d-Cube decompositions ofK n/K m

Necessary conditions onn, m andd are given for the existence of an edge-disjoint decomposition ofK _n/K _m into copies of the graph of ad-dimensional cube. Sufficiency is shown whend=3 and, in some cases, whend=2^t. We settle the problem of embedding 3-cube decompositions ofK _m into 3-cube decompositions ofK _n, wherenm.Research of P.A. and D.E.B. supported by Australian Research Council grant A49532750Research of D.E.B. supported by Australian Research Council grant ARCPDF015GResearch of S.I.E. and C.V.E. supported by Illinois State University Research Office     

A survey on the existence of G‐Designs

A G‐design of order n is a decomposition of the complete graph on n vertices into edge‐disjoint subgraphs isomorphic to G. We survey the current state of knowledge on the existence problem for G‐designs. This includes references to all the necessary designs and constructions, as well as a few new designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 373–410, 2008     

Integrating approaches to revitalise a church's mission strategy

A novel integration of elements of Strategic Options Development and Analysis (SODA) and Strategic Choice Approach (SCA) was used to assist a Church mission committee to overcome a crisis of direction. The intervention raised questions about the requirements for successful prosecution of community OR projects, and some answers to these are suggested.     

Packing cycles in complete graphs

We introduce a new technique for packing pairwise edge-disjoint cycles of specified lengths in complete graphs and use it to prove several results. Firstly, we prove the existence of dense packings of the complete graph with pairwise edge-disjoint cycles of arbitrary specified lengths. We then use this result to prove the existence of decompositions of the complete graph of odd order into pairwise edge-disjoint cycles for a large family of lists of specified cycle lengths. Finally, we construct new maximum packings of the complete graph with pairwise edge-disjoint cycles of uniform length.     

On Surfaces with Prescribed Shape Operator

The problem of immersing a simply connected surface with a prescribed shape operator is discussed. It is shown that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the space of such realizations is a convex set in an affine space of dimension at most 3. The cases where this maximum dimension of realizability is achieved are analyzed and it is found that there are two such families of shape operators, one depending essentially on three arbitrary functions of one variable and another depending essentially on two arbitrary functions of one variable. The space of realizations is discussed in each case, along with some of their remarkable geometric properties. Several explicit examples are constructed.     

Electrophoresis of colloidal α-alumina

Measurements of the electrophoretic mobility (u _E) of particles of colloidal α-alumina were made as a function of pH, electrolyte concentration and electrolyte type (NaCl, NaNO₃ and KCl) using two similar instrumental techniques. Significant differences (50% or less) in the values of u _E of particles in NaCl were obtained from the two instruments; however, the isoelectric points (IEPs) (the pH at which u _E=0), estimated from the two sets of measurements, occurred at 7.5 ± 0.3 and 7.8 ± 0.05 and were not significantly different. The latter estimate corresponds with those for particles in KCl and NaNO₃ of 8.05 ± 0.11 and 7.95 ± 0.18, respectively, made using the same instrument and indicate that the IEP was a weak function of electrolyte type. When cations acted as counterions (pH > IEP), the absolute magnitudes and the ranges of u _E with electrolyte concentration were found to be significantly less than when anions acted as counterions (IEP > pH). Estimates of the zeta potential (ζ), made using various procedures, showed variations of up to 25% at low ratios of electrical-double-layer thickness (κ ⁻¹) to particle radius (a) (κa∼10) and were of a similar scale to differences in u _E, but no significant variations (95% confidence) in ζ were obtained at high values (κa∼200). Received: 12 July 2000 Accepted: 17 October 2000