The grocery retailing industry is a key sector of the UK economy, accounting for 16% of consumer expenditure. As such developments in this sector will have an important effect on UK economic and social welfare. Arguably, competition in the sector has intensified in the last two decades. Yet significant consolidation in the sector has put it under the spotlight of the UK competition authorities. Previous research analysing factors affecting the performance and positioning of UK grocery retailers has tended to be restricted to a limited time period and a small sample size. This study extends this research by investigating industry evolution and factors affecting performance and industry structure with longitudinal panel data, covering a two-decade time period. The sample used covers a wide variety of grocery retailers with a range of different attributes over a significant period in the development of the industry, running from 1985 through to 2003. Strategic group theory is employed to study industry evolution and examine the effects of strategic positioning on performance. Specifically, fixed-effect panel econometric models are estimated at different levels of aggregation to analyse firm, industry and strategic group effects on the performance of retailers. Also, stochastic frontier models, in the form of Cobb-Douglas and Transcendental Logarithmic functions, are estimated at different levels of aggregation to analyse the efficiency of retailers in the sector. The results from cluster analysis on strategic groups suggest that industry structure is likely to become more concentrated, and the size of retailers will be a significant mobility barrier in the industry. The profitability analysis finds industry level factors and strategic group composition to be crucial in explaining performance differences. The efficiency analysis finds large retailers exhibiting economies of scale through operating large store formats to be significantly better positioned from smaller-format retailers with fewer outlets. Consistent and significant time dummies demonstrate the favourable macro environment enjoyed by the retailers for much of the 1990s. The analysis reveals potentially useful insights for retail managers, especially concerning the importance of positioning in the industry and the choice of strategic orientation. More generally, the study opens up further possibilities for future studies of performance and efficiency measurement over an even longer time as the sector continues to develop and shape the way consumers shop in the UK.

The main result of this thesis is an original proof that every word has finite width in a compact $p$-adic analytic group. The proof we give here is an alternative to Andrei Jaikin-Zapirain's recent proof of the same result, and utilises entirely group-theoretical ideas. We accomplish this by reducing the problem to a proof that every word has finite width in a profinite group which is virtually a polycyclic pro-$p$ group. To obtain this latter result we first establish that such a group can be embedded as an open subgroup of a group of the form $N_1M_1$, where $N_1$ is a finitely generated closed normal nilpotent subgroup, and $M_1$ is a finitely generated closed nilpotent-by-finite subgroup; we then adapt a method of V. A. Romankov. As a corollary we note that our approach also proves that every word has finite width in a polycyclic-by-finite group (which is not profinite). As a supplementary result we show that for finitely generated closed subgroups $H$ and $K$ of a profinite group the commutator subgroup $[H,K]$ is closed, and give examples to show that various hypotheses are necessary. This implies that the outer-commutator words have finite width in profinite groups of finite rank. We go on to establish some bounds for this width. In addition, we show that every word has finite width in a product of a nilpotent group of finite rank and a virtually nilpotent group of finite rank. We consider the possible application of this to soluble minimax groups.

This thesis is a study of the subgroup structure of some remarkable groups of automorphisms of rooted trees. It is divided into two parts. The main result of the first part is seemingly of an algorithmic nature, establishing that the Gupta--Sidki 3-group G has solvable membership problem. This follows the approach of Grigorchuk and Wilson who showed the same result for the Grigorchuk group. The proof, however, is not algorithmic, and it moreover shows a striking subgroup property of G: that all its infinite finitely generated subgroups are abstractly commensurable with either G or G × G. This is then used to show that G is subgroup separable which, together with some nice presentability properties of G, implies that the membership problem is solvable. The proof of the main theorem is also used to show that G satisfies a "strong fractal" property, in that every infinite finitely generated subgroup acts like G on some rooted subtree. The second part concerns the subgroup structure of branch and weakly branch groups in general. Motivated by a natural question raised in the first part, a necessary condition for direct products of branch groups to be abstractly commensurable is obtained. From this condition it follows that the Gupta--Sidki 3-group is not abstractly commensurable with its direct square. The first main result in the second part states that any (weakly) branch action of a group on a rooted tree is determined by the subgroup structure of the group. This is then applied to answer a question of Bartholdi, Siegenthaler and Zalesskii, showing that the congruence subgroup property for branch and weakly branch groups is independent of the actions on a tree. Finally, the information obtained on subgroups of branch groups is used to examine which groups have an essentially unique branch action and why this holds.

