A forest building process on simple graphs

Consider the following process on a simple graph without isolated vertices: order the edges randomly and keep an edge if and only if it contains a vertex which is not contained in some preceding edge. The resulting set of edges forms a spanning forest of the graph.The probability of obtaining $k$ components in this process for complete bipartite graphs is determined as well as a formula for the expected number of components in any graph. A generic recurrence and some additional basic properties are discussed.     

Semi-Interpenetrating Hydrogels of Polyelectrolytes,Polymer-Metal Complexes and Polymer-Protected Palladium Nanoparticles

Summary: Semi-interpenetrating hydrogels (SIHs) of polyelectrolytes consisting of poly(acrylamide) hydrogel (PAAH) as matrix and sodium poly(styrenesulfonate) (NaPSS), poly(N,N′-dimethyl-N,N′-diallylammonium chloride) (PDMDAAC), stoichiometric interpolyelectrolyte complex of NaPSS-PDMDAAC, and linear polyethyleneimine-metal complexes (PMC) as well as polymer-protected palladium nanoparticles were prepared by in situ polymerization, e.g. crosslinked acrylamide chains were formed in aqueous solutions of NaPSS, PDMDAAC, NaPSS-PDMDAAC (1:1 mol/mol), PMC and palladium nanoparticles protected by poly(N-vinylpyrrolidone) (PVP), poly-(vinyl alcohol) (PVA), poly(acrylic acid) (PAA), and linear poly(ethyleneimine) (LPEI) respectively in the presence of N,N′-methylenebisacrylamide. For each system the equilibrium swelling degree α and the parameters n and k describing the mechanism of water transport into SIHs volume was determined. Catalytic properties of SIHs were studied in allyl alcohol hydrogenation and cyclohexane oxidation reactions.     

Two-sided estimates for matrix operators on the cone of monotone sequences

Weighted estimates for a class of non-negative lower triangular matrices have been established on the cone of monotone sequences.