Extending paired multifunctions

As a rule, the classical Michael-type selection theorems for the existence of single-valued selections are analogues and, in certain respects, generalisations of ordinary extension theorems. In contrast to this, the theorems for the existence of multi-selections deal with natural generalisations of cover properties of topological spaces. This paper continues the study of the latter problem, and its main purpose is to furnish a mapping characterisation of a cover-extension property—the so-called Katětov spaces.     

Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces

Aequationes mathematicae - Quasimetric spaces have been an object of thorough investigation since Frink’s paper appeared in 1937 and various generalisations of the axioms of metric spaces are...     

Highest weight categories arising from Khovanov's diagram algebra II: Koszulity

This is the second of a series of four papers studying various generalisations of Khovanov's diagram algebra. In this paper we develop the general theory of Khovanov's diagrammatically defined “projective functors” in our setting. As an application, we give a direct proof of the fact that the quasi-hereditary covers of generalised Khovanov algebras are Koszul.     

Tilting equivalences: From hereditary algebras to symmetric groups

We study Koszul duality for finite dimensional hereditary algebras, and various generalisations to trivial extension algebras, to Schur algebras, to doubles of Schur bialgebras, and to deformations of doubles of Schur bialgebras. We describe applications to the modular representation theory of symmetric groups.     

Optimal compactification of a floorplan and its relation to other optimization problems-a dynamic programming approach

We examine an algorithm for the compactification of an arrangement of rectangles in the plane as it is used for floorplans in the automated design of electronic circuits (also called sizing of floorplans). We reformulate this problem as a multistage decision problem and show that the algorithm is in fact the optimal solution obtained by the backward induction procedure of dynamic programing. The model allows generalisations to non-geometrical applications in scheduling and reliability.     

Minimum Paths in Directed Graphs

This paper considers the problem of finding paths from a fixed node to all other nodes of a directed graph which minimise a function defined on the paths. Under certain assumptions a characterisation of optimal paths is derived. Two algorithms which are generalisations of standard shortest path methods are then given.     

Archimedean solids of genus two

The problem of classifying orientable vertex-transitive maps on a surface with genus two is considered. We construct and classify all simple orientable vertex-transitive maps, with face width at least 3 which can be viewed as generalisations of classical Archimedean solids. The proof is computer-aided. The developed method applies to higher genera as well.     

Some Generalisations of the Time Minimising Assignment Problem

The time minimising assignment problem is the problem of finding an assignment of n jobs to n facilities, one to each, which minimises the total time for completing all the jobs. The usual assumption made in these problems is that all the jobs are commenced simultaneously. In this paper two generalisations of this assumption are considered, and algorithms are presented to solve these general problems. Numerical examples are worked out illustrating the algorithms.     

Operator Theoretic Methods for the Eigenvalue Counting Function in Spectral Gaps

Using the notion of spectral flow, we suggest a simple approach to various asymptotic problems involving eigenvalues in the gaps of the essential spectrum of self-adjoint operators. Our approach employs some elements of the theory of the spectral shift function. Using this approach, we provide generalisations and streamlined proofs of two results in this area already existing in the literature. We also give a new proof of the generalised Birman–Schwinger principle. Submitted: March 10, 2009. Accepted: April 22, 2009.     

A Geometric Integration Approach to Nonsmooth,Nonconvex Optimisation

The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter optimisation problems. Bilevel optimisation of parameters is a standard setting in areas such as variational regularisation problems and supervised machine learning. We present efficient and robust derivative-free methods called randomised Itoh–Abe methods. These are generalisations of the Itoh–Abe discrete gradient method, a well-known scheme from geometric integration, which has previously only been considered in the smooth setting. We demonstrate that the method and its favourable energy dissipation properties are well defined in the nonsmooth setting. Furthermore, we prove that whenever the objective function is locally Lipschitz continuous, the iterates almost surely converge to a connected set of Clarke stationary points. We present an implementation of the methods, and apply it to various test problems. The numerical results indicate that the randomised Itoh–Abe methods can be superior to state-of-the-art derivative-free optimisation methods in solving nonsmooth problems while still remaining competitive in terms of efficiency.

