Symmetrized curve-straightening

The ‘traditional’ curve-straightening flow is based on one of the standard Sobolev inner products and it is known to break certain symmetries of reflection. The purpose of this paper is to show that there are alternative Riemannian structures on the space of curves that yield flows that preserve symmetries. This feature comes at a price. In one symmetrizing metric the gradient vector fields are considerably more demanding to compute. In another symmetrizing metric smoothness is lost. This investigation will also explain the phenomena of ‘spinning’ as observed in several examples in the traditional flow. Three classes of alternative Riemannian structures are examined. The first class includes the traditional metric as a special case and is shown to never preserve both rotation symmetries and symmetries of reflection. The second class consists of a single metric corresponding to one of the standard Sobolev metrics, and is shown to preserve both types of symmetries. The third class also includes the traditional metric but it is shown that there is a unique different metric in this class, which preserves both types of symmetries. This particular metric generally yields smooth vector fields, which when evaluated at a smooth function do not give a smooth element of the corresponding tangent space. The third class is nevertheless ‘preferred’ since it has the distinction that it ‘respects’ the projection induced by the derivative operator onto the tangent bundle of the space of derivatives. The paper concludes with a number of graphical illustrations that show preserved symmetry and removal of spinning.     

Prioritization of medical equipment for maintenance decisions

Clinical engineering departments in hospitals are responsible for establishing and regulating a Medical Equipment Management Program to ensure that medical devices are safe and reliable. In order to mitigate functional failures, significant and critical devices should be identified and prioritized. In this paper, we present a multi-criteria decision-making model to prioritize medical devices according to their criticality. Devices with lower criticality scores can be assigned a lower priority in a maintenance management program. However, those with higher scores should be investigated in detail to find the reasons for their higher criticality, and appropriate actions, such as ‘preventive maintenance’, ‘user training’, ‘redesigning the device’, etc, should be taken. In this paper,we also describe how individual score values obtained for each criterion can be used to establish guidelines for appropriate maintenance strategies for different classes of devices. The information of 26 different medical devices is extracted from a hospital's maintenance management system to illustrate an application of the proposed model.     

Higher index theory for certain expanders and Gromov monster groups,II

In this paper, the second of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs has girth tending to infinity, then the maximal coarse Baum–Connes assembly map is an isomorphism for the associated metric space X. As discussed in the first paper in this series, this has applications to the Baum–Connes conjecture for ‘Gromov monster’ groups.We also introduce a new property, ‘geometric property (T)’. For the metric space associated to a sequence of graphs, this property is an obstruction to the maximal coarse assembly map being an isomorphism. This enables us to distinguish between expanders with girth tending to infinity, and, for example, those constructed from property (T) groups.     

Metrics for bullwhip effect analysis

A bullwhip measurement system based on a two-criterion assessment—‘internal process efficiency’ and ‘customer service level’—is developed in this paper. The framework is designed to assess both individual (single member) and systemic (whole supply chain) performances. Data collection and calculation methods, update and monitoring mechanisms, as well as related procedures for each metric used, are detailed. A comparative analysis with a recent work by Barlas and Gunduz is performed, showing that the adoption of the proposed performance measurement system can help academics and practitioners to better understand, study and avoid the bullwhip effect. Such analysis also provides evidence on the relevance of considering when analysing the bullwhip effect in supply chains, the ‘customer importance’ aspect that is often forgotten in the published literature.     

Common fixed point theorems in intuitionistic fuzzy metric spaces

The purpose of this paper, using the idea of intuitionistic fuzzy set due to Atanassov [2], we define the notion of intuitionistic fuzzy metric spaces (see, [1]) due to Kramosil and Michalek [17] and Jungck’s common fixed point theorem ([11]) is generalized to intuitionistic fuzzy metric spaces. Further, we first formulate the definition of weakly commuting and R-weakly commuting mappings in intuitionistic fuzzy metric spaces and prove the intuitionistic fuzzy version of Pant’s theorem ([21]).     

