Reaping Numbers of Boolean Algebras

A subset A of a Boolean algebra B is said to be (n,m)-reapedif there is a partition of unity p B of size n such that |{b p:b a 0}| m for all a A. The reaping number r_n,m (B) ofa Boolean algebra B is the minimum cardinality of a set A B\{0}which cannot be (n,m)-reaped. It is shown that for each n, thereis a Boolean algebra B such that r_n+1,2(B) r_n,2(B). Also, {r_n,m(B):mn } consists of at most two consecutive cardinals. The existenceof a Boolean algebra B such that r_n,m (B) r_n',m' (B) is equivalentto a statement in finite combinatorics which is also discussed.     

Chaotic scattering in the gravitational three-body problem

We summarize some results of an ongoing study of the chaotic scattering interaction between a bound pair of stars (a binary) and an incoming field star. The stars are modeled as point masses and their equations of motion are numerically integrated for a large number of initial conditions. The global features of the resulting initial-value space maps are presented, and their evolution as a function of system parameters is discussed. We find that the maps contain regular regions separated by rivers of chaotic behavior. The probability of escape within the chaotic regions is discussed, and a straightforward explanation of the scaling present in these regions is reviewed. We investigate a statistical quantity of interest, namely the cross section for temporarily bound interactions, as a function of the third star's incoming velocity and mass. Finally, a new way of considering long-lived trajectories is presented, allowing long data sets to be qualitatively analyzed at a glance.     

On depth first search strees inm-out digraphs

We consider depth first search (DFS for short) trees in a class of random digraphs: am-out model. Let _i be thei ^th vertex encountered by DFS andL(i, m, n) be the height of _i in the corresponding DFS tree. We show that ifi/n asn, then there exists a constanta(,m), to be defined later, such thatL(i, m, n)/n converges in probability toa(,m) asn. We also obtain results concerning the number of vertices and the number of leaves in a DFS tree.     

Some aspects of the dependence of solid-state structure on electron count

The method of moments is used to show how the structures of molecules and solids is sensitive to the number of valence electrons.     

Transport processes in the sorption of colored ions by peat particles

A three-step model has been proposed for the adsorption of Astrazone Blue dye (Basic Blue 69) on peat. The initial rate of uptake of dye ions due to physical adsorption and chemisorption (ion exchange) has been correlated using a surface mass transfer coefficient. These coefficients have been determined and expressed in the dimensionless mass transfer form, Sh/Sc^0.33, as a function of agitation, initial dye concentration, peat particle size range, dye solution temperature, and mass of peat.     

Preparation of some substituted imidazo[1,2-a]pyridine derivatives

2-Bromopyridine derivatives 2a-2c were prepared. Compounds 2b and 2c and ammonia yielded aminopyridines 3b and 3c which were converted to imidazo[1,2-a]pyridine derivatives 4b and 4c . Compound 4b was nitrated giving the analogue 5b of metronidazole 1 .     

Including correlation effects in an improved spreadsheet calculation of combined standard uncertainties

A spreadsheet method allowing rapid calculation of combined standard uncertainties is described. The model used allows explicitly for correlation effects, and requires a user to enter only the parameters, the calculation used to obtain the final result (including relevant influence factors), the individual standard uncertainties for the parameters, and estimates of correlation coefficients where necessary. The estimation of correlation coefficients in common cases is discussed, and it is shown that correlation is likely to be practically significant only when the correlated contribution to individual standard uncertainties is significantly over about 30% of the relevant standard uncertainty, leading to correlation coefficients |r| greater than 0.1. The implementation includes a more robust differentiation algorithm than previously reported for spreadsheet use, and initial preparation of the spreadsheets has been automated. The principle is illustrated with a simple example. Electronic Supplementary Material  Supplementary material is available in the online version of this article at and is accessible for authorized users.