A series of novel, cyclic, volatile, N-alkylgallazanes (RNHGaH,), (R = Et, Pr", Pr' , Bun, Bu', Bu", or But), has been synthesised and characterised by analyses, molecular weight determinations, and i.r., l H n.m.r., and mass spectroscopic measurements. The effect of R group on ring size and conformational properties has been investigated.Normal alkyl groups lead to trimeric gallazanes whereas with branched-chained alkyl groups a preference for a dimeric structure has been demonstrated. lsopropyl gallazane and s-butyl gallazane are mobile liquids at room temperature and novel lH n.m.r. and i.r. spectra have been obtained on the neat liquids.

We explore the relationship between subgroups and the possible shifts of finite type (SFTs) which can be defined on a group. In particular, we investigate two group invariants, weak periodicity and strong periodicity, defined via symbolic dynamics on the group. We show that these properties are invariants of commensurability. Thus, many known results about periodic points in SFTs defined over groups are actually results about entire commensurability classes. Additionally, we show that the property of being not strongly periodic (i.e. the property of having a weakly aperiodic SFT) is preserved under extensions with finitely generated kernels. We conclude by raising questions and conjectures about the relationship of these invariants to the geometric notions of quasi-isometry and growth.

Single-case experimental designs allow practitioners to conduct clinical outcomes research without the large samples and substantial resources required by randomized clinical trials. Single-case designs have been used to conduct outcomes research for many decades; however, the statistical measurement of treatment effect sizes remains an unresolved issue. The percentage of nonoverlapping data (PND) is one widely used statistic for effect size measurement of single-case experimental designs. Despite its limitations, PND is useful because it is easy to calculate and interpret. However, null hypothesis significance testing (i.e., the use of p values) is not currently feasible with PND because it has an unknown sampling distribution. A method to calculate p values for PND is introduced and discussed. An online calculator and statistical computing code are also made available to single-case investigators who wish to calculate p values for their data. Calculating PND and its associated p values may provide practitioners with valuable insights about their treatment outcomes when PND is used appropriately and its statistical assumptions are not violated.

A series of novel, volatile, cyclogallata-azonianes [dH2*(CH2);h*GaH2],, (wherex = 1, 2, 3, or 4; n = 2 or 3). has been prepared and characterized. Analogous aluminium and boron compounds have been prepared and comparative studies have been made on the three series of compounds. Factors governing the degree of association, n, of the compounds are discussed and ring strain and additional arguments are invoked to explain the tendency of some of the compounds to undergo ring fission and polymerization.

Abstract. We prove that if G P is a finitely constrained group of binary rooted tree automorphisms (a group binary tree subshift of finite type) defined by an essential pattern group P of pattern size d, d ≥ 2, and if G P has maximal Hausdorff dimension (equal to 1−1/2 d−1 ), then G P is not topologically finitely generated. We describe precisely all essential pattern groups P that yield finitely constrained groups with maximal Haudorff dimension. For a given size d, d ≥ 2, there are exactly 2 d−1 such pattern groups and they are all maximal in the group of automorphisms of the finite rooted regular tree of depth d.