Eigenvalues of the Laplacian matrix of a graph have been widely used in studying connectivity and expansion properties of networks, and also in analyzing random walks on a graph. Independently, statisticians introduced various optimality criteria in experimental design, the goal being to obtain more accurate estimates of quantities of interest in an experiment. It turns out that the most popular of these optimality criteria for block designs are determined by the Laplacian eigenvalues of the concurrence graph, or of the Levi graph, of the design. The most important optimality criteria, called A (average), D (determinant) and E (extreme), are related to the conductance of the graph as an electrical network, the number of spanning trees, and the isoperimetric properties of the graphs, respectively. The number of spanning trees is also an evaluation of the Tutte polynomial of the graph, and is the subject of the Merino–Welsh conjecture relating it to acyclic and totally cyclic orientations, of interest in their own right. This chapter ties these ideas together, building on the work in [4] and [5].

In some experiments, the experimental units are all pairs of individuals who have to undertake a given task together. The set of such pairs forms a triangular association scheme. Appropriate randomization then gives two non-trivial strata. The design is said to have commutative orthogonal block structure (COBS) if the best linear unbiased estimators of treatment contrasts do not depend on the stratum variances. There are precisely three ways in which such a design can have COBS. We give a complete description of designs for which all treatment contrasts are in the same stratum. Then we give a very general construction for designs with COBS which have some treatment contrasts in each stratum.

Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over any group, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our axioms. However, for m>=3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions are equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group except in four small cases with m<=3. The class of diagonal graphs includes some well-known families, Latin-square graphs and folded cubes. We obtain partial results on the chromatic number of a diagonal graph, and mention an application to synchronization.

Semi-Latin rectangles are generalizations of Latin squares and semi-Latin squares. Although they are called rectangles, the number of rows and the number of columns are not necessarily distinct. There are k treatments in each cell (row-column intersection): these constitute a block. Each treatment of the design appears a definite number of times in each row and also a definite number of times in each column (these parameters also being not necessarily distinct). When k = 2, the design is said to have block size two. Regular-graph semi-Latin rectangles have the additional property that the treatment concurrences between any two pairs of distinct treatments differ by at most one. Constructions for semi-Latin rectangles of this class with k = 2 which have v treatments, v/2 rows and v columns, where v is even, are given in Bailey and Monod (2001). These give the smallest designs when v is even. Here we give constructions for smallest designs with k = 2 when v is odd. These are regular-graph semi-Latin rectangles where the numbers of rows, columns and treatments are identical. Then we extend the smallest designs in each case to obtain larger designs.

We examine the subgroup $D(G)$ of a transitive permutation group $G$ which is generated by the derangements in $G$. Our main results bound the index of this subgroup: we conjecture that, if $G$ has degree $n$ and is not a Frobenius group, then $|G:D(G)|\leqslant\sqrt{n}-1$; we prove this except when $G$ is a primitive affine group. For affine groups, we translate our conjecture into an equivalent form regarding $|H:R(H)|$, where $H$ is a linear group on a finite vector space and $R(H)$ is the subgroup of $H$ generated by elements having eigenvalue~$1$. If $G$ is a Frobenius group, then $D(G)$ is the Frobenius kernel, and so $G/D(G)$ is isomorphic to a Frobenius complement. We give some examples where $D(G)\ne G$, and examine the group-theoretic structure of $G/D(G)$; in particular, we construct groups $G$ in which $G/D(G)$ is not a Frobenius complement.

The expectation part of a linear model is often presented as a single equation with unknown parameters, and the reader is supposed to know that this is shorthand for a whole family of expectation models (for example, is there interaction or not?). It is helpful to list the whole family of models separately and then represent them on a Hasse diagram. This shows which models are sub-models of others, which helps the user to respect marginality when choosing the most parsimonious model to explain the data. Each row in an analysis-of-variance table corresponds to an edge in the Hasse diagram. In the scaled version of the Hasse diagram, the length of each edge is proportional to the appropriate mean square. This gives a visual display of the analysis of variance (ANOVA). For some people, this is easier to interpret than the standard analysis-of-variance table. Moreover, the scaled Hasse diagram makes clear the difficulties in model choice that can occur under non-orthogonality. The ideas are illustrated using some familiar families of models defined by crossed and nested factors, possibly including polynomial terms for quantitative factors, as well as some more recently introduced families of models for experiments in biodiversity.