<b>Asymptotic cones.</b> A finitely generated group has a word metric, which one can scale and thereby view the group from increasingly distant vantage points. The group coalesces to an "asymptotic cone" in the limit (this is made precise using techniques of non-standard analysis). The reward is that in place of the discrete group one has a continuous object "that is amenable to attack by geometric (e.g. topological, infinitesimal) machinery" (to quote Gromov). We give coarse geometric conditions for a metric space X to have N-connected asymptotic cones. These conditions are expressed in terms of certain filling functions concerning filling N-spheres in an appropriately coarse sense. We interpret the criteria in the case where X is a finitely generated group &Gamma; with a word metric. This leads to upper bounds on filling functions for groups with simply connected cones -- in particular they have linearly bounded filling length functions. We prove that if all the asymptotic cones of &Gamma; are N-connected then &Gamma; is of type F_N+1 and we provide N-th order isoperimetric and isodiametric functions. Also we show that the asymptotic cones of a virtually polycyclic group &Gamma; are all contractible if and only if &Gamma; is virtually nilpotent. <b>Combable groups and almost-convex groups.</b> A combing of a finitely generated group &Gamma; is a normal form; that is a choice of word (a combing line) for each group element that satisfies a geometric constraint: nearby group elements have combing lines that fellow travel. An almost-convexity condition concerns the geometry of closed balls in the Cayley graph for &Gamma;. We show that even the most mild combability or almost-convexity restrictions on a finitely presented group already force surprisingly strong constraints on the geometry of its word problem. In both cases we obtain an n! isoperimetric function, and upper bounds of ~ n² on both the minimal isodiametric function and the filling length function.

One might easily argue that the Classification of Finite Simple Groups is one of the most important theorems of group theory. Given that any finite group can be deconstructed into its simple composition factors, it is of great importance to have a detailed knowledge of the structure of finite simple groups. One of the classes of finite groups that appear in the classification theorem are the simple classical groups, which are matrix groups preserving some form. This thesis will shed some new light on almost simple classical groups in dimension 13, 14 and 15. In particular we will determine their maximal subgroups. We will build on the results by Bray, Holt, and Roney-Dougal who calculated the maximal subgroups of all almost simple finite classical groups in dimension less than 12. Furthermore, Aschbacher proved that the maximal subgroups of almost simple classical groups lie in nine classes. The maximal subgroups in the first eight classes, i.e. the subgroups of geometric type, were determined by Kleidman and Liebeck for dimension greater than 13. Therefore this thesis concentrates on the ninth class of Aschbacher's Theorem. This class roughly consists of subgroups which are almost simple modulo scalars and do not preserve a geometric structure. As our final result we will give tables containing all maximal subgroups of almost simple classical groups in dimension 13, 14 and 15.

An n-dimensional halved cube is a graph whose vertices are the binary strings of length n, where two vertices are adjacent if and only if they differ in exactly two positions. It can be regarded as the graph whose vertex set is one partite set of the n-dimensional hypercube, with an edge joining vertices at hamming distance two. In this thesis we compute the automorphism groups of the halved cubes by embedding them in R n and realizing the automorphism group as a subgroup of GLn(R). As an application we show that a halved cube is a circulant graph if and only if its dimension of is at most four.

