Robert Leslie Ellisʼs work on philosophy of science and the foundations of probability theory

The goal of this paper is to provide an extensive account of Robert Leslie Ellis?s largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellis?s reformulation of the metaphysical foundations of traditional probability theory. This parallel is assessed with reference to the disagreement between Ellis and Whewell on the nature of (pure) mathematics and its relation to scientific knowledge.     

On the parallel between normality and extremal disconnectedness

Several familiar results about normal and extremally disconnected (classical or pointfree) spaces shape the idea that the two notions are somehow dual to each other and can therefore be studied in parallel. This paper investigates the source of this ‘duality’ and shows that each pair of parallel results can be framed by the ‘same’ proof. The key tools for this purpose are relative notions of normality, extremal disconnectedness, semicontinuity and continuity (with respect to a fixed class of complemented sublocales of the given locale) that bring and extend to locale theory a variety of well-known classical variants of normality and upper and lower semicontinuities in an illuminating unified manner. This approach allows us to unify under a single localic proof all classical insertion, as well as their corresponding extension results.     

Eigenvalue computation for unitary rank structured matrices

In this paper we describe how to compute the eigenvalues of a unitary rank structured matrix in two steps. First we perform a reduction of the given matrix into Hessenberg form, next we compute the eigenvalues of this resulting Hessenberg matrix via an implicit QR-algorithm. Along the way, we explain how the knowledge of a certain ‘shift’ correction term to the structure can be used to speed up the QR-algorithm for unitary Hessenberg matrices, and how this observation was implicitly used in a paper due to William B. Gragg. We also treat an analogue of this observation in the Hermitian tridiagonal case.     

Bayesian modeling of several covariance matrices and some results on propriety of the posterior for linear regression with correlated and/or heterogeneous errors

We explore simultaneous modeling of several covariance matrices across groups using the spectral (eigenvalue) decomposition and modified Cholesky decomposition. We introduce several models for covariance matrices under different assumptions about the mean structure. We consider ‘dependence’ matrices, which tend to have many parameters, as constant across groups and/or parsimoniously modeled via a regression formulation. For ‘variances’, we consider both unrestricted across groups and more parsimoniously modeled via log-linear models. In all these models, we explore the propriety of the posterior when improper priors are used on the mean and ‘variance’ parameters (and in some cases, on components of the ‘dependence’ matrices). The models examined include several common Bayesian regression models, whose propriety has not been previously explored, as special cases. We propose a simple approach to weaken the assumption of constant dependence matrices in an automated fashion and describe how to compute Bayes factors to test the hypothesis of constant ‘dependence’ across groups. The models are applied to data from two longitudinal clinical studies.     

Qualitative and numerical analysis of a class of prey-predator models

We consider a problem of the dynamics of prey-predator populations suggested by the content of a letter of the biologist Umberto D'Ancona to Vito Volterra. The main feature of the problem is the special type of competition between predators of the same species as well as of different species. Two classes of cases are investigated: a first class in which the behaviour of the predator is blind and the second one in which the behaviour is intelligent. A qualitative analysis of the dynamical systems under consideration is followed by a numerical analysis of the most significant cases.     

Moving zeros among matrices

We investigate the zero-patterns that can be created by unitary similarity in a given matrix, and the zero-patterns that can be created by simultaneous unitary similarity in a given sequence of matrices. The latter framework allows a “simultaneous Hessenberg” formulation of Pati’s tridiagonal result for 4 × 4 matrices. This formulation appears to be a strengthening of Pati’s theorem. Our work depends at several points on the simplified proof of Pati’s result by Davidson and Djokovi?. The Hessenberg approach allows us to work with ordinary similarity and suggests an extension from the complex to arbitrary algebraically closed fields. This extension is achieved and related results for 5 × 5 and larger matrices are formulated and proved.     

Enhancements to the von Neumann trace inequality

Upper trace bounds for the product of two n × n complex matrices are presented. The real component of the trace inequality is tighter than von Neumann’s inequality, and the imaginary component is new.     

A dichotomy for the convex spaces of probability measures

We show that every nonempty compact and convex space M of probability Radon measures either contains a measure which has ‘small’ local character in M or else M contains a measure of ‘large’ Maharam type. Such a dichotomy is related to several results on Radon measures on compact spaces and to some properties of Banach spaces of continuous functions.     

On Klein’s icosahedral solution of the quintic

We present an exposition of the icosahedral solution of the quintic equation first described in Klein’s classic work ‘Lectures on the icosahedron and the solution of equations of the fifth degree’. Although we are heavily influenced by Klein we follow a slightly different approach which enables us to arrive at the solution more directly.     

Prefilter spaces and a precompletion of preuniform convergence spaces related to some well-known completions

In non-symmetric Convenient Topology the notion of pre-Cauchy filter is introduced and the construction of a precompletion of a preuniform convergence space is given from which Wyler's completion of a separated uniform limit space [O. Wyler, Ein Komplettierungsfunktor für uniforme Limesräume, Math. Nachr. 46 (1970) 1-12] as well as Weil's Hausdorff completion of a separated uniform space [A. Weil, Sur les Espaces à Structures Uniformes et sur la Topologie Générale, Hermann, Paris, 1937] can be derived (up to isomorphism). By the way, the construct PFil of prefilter spaces, i.e. of those preuniform convergence space which are ‘generated’ by their pre-Cauchy filters, is a strong topological universe filling in a gap in the theory of preuniform convergence spaces.     

Efficient numerical solution of the generalized Dirichlet-Neumann map for linear elliptic PDEs in regular polygon domains

A new and novel approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was recently introduced. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet-Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. For this computation, a collocation-type numerical method has been recently developed. Here, we study the collocation’s coefficient matrix properties. We prove that, for the Laplace’s equation on regular polygon domains with the same type of boundary conditions on each side, the collocation matrix is block circulant, independently of the choice of basis functions. This leads to the deployment of the FFT for the solution of the associated collocation linear system, yielding significant computational savings. Numerical experiments are included to demonstrate the efficiency of the whole computation.     