According to the O'Nan--Scott Theorem, a finite primitive permutation group either preserves a structure of one of three types (affine space, Cartesian lattice, or diagonal semilattice), or is almost simple. However, diagonal groups are a much larger class than those occurring in this theorem. For any positive integer $m$ and group $G$ (finite or infinite), there is a diagonal semilattice, a sub-semilattice of the lattice of partitions of a set $\Omega$, whose automorphism group is the corresponding diagonal group. Moreover, there is a graph (the diagonal graph), bearing much the same relation to the diagonal semilattice and group as the Hamming graph does to the Cartesian lattice and the wreath product of symmetric groups. Our purpose here, after a brief introduction to this semilattice and graph, is to establish some properties of this graph. The diagonal graph $\Gamma_D(G,m)$ is a Cayley graph for the group~$G^m$, and so is vertex-transitive. We establish its clique number in general and its chromatic number in most cases, with a conjecture about the chromatic number in the remaining cases. We compute the spectrum of the adjacency matrix of the graph, using a calculation of the M\"obius function of the diagonal semilattice. We also compute some other graph parameters and symmetry properties of the graph. We believe that this family of graphs will play a significant role in algebraic graph theory.

A new class of designs is introduced for both estimating the variance components of nested factors and testing hypotheses about those variance components. These designs are flexible, and can be chosen so that the degrees of freedom are more evenly spread among the factors than they are in balanced nested designs. The variances of the estimators are smaller than those in stair nested designs of comparable size. The mean squares used in the estimation process are mutually independent, which avoids some of the problems with staggered nested designs.

Microcosm studies are a useful tool when it comes to studying leaf litter decomposition but designing and analysing them can be a tricky path with many pitfalls. Because there is a plethora of drivers of leaf decomposition, it is important to be precise about the scientific questions that can be addressed with microcosm set-ups, and to use experimental designs that have minimal logistic implications but, at the same time, high statistical power. In this chapter, we first set the scene by introducing a hypothetical study that has the aim to estimate how leaf decomposition is driven by different decomposers and abiotic conditions. Following from this scenario, we give an overview of the main biotic and abiotic drivers of leaf decomposition that will play a role in laboratory settings (with special attention to consumer species identity, species richness, body size and metabolic capacity, and also temperature, time scales and stressors). We then explain how to design and analyse laboratory experiments on aquatic leaf litter decomposition including the mathematics for calculating the metabolic power of leaf decomposers and some statistical models. Further three case studies are given—highly controlled experiment that can be analysed by analysis of variance.

Declines in species diversity carry profound implications for ecosystem functioning. Communities of primary producers and consumers interact on evolutionary as well as ecological time scales, shaping complex relationships between biodiversity and ecosystem functioning. In subsidized ecosystems, resource inputs are independent of consumer actions, offering a simplified view of the relationship between species diversity and function for higher trophic levels. With food webs supported by substantial but variable inputs of detritus from adjacent marine ecosystems, sandy beaches are classic examples of subsidized ecosystems. We investigated effects of consumer species diversity and identity on a key ecological function, consumption of kelp wrack from nearshore giant kelp (Macrocystis pyrifera) forests. We assessed effects of species richness on kelp consumption by experimentally manipulating richness of six common species of invertebrate detritivores in laboratory mesocosms and conducting field assays of kelp consumption on beaches. Consumer richness had no effect on kelp consumption in the field and a slight negative effect in laboratory experiments. Kelp consumption was most strongly affected by the species composition of the detritivore community. Species identity and body size of intertidal detritivores drove variation in kelp consumption rates in both experiments and field assays. Our results provide further evidence that species traits, rather than richness per se, influence ecosystem function most, particularly in detrital-based food webs with high functional redundancy across species. On sandy beaches, where biodiversity is threatened by rising sea levels and expanding development, our findings suggest that loss of large-bodied consumer species could disproportionally impact ecosystem function.

