Relation modules and identities for presentations of inverse monoids

We investigate the Squier complexes of presentations of groups and inverse monoids using the theory semiregular, regular, and pseudoregular groupoids. Our main interest is the class of regular groupoids, and the new class of pseudoregular groupoids. Our study of group presentations uses monoidal, regular groupoids. These are equivalent to crossed modules, and we recover the free crossed module usually associated to a group presentation, and a free presentation of the relation module with kernel the fundamental group of the Squier complex, the module of identities among the relations. We carry out a similar study of inverse monoid presentations using pseudoregular groupoids. The relation module is defined via an intermediate construction – the derivation module of a homomorphism, – and a key ingredient is the factorisation of the presentation map from a free inverse monoid as the composition of an idempotent pure map and an idempotent separating map. We can then use the properties of idempotent separating maps, and properties of the derivation module as a left adjoint, to derive a free presentation of the relation module. The construction of its kernel – the module of identities – uses further key facts about pseudoregular groupoids

The pullback closure, perfect morphisms and completions

Bibliography: pages 92-97.Closure operations within objects of various categories have played an important role in the development of Categorical Topology. Notably they have been used to characterise epimorphisms and investigate cowellpoweredness in specific categories, to generalise Hausdorff separation through diagonal theorems, and to extend topological notions such as compactness of objects and perfectness of morphisms to abstract categories. The categorical theory of factorisation structures for families of morphisms which developed in the 1970's laid the foundation for an axiomatic theory of categorical closure operators. This theory drew together many endeavours involving closure operations, and was coalesced in [Dikranjan, Giuli 1987]. The literature on categorical closure operators continues to extend the theory as well as apply it to problems in Category Theory. Central to our thesis is a particular closure operator (in the sense of [Dikranjan, Giuli 1987]) which we name the "pullback closure operator". Its construction is not entirely new, but no author has studied this operator in its own right. We investigate some of the operator's properties, present several examples and then apply it in two areas of Categorical Topology. First we use the pullback closure operator to establish links between two previously disjoint theories of perfect morphisms. One theory, which developed in the 1970's, exploits the orthogonality properties and functor related properties of perfect continuous maps. Another theory, which has developed more recently, generalises the closure and compactness properties of perfect continuous maps. (We should note that this does not include the recent work in [Clementino, Giuli, Tholen 1995] which takes another approach to perfect morphisms via closure operators.) Our investigations centre around finding conditions that are sufficient to ensure that the links between these two theories can be utilised. Our second use of the pullback closure operator is in pursuing the precategorical ideas expressed in [Birkhoff 1937], and some developments of these ideas in [Brummer, Giuli, Herrlich 1992] and [Brummer, Giuli 1992], to build a theory of completion of objects in an abstract category. In this context the pullback closure operator is shown to be appropriate in characterising complete objects, illuminating links with previously studied completion notions and describing epimorphisms in the category in which we are working. (In fact the pullback closure operator can be used to describe epimorphisms in even wider contexts.) Our methodology is what has been termed colloquially as "doing topology in categories". Topological notions and results are expressed in the language of category theory. Using these reformulations, new results are pursued at the level of categories, and are then applied in specific topological or algebraic contexts. Within this, our approach has been to make as few global assumptions as possible. The pullback closure operator is strictly a tool, in the sense that when assumptions are made, they concern the underlying categories, functors and classes of morphisms and objects and not the operator itself

Polynomials for Crystal Frameworks and the Rigid Unit Mode Spectrum

To each discrete translationally periodic bar-joint framework \C in \bR^d we associate a matrix-valued function \Phi_\C(z) defined on the d-torus. The rigid unit mode spectrum \Omega(\C) of \C is defined in terms of the multi-phases of phase-periodic infinitesimal flexes and is shown to correspond to the singular points of the function z \to \rank \Phi_\C(z) and also to the set of wave vectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium, which corresponds to \Phi_\C(z) being square, the determinant of \Phi_\C(z) gives rise to a unique multi-variable polynomial p_\C(z_1,\dots,z_d). For ideal zeolites the algebraic variety of zeros of p_\C(z) on the d-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealised framework rigidity and flexibility and in particular leads to an explicit formula for the number of supercell-periodic floppy modes. In the case of certain zeolite frameworks in dimensions 2 and 3 direct proofs are given to show the maximal floppy mode property (order $N$). In particular this is the case for the cubic symmetry sodalite framework and some other idealised zeolites.Comment: Final version with new examples and figures, and with clearer streamlined proof