Network Criticality with Multiple Work-Patterns

When multiple work-patterns (also called calendars) are present in a network, it is quite common that there is no critical path spanning the network. This paper explains why this is so. To address the problem, the normal definition of criticality in project networks is split into two-delay criticality, and compress criticality.     

Contagion in cyber security attacks

Systems security is essential for the efficient operation of all organizations. Indeed, most large firms employ a designated ‘Chief Information Security Officer’ to coordinate the operational aspects of the organization’s information security. Part of this role is in planning investment responses to information security threats against the firm’s corporate network infrastructure. To this end, we develop and estimate a vector equation system of threats to 10 important IP services, using industry standard SANS data on threats to various components of a firm’s information system over the period January 2003 – February 2011. Our results reveal strong evidence of contagion between such attacks, with attacks on ssh and Secure Web Server indicating increased attack activity on other ports. Security managers who ignore such contagious inter-relationships may underestimate the underlying risk to their systems’ defence of security attributes, such as sensitivity and criticality, and thus delay appropriate information security investments.     

Krasnoselskii's iteration process in hyperbolic space

A well-known iteration scheme due to Krasnoselskii for approximation of fixed points of nonexpansive mappings in Banach spaces is extended to a wider class of spaces. This class includes convex metric spaces of ‘hyperbolic’ type, and the results apply to the study of holomorphic self-mappings of the unit ball in complex Hilbert space.     

Pixley-Roy spaces over subsets of the reals

E. van Douwen, and later M. Wage, asked whether the Pixley-Roy space over the irrationals is homeomorphic to the Pixley-Roy space over the reals. We show that the answer is ‘no’ in the following way: the reals satisfy the Hurewicz property, while the irrationals do not; we define ‘weakly Hurewicz’ in a natural way and show that this property distinguishes the corresponding Pixley-Roy spaces. Similar methods are used to distinguish other special subsets of metric spaces such as C″-sets, γ-sets, and those satisfying a certain game characterization.     

Regularity and Conditioning of Solution Mappings in Variational Analysis

Concepts of conditioning have long been important in numerical work on solving systems of equations, but in recent years attempts have been made to extend them to feasibility conditions, optimality conditions, complementarity conditions and variational inequalities, all of which can be posed as solving ‘generalized equations’ for set-valued mappings. Here, the conditioning of such generalized equations is systematically organized around four key notions: metric regularity, subregularity, strong regularity and strong subregularity. Various properties and characterizations already known for metric regularity itself are extended to strong regularity and strong subregularity, but metric subregularity, although widely considered, is shown to be too fragile to support stability results such as a radius of good behavior modeled on the Eckart–Young theorem.     

Optimal approximations with Rough Sets and similarities in measure spaces

When arbitrary sets are approximated by more structured sets, it may not be possible to obtain an exact approximation that is equivalent to a given set. Presented here, is a new proposal for a ‘metric’ approach to Rough Sets. We assume some finite measure space is defined on a given universe, and then use it to define various similarity indexes. A set of axioms and the concept of consistency for similarity indexes are also proposed. The core of the paper is a definition of the ‘optimal’ or ‘best’ approximation with respect to any particular similarity index, and an algorithm to find this optimal approximation by using the Marczewski–Steinhaus Index. This algorithm is also shown to hold for a class of similarity indexes that are consistent with the Marczewski–Steinhaus Index.     

Hirschman versus Herfindahl: Some topological properties for the use of concentration indexes

The properties of the Hirschman index of concentration are examined, and shown to be superior to those of the closely related, but better known, Herfindahl index. Specifically, Hirschman's index satisfies the requirements of a norm (and hence of a metric), thus allowing the measurement of ‘distances’ for static and dynamic comparisons.     

A pentatonic classification of extreme events

In this paper we present a classification of the extreme events – very small and very large outcomes – of positive-valued random variables. The classification distinguishes five different categories of randomness, ranging from the very ‘mild’ to the very ‘wild’. In analogy with the common five-tone musical scale we term the classification ‘pentatonic’. The classification is based on the analysis of the inherent Gibbsian ‘forces’ and ‘temperatures’ existing on the logarithmic scale of the random variables under consideration, and provides a statistical-physics insight regarding the nature of these random variables. The practical application of the pentatonic classification is remarkably straightforward, it can be performed by non-experts, and it is demonstrated via an array of examples.     

