On Rado's Boundedness Conjecture

We prove that Rado's Boundedness Conjecture from Richard Rado's 1933 famous dissertation Studien zur Kombinatorik is true if it is true for homogeneous equations. We then prove the first nontrivial case of Rado's Boundedness Conjecture: if a₁,a₂, and a₃ are integers, and if for every 24-coloring of the positive integers (or even the nonzero rational numbers) there is a monochromatic solution to the equation a₁x₁+a₂x₂+a₃x₃=0, then for every finite coloring of the positive integers there is a monochromatic solution to a₁x₁+a₂x₂+a₃x₃=0.     

Inhibition of the reduction of Cr(VI) at the magnetite–water interface by calcium carbonate coatings

The effect of calcium carbonate coatings on the reduction of aqueous chromate on the magnetite(1 1 1) surface has been investigated using a combination of synchrotron based X-ray photoemission spectroscopy (PES) and X-ray absorption near edge structure (XANES) spectroscopy, along with laboratory-based powder X-ray diffraction (XRD) and scanning electron microscopy (SEM). CaCO₃ coatings (dominantly calcite with minor quantities of aragonite and vaterite) of thicknesses ranging from 10 Å to 20 m were grown on magnetite(1 1 1) surfaces by exposure to supersaturated aqueous solutions followed by evaporation of the solution—a process that mimics pore-water evaporation in vadose zones leading to the formation of caliche and calcium carbonate coatings on mineral grains. Coating thicknesses were determined from attenuation of the Fe 2p photoemission signal by the carbonate coating. For coatings less than 15 Å thick, Cr 2p photoemission and Cr L_II, L_III-edge XANES spectra show that chromate is reduced by the underlying magnetite surface; however, as the minimum coating thickness increases beyond 15 Å, the magnetite surface becomes passivated and further chromate reduction ceases. Our findings suggest that carbonate coatings on natural magnetite grains can significantly reduce or eliminate their ability to reduce Cr(VI), which is a toxic and highly mobile environmental contaminant.     

Fluorinations with complex metal fluorides. Part 6 [1] fluorination of yridine and related compounds with caesium tetrafluorocobaltate(III) [2]

Pyridine has been fluorinated over caesium tetrafluorocobaltate(III) (CsCo^IIIF₄) at 300–400°C to give a mixture of undecafluoro-N-methylpyrrolidine, bis(trifluoromethyl)amine, pentafluoropyridine and several polyfluoropyridines; the product composition depended to some extent on the geometry of the reactor. The fluorinations of pentafluoropyridine, piperidine and undecafluoropiperidine were also investigated.     

On the evolution of islands

Letn cells be arranged in a ring, or alternatively, in a row. Initially, all cells are unmarked. Sequentially, one of the unmarked cells is chosen at random and marked until, aftern steps, each cell is marked. After thekth cell has been marked the configuration of marked cells defines some number of islands: maximal sets of adjacent marked cells. Let ξ _k denote the random number of islands afterk cells have been marked. We give explicit expressions for moments of products of ξ _k ’s and for moments of products of 1/ξ _k ’s. These are used in a companion paper to prove that if a random graph on the natural number is made by drawing an edge betweeni≧1 andj>i with probabilityλ/j, then the graph is almost surely connected ifλ>1/4 and almost surely disconnected ifλ≦1/4.     

Concentration for self‐bounding functions and an inequality of Talagrand

We see that the entropy method yields strong concentration results for general self‐bounding functions of independent random variables. These give an improvement of a concentration result of Talagrand much used in discrete mathematics. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Optimal Server Acquisition in Open Queueing Networks

Several production and flexible manufacturing systems can naturally be modelled using queueing networks. In this paper, we consider the problem of acquiring servers for the nodes of an open queueing network, so as to optimize the steady-state mean virtual system parameters subject to a budget constraint. A partial enumeration scheme and a heuristic method have been proposed to solve this problem. Empirical results based on randomly generated test problems are used to identify a class of problems for which the heuristic performs well.