Massively parallel solvers for elliptic partial differential equations in numerical weather and climate prediction:scalability of elliptic solvers in NWP

The demand for substantial increases in the spatial resolution of global weather- and climate- prediction models makes it necessary to use numerically efficient and highly scalable algorithms to solve the equations of large scale atmospheric fluid dynamics. For stability and efficiency reasons several of the operational forecasting centres, in particular the Met Office and the ECMWF in the UK, use semi-implicit semi-Lagrangian time stepping in the dynamical core of the model. The additional burden with this approach is that a three dimensional elliptic partial differential equation (PDE) for the pressure correction has to be solved at every model time step and this often constitutes a significant proportion of the time spent in the dynamical core. To run within tight operational time scales the solver has to be parallelised and there seems to be a (perceived) misconception that elliptic solvers do not scale to large processor counts and hence implicit time stepping can not be used in very high resolution global models. After reviewing several methods for solving the elliptic PDE for the pressure correction and their application in atmospheric models we demonstrate the performance and very good scalability of Krylov subspace solvers and multigrid algorithms for a representative model equation with more than $10^{10}$ unknowns on 65536 cores on HECToR, the UK's national supercomputer. For this we tested and optimised solvers from two existing numerical libraries (DUNE and hypre) and implemented both a Conjugate Gradient solver and a geometric multigrid algorithm based on a tensor-product approach which exploits the strong vertical anisotropy of the discretised equation. We study both weak and strong scalability and compare the absolute solution times for all methods; in contrast to one-level methods the multigrid solver is robust with respect to parameter variations.Comment: 24 pages, 7 figures, 7 table

Cooperative Interactions in Lattices of Atomic Dipoles

Coherent radiation by an ensemble of scatterers can dramatically modify the ensemble's optical response. This can include, for example, enhanced and suppressed decay rates (superradiance and subradiance respectively), energy level shifts, and highly directional scattering. This behaviour is referred to as cooperative, since the scatterers in the ensemble behave as a collective rather than independently. In this Thesis, we investigate the cooperative behaviour of one- and two-dimensional arrays of interacting atoms. We calculate the extinction cross-section of these arrays, analysing how the cooperative eigenmodes of the ensemble contribute to the overall extinction. Typically, the dominant eigenmode if the atoms are driven by a uniform or Gaussian light beam is the mode in which the atomic dipoles oscillate in phase together and with the same polarisation as the driving field. The eigenvalues of this mode become strongly resonant as the atom number is increased. For a one-dimensional array, the location of these resonances occurs when the atomic spacing is an integer or half integer multiple of the wavelength, thus behaving analogously to a single atom in a cavity. The interference between this mode and additional eigenmodes can result in Fano-like asymmetric lineshapes in the extinction. We find that the kagome lattice in particular exhibits an exceptionally strong interference lineshape, like a cooperative analog of electromagnetically induced transparency. Triangular, square and hexagonal lattices however are typically dominated by one single mode which, for lattice spacings of the order of a wavelength, can be highly subradiant. This can result in near-perfect extinction of a resonant driving field, signifying a significant increase in the atom-light coupling efficiency. We show that this extinction is robust to possible experimental imperfections

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Matching, entropy, holes and expansions