On the sharp singular limit for slightly compressible fluids

We consider the equations of motion to slightly compressible fluids and we prove that solutions converge, in the strong norm, to the solution of the equations of motion of incompressible fluids, as the Mach number goes to zero. From a physical point of view this means the following. Assume that we are dealing with a well-specified fluid, so slightly compressible that we assume it to be incompressible. Our result means that the distance between the (continuous) trajectories of the real and of the idealized solution is ‘small’ with respect to the natural metric, i.e. the metric that endows the data space.     

Functionals strictly subjected to convergent series and search for singularities of mappings

This paper is a continuation of the author’s previous works where the concepts of search functionals were introduced and used to develop methods of cascade search for singularities of mappings between metric spaces. Such singularities of mappings were considered as coincidences, common preimages, common fixed points and common roots of finite sets of mappings. In this paper, a new class of functionals strictly subjected to convergent series is introduced. With the help of such functionals the problems of local search for the above-mentioned singularities of mappings between metric spaces are investigated. Some new iteration schemes are suggested. Generalizations of several known results are obtained.     

RESULTS AND CONJECTURES ON COVERING METRIC SPACES

Abstract

A semi-quantitative theory of covering metric spaces is studied; the theory, involving the concept of a ‘fine cover’, has proved useful in complex analysis. We obtain general results on fine covers, and put forward some conjectures. Most of these remain unanswered (and indeed seem most difficult to answer) for subsets of the real line.     

Making mathematics and science integration happen: key aspects of practice

The integration of mathematics and science teaching and learning facilitates student learning, engagement, motivation, problem-solving, criticality and real-life application. However, the actual implementation of an integrative approach to the teaching and learning of both subjects at classroom level, with in-service teachers working collaboratively, at second-level education, is under-researched due to the complexities of school-based research. This study reports on a year-long case study on the implementation of an integrated unit of learning on distance, speed and time, within three second-level schools in Ireland. This study employed a qualitative approach and examined the key aspects of practice that impact on the integration of mathematics and science teaching and learning. We argue that teacher perspective, teacher knowledge of the ‘other subject’ and of technological pedagogical content knowledge (TPACK), and teacher collaboration and support all impact on the implementation of an integrative approach to mathematics and science education.     

Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows—provided the sequence is tight—from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Λ-coalescents. We show that the Λ-coalescent defines an infinite (random) metric measure space if and only if the so-called “dust-free”-property holds.     

Fuzzy perturbation analysis,Part 2: Undirectional perturbation

In this paper, the stability of the solutions of fuzzy relation equations is discussed by the method of undirectional fuzzy perturbation. First, a general definition of ‘fuzzy perturbation’ is given. From this, the concept of fuzzy perturbation equation is introduced, and thus stability of the solutions is defined. Next, the degree of stability of the solutions, a kind of metric of the stability, is advanced. Three ordered quotient sets are induced by use of the degree of stability. These quotient sets have a well chained characteristic.     

Improving credit scoring by differentiating defaulter behaviour

We present a methodology for improving credit scoring models by distinguishing two forms of rational behaviour of loan defaulters. It is common knowledge among practitioners that there are two types of defaulters, those who do not pay because of cash flow problems (‘Can’t Pay’), and those that do not pay because of lack of willingness to pay (‘Won’t Pay’). This work proposes to differentiate them using a game theory model that describes their behaviour. This separation of behaviours is represented by a set of constraints that form part of a semi-supervised constrained clustering algorithm, constructing a new target variable summarizing relevant future information. Within this approach the results of several supervised models are benchmarked, in which the models deliver the probability of belonging to one of these three new classes (good payers, ‘Can’t Pays’, and ‘Won’t Pays’). The process improves classification accuracy significantly, and delivers strong insights regarding the behaviour of defaulters.