-dienes that employs TMSN3... 

Polaritons in two-dimensional (2D) materials provide unique opportunities for controlling light at nanoscales. Tailoring these polaritons via gradient polaritonic surfaces with space-variant response can enable versatile light-matter interaction platforms with advanced functionalities. However, experimental progress has been hampered by the optical losses and poor light confinement of conventionally used artificial nanostructures. Here, we demonstrate natural gradient polaritonic surfaces based on superlattices of solitons—localized structural deformations—in a prototypical moiré system, twisted bilayer graphene on boron nitride. We demonstrate on-off switching and continuous modulation of local polariton-soliton interactions, which results from marked modifications of topological and conventional soliton states through variation of local strain direction. Furthermore, we reveal the capability of these structures to spatially modify the near-field profile, phase, and propagation direction of polaritons in record-small footprints, enabling generation and electrical switching of directional polaritons. Our findings open up new avenues toward nanoscale manipulation of light-matter interactions and spatial polariton engineering through gradient moiré superlattices. 

We study the problem of fair k-median where each cluster is required to have a fair representation of individuals from different groups. In the fair representation k-median problem, we are given a set of points X in a metric space. Each point x ∈ X belongs to one of ℓ groups. Further, we are given fair representation parameters αj and β_j for each group j ∈ [ℓ]. We say that a k-clustering C_1, ⋅⋅⋅, C_k fairly represents all groups if the number of points from group j in cluster C_i is between α_j |C_i| and β_j |C_i| for every j ∈ [ℓ] and i ∈ [k]. The goal is to find a set of k centers and an assignment such that the clustering defined by fairly represents all groups and minimizes the ℓ_1-objective ∑_{x ∈ X} d(x, ϕ(x)). We present an O(log k)-approximation algorithm that runs in time n^{O(ℓ)}. Note that the known algorithms for the problem either (i) violate the fairness constraints by an additive term or (ii) run in time that is exponential in both k and ℓ. We also consider an important special case of the problem where and for all j ∈ [ℓ]. For this special case, we present an O(log k)-approximation algorithm that runs in time.