On the structure of a random sum-free set

Summary There is a natural probability measure on the set of all sum-free sets of natural numbers. If we represent such a set by its characteristic functions, then the zero-one random variabless(i) are far from independent, and we cannot expect a law of large numbers to hold for them. In this paper I conjecture a decomposition of into countably many more tractible pieces (up to a null set). I prove that each piece has positive measure, and show that, within each piece, a random set almost surely has a density which is a fixed rational number depending only on the piece. For example, the first such piece is made up of sets consisting entirely of odd numbers; it has probability 0.218 ..., and its members almost surely have density 1/4.     

Self-avoiding random walk: A Brownian motion model with local time drift

Summary A natural model for a self-avoiding Brownian motion inR ^d, when specialised and simplified tod=1, becomes the stochastic differential equation , where {L(t, x):t0,xR} is the local time process ofX. ThoughX is not Markovian, an analogue of the Ray-Knight theorem holds for {L(,x):xR}, which allows one to prove in many cases of interest that exists almost surely, and to identify the limit.     

Compatible additive permutations of finite integral bases

Let be an additive permutation of a finite integral base. It is shown that ifB is symmetric, then there is a unique additive permutation ofB which is compatible with in the sense that ^–1 is also an additive permutation; and that, further, ifB is asymmetric, then there is no additive permutation ofB which is compatible with. Thus, in the symmetric case, there are no additively compatible sets (of permutations) forB of size greater than 3. This contrasts with the situation for completely compatible sets (equivalently, additive sequences of permutations) where for certainB compatible sets of size (resp. length) 4 or less are known, but where nothing is known of sets of greater size (resp. length). It is also noted how this result restricts the possibility of a useful multiplication theorem for the additive analogue of perfect systems of difference sets and graceful graphs.     

Stamp collapse in soft lithography

We have studied the so-called roof collapse in soft lithography. Roof collapse is due to the adhesion between the PDMS stamp and substrate, and it may affect the quality of soft lithography. Our analysis accounts for the interactions of multiple punches and the effect of elastic mismatch between the PDMS stamp and substrate. A scaling law among the stamp modulus, punch height and spacing, and work of adhesion between the stamp and substrate is established. Such a scaling law leads to a simple criterion against the unwanted roof collapse. The present study agrees well with the experimental data.     

Polymerised bicontinuous microemulsions as stationary phases for capillary electrochromatography

Summary Polymerisation of bicontinuous microemulsions yields porous monolithic structures with well defined pore sizes that are potentially suitable for use as stationary phases for capillary electrochromatography (CEC). A variety of pore sizes can be achieved by altering the composition of the microemulsion, which typically consists of butyl methacrylate (BMA) and ethylene glycol dimethacrylate (EGDMA) as the polymerisable oil phase. The aqueous phase consists of water, a surfactant (sodium dodecyl sulphate, SDS) and a co-surfactant (1-propanol). 2-acrylamido-2-methyl-1-propane sulfonic acid (AMPS) is also added to provide charges along the polymer backbone to allow electroosmotic flow (EOF) to occur. SEM analysis shows that in-situ polymerisation yields a monolithic structure with a porous topography. Investigations have shown that these monoliths are easy to prepare, robust and suitable for the separation of phthalates. They generate higher linear velocities than are achieved using the silica based HPLC packings normally used for CEC.     

Preparation of 4-hydroxy-3-substituted, 2H-1-benzopyran-2-ones and 2H-1-benzothiopyran-2-ones from carboxylic esters and methyl salicylates or methyl thiosalicylate

C(α)-Carboxylic acid esters were treated with excess lithium diisopropylamide, condensed with methyl salicylates or methyl thiosalicylate, followed by acid cyclization to either 4-hydroxy-3-substituted, 2H-1-benzopyran-2-ones (coumarins), or 2H-1-benzothiopyran-2-ones (thiocoumarins).     

Sterically controlled reductive oligomerisations of CO by activated magnesium(i) compounds: deltate vs. ethenediolate formation

An extremely bulky, symmetrical three-coordinate magnesium(i) complex, [{(^TCHPNacnac)Mg}₂] (^TCHPNacnac = [{(TCHP)NCMe}₂CH]⁻, TCHP = 2,4,6-tricyclohexylphenyl) has been prepared and shown to have an extremely long Mg–Mg bond (3.021(1) Å) for such a complex. It was shown not to react with either DMAP (4-dimethylaminopyridine) or CO. Three unsymmetrical 1 : 1 DMAP adducts of less bulky Mg–Mg bonded species have been prepared, viz. [(^ArNacnac)Mg–Mg(DMAP)(^ArNacnac)] (^ArNacnac = [(ArNCMe)₂CH]⁻ Ar = 2,6-xylyl (Xyl), mesityl (Mes) or 2,6-diethylphenyl (Dep)), and their reactivity toward CO explored. Like the previously reported bulkier complex, [(^DipNacnac)Mg–Mg(DMAP)(^DipNacnac)] (Dip = 2,6-diisopropylphenyl), [(^DepNacnac)Mg–Mg(DMAP)(^DepNacnac)] reductively trimerises CO to give a rare example of a deltate complex, [{(^DepNacnac)Mg(μ-C₃O₃)Mg(DMAP)(^DepNacnac)}₂]. In contrast, the two smaller adduct complexes react with only two CO molecules, ultimately giving unusual ethenediolate complexes [{(^ArNacnac)Mg{μ-OC(H) C(DMAP^−H)O}Mg(^ArNacnac)}₂] (Ar = Xyl or Mes). DFT calculations show the latter reactions to proceed via reductive dimerizations of CO, and subsequent intramolecular C–H activation of Mg-ligated DMAP by “zig–zag” [C₂O₂]²⁻ fragments of reaction intermediates. Calculations also suggest that magnesium deltate complexes are kinetic products in these reactions, while the magnesium ethenediolates are thermodynamic products. This study shows that subtle changes to the bulk of the reacting 1 : 1 DMAP–magnesium(i) adduct complexes can lead to fine steric control over the products arising from their CO reductive oligomerisations. Furthermore, it is found that the more activated nature of the adduct complexes, relative to their symmetrical, three-coordinate counterparts, [{(^ArNacnac)Mg}₂], likely derives more from the polarisation of the Mg–Mg bonds of the former, than the elongated nature of those bonds.

Subtle changes to the bulk of 1 : 1 adducts of DMAP with magnesium(i) complexes leads to steric control over the products arising from their reductive oligomerisations of carbon monoxide.