On extremum properties of orthogonal quotients matrices

In this paper we explore the extremum properties of orthogonal quotients matrices. The orthogonal quotients equality that we prove expresses the Frobenius norm of a difference between two matrices as a difference between the norms of two matrices. This turns the Eckart-Young minimum norm problem into an equivalent maximum norm problem. The symmetric version of this equality involves traces of matrices, and adds new insight into Ky Fan’s extremum problems. A comparison of the two cases reveals a remarkable similarity between the Eckart-Young theorem and Ky Fan’s maximum principle. Returning to orthogonal quotients matrices we derive “rectangular” extensions of Ky Fan’s extremum principles, which consider maximizing (or minimizing) sums of powers of singular values.     

An alternative approach to unitoidness

In a recent paper [C.R. Johnson, S. Furtado, A generalization of Sylvester’s law of inertia, Linear Algebra Appl. 338 (2001) 287-290], Sylvester’s law of inertia is generalized to any matrix that is ∗-congruent to a diagonal matrix. Such a matrix is called unitoid. In the present paper, an alternative approach to the subject of unitoidness is offered. Specifically, Sylvester’s law of inertia states that a Hermitian n × n matrix of rank r with inertia (p, q, n − r) is ∗-congruent to the direct sum

eⁱ⁰I_p⊕e^iπI_q⊕0I_n-r.     

A group based solution strategy for multi-physics simulations in parallel

Multi-physics simulation often requires the solution of a suite of interacting physical phenomena, the nature of which may vary both spatially and in time. For example, in a casting simulation there is thermo-mechanical behaviour in the structural mould, whilst in the cast, as the metal cools and solidifies, the buoyancy induced flow ceases and stresses begin to develop. When using a single code to simulate such problems it is conventional to solve each ‘physics’ component over the whole single mesh, using definitions of material properties or source terms to ensure that a solved variable remains zero in the region in which the associated physical phenomenon is not active. Although this method is secure, in that it enables any and all the ‘active’ physics to be captured across the whole domain, it is computationally inefficient in both scalar and parallel. An alternative, known as the ‘group’ solver approach, involves more formal domain decomposition whereby specific combinations of physics are solved for on prescribed sub-domains. The ‘group’ solution method has been implemented in a three-dimensional finite volume, unstructured mesh multi-physics code, which is parallelised, employing a multi-phase mesh partitioning capability which attempts to optimise the load balance across the target parallel HPC system. The potential benefits of the ‘group’ solution strategy are evaluated on a class of multi-physics problems involving thermo-fluid–structural interaction on both a single and multi-processor systems. In summary, the ‘group’ solver is a third faster on a single processor than the single domain strategy and preserves its scalability on a parallel cluster system.     

Block elimination with one refinement solves bordered linear systems accurately

Linear systems with a fairly well-conditioned matrixM of the form , for which a black box solver forA is available, can be accurately solved by the standard process of Block Elimination, followed by just one step of Iterative Refinement, no matter how singularA may be — provided the black box has a property that is possessed by LU- and QR-based solvers with very high probability. The resulting Algorithm BE + 1 is simpler and slightly faster than T.F. Chan's Deflation Method, and just as accurate. We analyse the case where the black box is a solver not forA but for a matrix close toA. This is of interest for numerical continuation methods.Dedicated to the memory of J. H. Wilkinson     

Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem

We introduce a family of algebras which are multiplicative analogues of preprojective algebras, and their deformations, as introduced by M.P. Holland and the first author. We show that these algebras provide a natural setting for the ‘middle convolution’ operation introduced by N.M. Katz in his book ‘Rigid local systems’, and put in an algebraic setting by M. Dettweiler and S. Reiter, and H. Völklein. We prove a homological formula relating the dimensions of Hom and Ext spaces, study varieties of representations of multiplicative preprojective algebras, and use these results to study simple representations. We apply this work to the Deligne-Simpson problem, obtaining a sufficient (and conjecturally necessary) condition for the existence of an irreducible solution to the equation A₁A₂…A_k=1 with the A_i in prescribed conjugacy classes in GL_n(C).     

Almost disjoint families of connected sets

We investigate the question which (separable metrizable) spaces have a ‘large’ almost disjoint family of connected (and locally connected) sets. Every compact space of dimension at least 2 as well as all compact spaces containing an ‘uncountable star’ have such a family. Our results show that the situation for 1-dimensional compacta is unclear.     

A multivariate version of the Benjamini-Hochberg method

We propose a multivariate method for combining results from independent studies about the same ‘large scale’ multiple testing problem. The method works asymptotically in the number of hypotheses and consists of applying the Benjamini-Hochberg procedure to the p-values of each study separately by determining the ‘individual false discovery rates’ which maximize power subject to a restriction on the (global) false discovery rate. We show how to obtain solutions to the associated optimization problem, provide both theoretical and numerical examples, and compare the method with univariate ones.     

Equations ax = c and xb = d in rings and rings with involution with applications to Hilbert space operators

This paper reviews the equations ax = c and xb = d from a new perspective by studying them in the setting of associative rings with or without involution. Results for rectangular matrices and operators between different Banach and Hilbert spaces are obtained by embedding the ‘rectangles’ into rings of square matrices or rings of operators acting on the same space. Necessary and sufficient conditions using generalized inverses are given for the existence of the hermitian, skew-hermitian, reflexive, antireflexive, positive and real-positive solutions, and the general solutions are described in terms of the original elements or operators. New results are obtained, and many results existing in the literature are recovered and corrected.