The Waring problem for Lie groups and Chevalley groups

The classical Waring problem deals with expressing every natural number as a sum of g(k) k-th powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given non-trivial word w. In this paper we study this problem for Lie groups and Chevalley groups over infinite fields. We show that for a fixed non-trivial word w and for a classical connected real compact Lie group G of sufficiently large rank we have w(G)^2=G, namely every element of G is a product of 2 values of w. We prove a similar result for non-compact Lie groups of arbitrary rank, arising from Chevalley groups over R or over a p-adic field. We also study this problem for Chevalley groups over arbitrary infinite fields, and show in particular that every element in such a group is a product of two squares

Non-leaching antimicrobial surfaces through polydopamine bio-inspired coating of quaternary ammonium salts or an ultrashort antimicrobial lipopeptide.

Bacterial fouling on surfaces significantly increases the resistance of bacteria toward antibiotics, which leads to medical complications and a corresponding financial burden. Here, we report on a general and robust technique for facile modification of various surfaces with different antibacterial agents. Our approach in this study was inspired by the strong adhesion of mussel adhesion proteins (MAPs) to many types of surfaces, including metals, polymers, and inorganic materials. Thus, glass and polymeric slides were dip-coated with dopamine, as a MAP mimic, and the resulting surfaces were characterized. The reactivity of dopamine-coated surfaces toward nucleophilic addition was then confirmed by reacting them with fluorescent probes containing either a free amino or a free thiol group. Laser scanning confocal microscopy (LSCM), X-ray photoelectron spectroscopy (XPS), confocal Raman microscopy, matrix-assisted laser desorption/ionization-time of flight (MALDI-TOF) mass spectroscopy, and cyclic voltammetry studies collectively suggested that the probes had covalently attached to the surfaces. Fabrication of dopamine-coated surfaces with an antibacterial quaternary amine or an ultrashort lipopeptide analog generated surfaces that effectively kill Escherichia coli and Staphylococcus aureuscells on contact. Moreover, minimal leaching of the fabricated agent was detected after prolonged incubation. This technique could be further developed to a "paint-like" or self-assembling monolayer-like procedure for the preparation of antibacterial surfaces on various materials