In this dissertation, matching, entropy, holes and expansions come together. The first chapter is an introduction to ergodic theory and dynamical systems. The second chapter is on, what we called Flipped $\alpha$-expansions. For this family we have an invariant measure that is $\sigma$-finite infinite. We calculate the Krengel entropy for a large part of the parameter space and find an explicit expression for the density by using the natural extension. In Chapter 3 Ito Tanaka's $\alpha$-continued fractions are studied. We prove that matching holds almost everywhere and that the non-matching set has full Hausdorff dimension. In the fourth chapter we study $N$-expansions with flips. We use a Gauss-Kuzmin-Levy method to approximate the density for a large family and use this to give an estimation for the entropy. In the last Chapter we look at greedy $\beta$-expansions. We show that for almost every $\beta\in(1,2]$ the set of points $t$ for which the forward orbit avoids the hole $[0,t)$ has infinitely many isolated and infinitely many accumulation points in any neighborhood of zero. Furthermore, we characterize the set of $\beta$ for which there are no accumulation points and show that this set has Hausdorff dimension zero.Number theory, Algebra and Geometr

Coarse geometry: a foundational and categorical approach with applications to groups and hyperspaces

The topic of the manuscript is coarse geometry, also known as large-scale geometry, which is the study of large-scale properties of spaces. It found applications in geometric group theory after the work of Gromov, and in Novikov and coarse Baum-Connes conjectures. The thesis is divided into three parts. In the first one, we provide a foundational and categorical approach to coarse geometry. Large-scale geometry was originally developed for metric spaces and then Roe introduced coarse structures as a large-scale counterpart of uniformities. However, coarse spaces are innerly symmetric objects, and thus are not suitable to parametrise asymmetric objects such as monoids and quasi-metric spaces. In order to fill the gap, we introduce quasi-coarse spaces. Moreover, we consider also semi-coarse spaces and entourage spaces. These objects induce para-bornologies, quasi-bornologies, semi-bornologies, pre-bornologies (also known as bounded structures) and bornologies, and this process is similar to the definition of uniform topology from a (quasi-)uniform space. We study all the notions introduced and recalled to find extensions of classical results proved for metric or coarse spaces, and similarities with notions and properties for general topology. Furthermore, we study the categories of those objects and the relations among them. In particular, since all of them are topological categories, we have a complete understanding of their epimorphisms and monomorphisms, and the description of many categorical constructions. Among them, of particular interest are quotients. We then focus our attention on Coarse, the category of coarse spaces and bornologous maps, discussing its closure operators and the cowellpoweredness of its epireflective subcategories, and its quotient category Coarse/~, which turns out to be balanced and cowellpowered. The second part is dedicated to study the large-scale geometry of algebraic objects, such as unitary magmas, monoids, loops and groups. In particular, we focus on coarse groups (groups endowed with suitable coarse structures) and we investigate their category. We study different choices, underlining advantages and drawbacks. With some restrictions on the coarse groups that we are considering, if we enlarge the class of morphisms to contain bornologous quasi-homomorphisms (and not just bornologous homomorphisms), every coarse inverse of a homomorphism which is a coarse equivalence is a quasi-homomorphism. This observation is connected to the notion of localisation of a category and could provide a categorical justification to the notion of quasi-homomorphism. Once the categories of coarse groups are fixed, inspired by the notion of functorial topologies, we can introduce functorial coarse structures on Grp, the category of groups and homomorphisms, and on TopGrp, the category of topological groups and continuous homomorphisms. Among them, we pay attention to the ones induced by cardinal invariants, and to those associated to the family of relatively compact subsets. As for the latter functorial coarse structure, we study the transformation of large-scale properties along Pontryagin and Bohr functors. The third part is devoted to coarse hyperspaces, which are suitable coarse structures on power sets of coarse spaces. This construction was introduced following the work of Protasov and Protasova and miming the classical notion of uniform hyperspace. We see how properties of the initial coarse space are reflected on the hyperspace. Since the coarse hyperspace is highly disconnected, it is convenient to consider some special subspaces of it. For example, if the base space is a coarse group, it is natural to consider the subspace structure induced on the lattice of subgroups, called subgroup exponential hyperballean. We show that both the subgroup exponential hyperballean and the subgroup logarithmic hyperballean, another coarse structure on the subgroup lattice, capture many properties of the group