Geometric semigroup theory means different things to different people, but it is agreed that it involves associating a geometric structure to a semigroup and deducing properties of the semigroup based on that structure. One such property is finite presentability. In geometric group theory, the geometric structure of choice is the Cayley graph of the group. It is known that in group theory finite presentability is an invariant under quasi-isometry of Cayley graphs. We choose to associate a metric space to a semigroup based on a Cayley graph of that semigroup. This metric space is constructed by removing directions, multiple edges and loops from the Cayley graph. We call this a skeleton of the semigroup. We show that finite presentability of certain types of direct products, completely (0-)simple, and Clifford semigroups is preserved under isomorphism of skeletons. A major tool employed in this is the Švarc-Milnor Lemma. We present an example that shows that in general, finite presentability is not an invariant property under isomorphism of skeletons of semigroups, and in fact is not an invariant property under quasi-isometry of Cayley graphs for semigroups. We give several skeletons and describe fully the semigroups that can be associated to these.

A logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a group G of order Qk j=1 pj mj , the length of any LS for G satises `() Pk j=1mjpj . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS). The MLS conjecture states that every finite simple group has an MLS. If the conjecture is true then every finite group will have an MLS. The conjecture was shown to be true by a number of researchers for a few classes of finite simple groups. However, the problem is still wide open. This dissertation addresses the MLS conjecture for the classical simple groups. In particular, it is shown that MLS's exist for the symplectic groups Sp2n(q), the orthogonal groups O 2n(q0) and the corresponding simple groups PSp2n(q) and 2n(q0) for all n 2 N, prime power q and even prime power q0. The existence of an MLS is also shown for all unitary groups GUn(q) for all odd n and q = 2s under the assumption that an MLS exists for GUn 1(q). The methods used are very general and algorithmic in nature and may be useful for studying all nite simple groups of Lie type and possibly also the sporadic groups. The blocks of logarithmic signatures constructed in this dissertation have cyclic structure and provide a sort of cyclic decomposition for these classical groups. / by Nikhil Singhi. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.

The minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups, An, PSLn(q) when gcd(n; q 1) is 1, 4 or a prime and several sporadic groups [10, 9, 12, 14, 18]. This dissertation is devoted to proving that the conjecture is true for a large class of simple groups of Lie type called classical groups. The methods developed use the structure of these groups as isometry groups of bilinear or quadratic forms. A large part of the construction is also based on the Bruhat and Levi decompositions of parabolic subgroups of these groups. In this dissertation the conjecture is shown to be true for the following families of simple groups: the projective special linear groups PSLn(q), the projective symplectic groups PSp2n(q) for all n and q a prime power, and the projective orthogonal groups of positive type + 2n(q) for all n and q an even prime power. During the process, the existence of minimal logarithmic signatures (MLS's) is also proven for the linear groups: GLn(q), PGLn(q), SLn(q), the symplectic groups: Sp2n(q) for all n and q a prime power, and for the orthogonal groups of plus type O+ 2n(q) for all n and q an even prime power. The constructions in most of these cases provide cyclic MLS's. Using the relationship between nite groups of Lie type and groups with a split BN-pair, it is also shown that every nite group of Lie type can be expressed as a disjoint union of sets, each of which has an MLS. / by NIdhi Singhi. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.

Collaborative filtering (CF), a very successful recommender system, is one of the applications of data mining for incomplete data. The main objective of CF is to make accurate recommendations from highly sparse user rating data. My contributions to this research topic include proposing the frameworks of imputation-boosted collaborative filtering (IBCF) and imputed neighborhood based collaborative filtering (INCF). We also proposed a model-based CF technique, TAN-ELR CF, and two hybrid CF algorithms, sequential mixture CF and joint mixture CF. Empirical results show that our proposed CF algorithms have very good predictive performances. In the investigation of applying imputation techniques in mining incomplete data, we proposed imputation-helped classifiers, and VCI predictors (voting on classifications from imputed learning sets), both of which resulted in significant improvement in classification performance for incomplete data over conventional machine learned classifiers, including kNN, neural network, one rule, decision table, SVM, logistic regression, decision tree (C4.5), random forest, and decision list (PART), and the well known Bagging predictors. The main imputation techniques involved in these algorithms include EM (expectation maximization) and BMI (Bayesian multiple imputation). / by Xiaoyuan Su. / Vita. / Thesis (Ph.D.)--Florida Atlantic University, 2008. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2008. Mode of access: World Wide Web.