     

Variations on a theme: introducing new representations of fraction into two classrooms

Abstract

The interplay between generalisations and particular instances—examples—is an essential feature of mathematics teaching and learning. In this paper, we bring together our experiences of personal and classroom mathematics activity, and demonstrate that examples do not always fulfil their intended purpose (to point to generalisations). A distinction is drawn between ‘empirical’ and ‘structural’ generalisation, and the role of generic examples is discussed as a means of supporting the second of these qualities of generalisation.     

Travelling wave and similarity solutions to nonlinear evolution type equations through inverse variational and symmetry methods

Shallow water equations are usually modelled by nonlinear KdV type equations of which various generalisations now exist. For example there are vector versions of the modified KdV equation and shallow water equations with nonlinear internal waves. We discuss the reduction and solutions of these and other large classes of such type of equations using inverse variational and symmetry methods.     

The Tilting Tensor Product Theorem and Decomposition Numbers for Symmetric Groups

We show how the tilting tensor product theorem for algebraic groups implies a reduction formula for decomposition numbers of the symmetric group. We use this to prove generalisations of various theorems of Erdmann and of James and Williams. Supported by Nuffield grant scheme NUF-NAL 02. Preliminary work on this paper was undertaken at the Isaac Newton Institute as part of the programme on Symmetric Functions and Macdonald Polynomials.     

Rectangles as sums of squares

In this paper we examine generalisations of the following problem posed by Laczkovich: Given an n×m rectangle with n and m integers, it can be written as a disjoint union of squares; what is the smallest number of squares that can be used? He also asked the corresponding higher dimensional analogue. For the two dimensional case Kenyon proved a tight logarithmic bound but left open the higher dimensional case. Using completely different methods we prove good upper and lower bounds for this case as well as some other variants.     

On generalisations of Stieltjes transform

In this paper two different generalisations of the Stieltjes transform have been given along with their inversion formulae. The results are given in the form of two theorems. These two theorems yield many known generalisations of the transform in question as particular cases on specializing the parameters involved therein.     

Exact arithmetic on the Stern–Brocot tree

In this paper we present the Stern–Brocot tree as a basis for performing exact arithmetic on rational numbers. There exists an elegant binary representation for positive rational numbers based on this tree [Graham et al., Concrete Mathematics, 1994]. We will study this representation by investigating various algorithms to perform exact rational arithmetic using an adaptation of the homographic and the quadratic algorithms that were first proposed by Gosper for computing with continued fractions. We will show generalisations of homographic and quadratic algorithms to multilinear forms in n variables. Finally, we show an application of the algorithms for evaluating polynomials.     

Rate control for communication networks: shadow prices,proportional fairness and stability

This paper analyses the stability and fairness of two classes of rate control algorithm for communication networks. The algorithms provide natural generalisations to large-scale networks of simple additive increase/multiplicative decrease schemes, and are shown to be stable about a system optimum characterised by a proportional fairness criterion. Stability is established by showing that, with an appropriate formulation of the overall optimisation problem, the network's implicit objective function provides a Lyapunov function for the dynamical system defined by the rate control algorithm. The network's optimisation problem may be cast in primal or dual form: this leads naturally to two classes of algorithm, which may be interpreted in terms of either congestion indication feedback signals or explicit rates based on shadow prices. Both classes of algorithm may be generalised to include routing control, and provide natural implementations of proportionally fair pricing.     

Subexponential parameterized algorithms for degree-constrained subgraph problems on planar graphs

We present subexponential parameterized algorithms on planar graphs for a family of problems of the following shape: given a graph, find a connected (induced) subgraph with bounded maximum degree and with maximum number of edges (or vertices). These problems are natural generalisations of the Longest Path problem. Our approach uses bidimensionality theory combined with novel dynamic programming techniques over branch decompositions of the input graph. These techniques can be applied to a more general family of problems that deal with finding connected subgraphs under certain degree constraints.     

Sharper bounds on four lattice constants

Designs, Codes and Cryptography - The Korkine–Zolotareff (KZ) reduction and its generalisations, are widely used lattice reduction strategies in communications and cryptography. The KZ...