Let rk(n) and tk(n) denote the number of representations of n as a sum of k squares, and as a sum of k triangular numbers, respectively. We give a generalization of the result rk(8n + k) = cktk(n), which holds for 1 ≤ k ≤ 7, where ck is a constant that depends only on k. Two proofs are provided. One involves generating functions and the other is combinatorial.
A new 'graft-onto' method to attach poly(ethylene glycol) (PEG) onto multi-walled carbon nanotubes (MWNTs) has been developed. The method is based on the coupling reaction of radicals formed at the chain end of PEG onto the surface of MWNTs. The polymeric radicals are generated by atom (halogen) transfer reaction between chloroacetyl-terminated PEG and transition metal catalysts. The method allows direct covalent attachment of PEG to pristine MWNTs without pretreatment that could alter their original structure. The resulting PEG-grafted MWNTs showed improved dispersion stability in isopropanol and methanol.
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