R.A. Fisher was one of the greatest scientists of the 20th century, whose scientific contributions ranged over a very wide area of science. He was, however, from an early age a supporter of eugenic ideas, and for this reason has been accused of being a racist and an advocate of forced sterilisation of certain kinds of people. Concerns are now increasingly being raised about historical figures such as Fisher, stimulated by recent abhorrent racially motivated events. A primary aim of this paper is to conduct a careful analysis of Fisher’s writings on eugenics and related areas, in the context of these adverse views. We first review his scientific achievements and then describe his early development of interests in genetics and eugenic ideas. We outline his concerns about the inverse relationship between intellectual ability and fertility, his views on how this might adversely affect the development of human societies, his eugenic proposals for differential family allowances, and his support for voluntary sterilisation for the “feebleminded”. We follow this with an account of Fisher’s interaction with the Nazi-supporting German human geneticist Verschuer, and Fisher’s response to the 1952 UNESCO statement on “The Race Concept”. We finally discuss attitudes to Fisher’s eugenics related activities in a modern context. Our conclusion is that, however much people now reasonably disagree with his stated ideas on eugenics and racial issues, there is no case for dishonouring him and allowing these concerns to outweigh the appreciation of his enormous scientific and humanitarian contributions.

For integers n > 2 and k > 0, an (n x n)/k semi-Latin square is an n x n array of k-subsets (called blocks) of an nk-set (of treatments), such that each treatment occurs once in each row and once in each column of the array. A semi-Latin square is uniform if every pair of blocks, not in the same row or column, intersect in the same positive number of treatments. It is known that a uniform (n x n)/k semi-Latin square is Schur optimal in the class of all (n x n)/k semi-Latin squares, and here we show that when a uniform (n x n)/k semi-Latin square exists, the Schur optimal (n x n)/k semi-Latin squares are precisely the uniform ones. We then compare uniform semi-Latin squares using the criterion of pairwise-variance (PV) aberration, introduced by J. P. Morgan for affine resolvable designs, and determine the uniform (n x n)/k semi-Latin squares with minimum PV aberration when there exist n - 1 mutually orthogonal Latin squares of order n. These do not exist when n = 6, and the smallest uniform semi-Latin squares in this case have size (6 x 6)/10. We present a complete classification of the uniform (6 x 6)/10 semi-Latin squares, and display the one with least PV aberration. We give a construction producing a uniform ((n + 1) x (n + 1))/((n - 2)n) semi-Latin square when there exist n -1 mutually orthogonal Latin squares of order n, and determine the PV aberration of such a uniform semi-Latin square. Finally, we describe how certain affine resolvable designs and balanced incomplete-block designs can be constructed from uniform semi-Latin squares. From the uniform (6 x 6)/10 semi-Latin squares we classified, we obtain (up to block design isomorphism) exactly 16875 affine resolvable designs for 72 treatments in 36 blocks of size 12 and 8615 balanced incomplete-block designs for 36 treatments in 84 blocks of size 6. In particular, this shows that there are at least 16875 pairwise non-isomorphic orthogonal arrays OA(72, 6, 6, 2). (c) 2020 Elsevier B.V. All rights reserved.

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for m ≥ 2 , a set of m + 1 partitions of a set Ω , any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if m = 2 ), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G . In this paper we investigate what happens if we have m + r partitions with r ≥ 2 , any m of which are minimal elements of a Cartesian lattice. If m = 2 , this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For m > 2 , things are more restricted. Any m + 1 of the partitions generate a join-semilattice admitting a diagonal group over a group G . It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p , then the structure corresponds exactly to an arc of cardinality m + r in the ( m - 1 ) -dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q . These examples can be lifted to non-elementary abelian groups using p -adic techniques.

There exists a set of designs which form a subclass of semi-Latin rectangles.These designs, besides being semi-Latin rectangles, exhibit an additional property of balance, where no two distinct pairs of symbols (treatments) differ in their concurrences, that is, each pair of distinct treatments concurs a constant number of times in the design. Such a design exists for a limited set of parameter combinations. We designate it a balanced semi-Latin rectangle and give some properties and necessary conditions for its existence. Furthermore, algorithms for constructing the design for experimental situations where there are two treatments in each row–column intersection (block) are also given.

Square lattice designs are often used in trials of new varieties of various agricultural crops. However, there are no square lattice designs for 36 varieties in blocks of size six for four or more replicates. Here, we use three different approaches to construct designs for up to eight replicates. All the designs perform well in terms of giving a low average variance of variety contrasts. Supplementary materials accompanying this paper appear online.

We study equitable partitions of Latin-square graphs, and give a complete classification of those whose quotient matrix does not have an eigenvalue $-3$.

Instructional design is a structured, systematic process of designing and developing an instructional interface. Correct application of instructional design principles influences user behavior and promotes safe and effective use of medical devices and drug delivery products. There are many resources available to the person or team responsible for developing the instructional interface. These resources guide the application of instructional design principles for the best possible design of instructional and training materials. This chapter provides a practical guide to applying design principles during initial development of instructional materials, or when identifying root causes and recommended improvements based on human factors study data.

A triple array is a rectangular array containing letters, each letter occurring equally often with no repeats in rows or columns, such that the number of letters common to two rows, two columns, or a row and a column are (possibly different) non-zero constants. Deleting the condition on the letters common to a row and a column gives a double array. We propose the term \emph{sesqui-array} for such an array when only the condition on pairs of columns is deleted. Thus all triple arrays are sesqui-arrays. In this paper we give three constructions for sesqui-arrays. The first gives $(n+1)\times n^2$ arrays on $n(n+1)$ letters for $n\geq 2$. (Such an array for $n=2$ was found by Bagchi.) This construction uses Latin squares. The second uses the \emph{Sylvester graph}, a subgraph of the Hoffman--Singleton graph, to build a good block design for $36$ treatments in $42$ blocks of size~$6$, and then uses this in a $7\times 36$ sesqui-array for $42$ letters. We also give a construction for $K\times(K-1)(K-2)/2$ sesqui-arrays on $K(K-1)/2$ letters. This construction uses biplanes. It starts with a block of a biplane and produces an array which satisfies the requirements for a sesqui-array except possibly that of having no repeated letters in a row or column. We show that this condition holds if and only if the \emph{Hussain chains} for the selected block contain no $4$-cycles. A sufficient condition for the construction to give a triple array is that each Hussain chain is a union of $3$-cycles; but this condition is not necessary, and we give a few further examples. We also discuss the question of which of these arrays provide good designs for experiments.

A triple array is a rectangular array containing letters, each letter occurring equally often with no repeats in rows or columns, such that the number of letters common to two rows, two columns, or a row and a column are (possibly different) non-zero constants. Deleting the condition on the letters common to a row and a column gives a double array. We propose the term sesqui-array for such an array when only the condition on pairs of columns is deleted. Thus all triple arrays are sesqui-arrays. In this paper we give three constructions for sesqui-arrays. The first gives (n + 1) × n2 arrays on n(n + 1) letters for n ≥ 2. (Such an array for n = 2 was found by Bagchi.) This construction uses Latin squares. The second uses the Sylvester graph, a subgraph of the Hoffman–Singleton graph, to build a good block design for 36 treatments in 42 blocks of size 6, and then uses this in a 7×36 sesquiarray for 42 letters. We also give a construction for K × (K − 1)(K − 2)/2 sesquiarrays on K(K − 1)/2 letters. This construction uses biplanes. It starts with a block of a biplane and produces an array which satisfies the requirements for a sesqui-array except possibly that of having no repeated letters in a row or column. We show that this condition holds if and only if the Hussain chains for the selected block contain no 4cycles. A sufficient condition for the construction to give a triple array is that each Hussain chain is a union of 3-cycles; but this condition is not necessary, and we give a few further examples. We also discuss the question of which of these arrays provide good designs for experiments. School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, U.K. Department of Science Education and Mathematics, Mid Sweden University, SE-851 70 Sundsvall, Sweden