Hypernumbers—II. further concepts and computational applications

In a previous paper [1] on hypernumbers it was suggested that there were significant ramifications to be explored and exemplified. In particular, the following important areas were only mentioned in passing, but are now more fully examined: the hypernumber use of idempotents, zero divisors, and nilpotents; the concept of bimatrices; and a survey, necessarily brief, of hypernumbers beyond ε and i, that is, beyond the proper square and fourth roots of unity respectively. The term hypernumber was introduced into mathematics by the present author in 1966 [2] to denote domains of number including or beyond the arithmetic of ordinary numbers, complex numbers, quaternions, matrices, or octaves (Cayley-Graves numbers); and the term has since been used by Kline [3] and by Spencer and Moon [4], but unnecessarily restricted to forms of i and ε, and too often only to matrix forms.     

Analysis of the development concepts and methods of visual data representation in computational physics

Main steps in the development of scientific visualization as a branch of science are discussed. The evolution and prospects of the development of concepts, methods, and approaches of visual representation of numerical results obtained in computational physics (mainly, in computational fluid dynamics) are discussed.     

Membrane fission: A computational complexity perspective

Membrane fission is a process by which a biological membrane is split into two new ones in the manner that the content of the initial membrane is separated and distributed between the new membranes. Inspired by this biological phenomenon, membrane separation rules were considered in membrane computing. In this work, we investigate cell‐like P systems with symport/antiport rules and membrane separation rules from a computational complexity perspective. Specifically, we establish a limit on the efficiency of such P systems which use communication rules of length at most two, and we prove the computational efficiency of this kind of models when using communication rules of length at most three. Hence, a sharp borderline between tractability and NP –hardness is provided in terms of the length of communication rules. © 2015 Wiley Periodicals, Inc. Complexity 21: 321–334, 2016     

Acoustic gravity waves: a computational approach

This paper discusses numerical solutions of a hyperbolic initial boundary value problem that arises from acoustic wave propagation in the atmosphere. Field equations are derived from the atmospheric fluid flow governed by the Euler equations. The resulting original problem is nonlinear. A first-order linearized version of the problem is used for computational purposes. The main difficulty in the problem as with any open boundary problem is in obtaining stable boundary conditions. Approximate boundary conditions are derived and shown to be stable. Numerical results are presented to verify the effectiveness of these boundary conditions.     

Multi-scale computational homogenization: Trends and challenges

In the past decades, considerable progress had been made in bridging the mechanics of materials to other disciplines, e.g. downscaling to the field of materials science or upscaling to the field of structural engineering. Within this wide context, this paper reviews the state-of-the-art of a particular, yet powerful, method, i.e. computational homogenization. The paper discusses the main trends since the early developments of this approach up to the ongoing contributions and upcoming challenges in the field.     

Evaluating computational efficiency: A stochastic approach

This paper introduces the use of stochastic models for the evaluation of relative computational efficiency of algorithms. Such an approach is used for the comparison of computational efficiency of three algorithms for quadratic programming.     

Moral concepts: From thickness to response-dependence

The paper examines three tenets of Dancy's meta-ethics, finds them incompatible, and proposes a response-dependentist (or response-dispositional) solution. The first tenet is the central importance of thick concepts and properties. The second is that such concepts essentially involve response(s) of observers, which Dancy interprets in a way that fits the pattern of context-dependent resultance: thick concepts are well suited for the particularist grounding of moral theory. However, and this is the third tenet, in his earlier paper (1986) Dancy forcefully argues against response-dispositional accounts of moral concepts and properties. The present paper argues that an anti-dispositional view is incompatible with the first two points concerning thick concepts. If thick concepts and properties are paramount and ubiquitous in moral thought and reality, and if they are essentially tied to human responses, then anti-dispositionalism is false. Dancy himself avoids obvious contradiction by characterizing thick items (concepts) differently from the usual characterization of response-dependent items. Actions that satisfy thick concepts do so in virtue of meriting a determinate response. The (non-reductionist) response-dependentist usually puts it slightly differently: such actions satisfy a given moral concepts in virtue of eliciting a merited response. I have argued at length that this tenuous difference in formulation is too weak to support a relevant difference in rebus. If the argument is right, Dancy is implicitly committed to a kind of response-dependentism. Finally, the particularist should embrace thick concepts and properties, and reject anti-dispositionalism. However, this would bring back the analogy with color and other secondary qualities. Since there are ceteris paribus laws governing such properties, the analogy suggests that moral properties might also be best accounted for by a ceteris paribus, or hedged account, a compromise between traditional generalism and the particularism of Dancy's variety.     

Two-dimensional integral-algebraic systems: Analysis and computational methods

In this article we formulate sufficient conditions for the existence and uniqueness of solution to systems of two-dimensional Volterra integral equations, in which the coefficient of the main term is a singular matrix. A numerical method is introduced which can be applied to approximate the solution when the given conditions are satisfied. The convergence of this method is proved and illustrated by numerical examples.     

Penalised spline support vector classifiers: computational issues

We study computational issues for support vector classification with penalised spline kernels. We show that, compared with traditional kernels, computational times can be drastically reduced in large problems making such problems feasible for sample sizes as large as ~10⁶. The optimisation technology known as interior point methods plays a central role. Penalised spline kernels are also shown to allow simple incorporation of low-dimensional structure such as additivity. This can aid both interpretability and performance.     

Stochastic lot-sizing with backlogging: computational complexity analysis

For a multi-stage stochastic programming problem, one approach is to explore a scenario tree based formulation for the problem and solve the formulation efficiently. There has been significant research progress on how to generate scenario trees in the literature. However, there is limited work on analyzing the computational complexity of the scenario-tree based formulation that considers the number of tree nodes as the input size. In this paper, we use stochastic lot-sizing problems as examples to study the computational complexity issues for the scenario-tree based formulations. We develop production path properties and a general dynamic programming framework based on these properties. The dynamic programming framework allows us to show that the optimal value function is piecewise linear and continuous, which enables us to develop polynomial time algorithms for several different problems, including those with backlogging and varying capacities under certain conditions. As special cases, we develop polynomial time algorithms that run in O(n²){\mathcal{O}(n^2)} and O(n²T log n){\mathcal{O}(n^2T\,{\rm log}\,n)} times, respectively for stochastic uncapacitated and constant capacitated lot-sizing problems with backlogging, regardless of the scenario tree structure.     

Implementing PLS for distance-based regression: computational issues

Distance-based regression allows for a neat implementation of the partial least squares recurrence. In this paper, we address practical issues arising when dealing with moderately large datasets (n ~ 10⁴) such as those typical of automobile insurance premium calculations.     

LCNO Sturm-Liouville problems: computational difficulties and examples

Summary. In this paper we give a new proof of a theorem of Bailey, Everitt and Zettl on the convergence of truncated approximations to limit circle (LC) Sturm-Liouville problems, both non-oscillatory (LCNO) and oscillatory (LCO). The proof gives an error bound not previously available. We prove a theorem on the conditioning of LCNO problems with respect to non-Friedrichs boundary conditions. We present numerical experiments which illustrate how the theorem successfully predicts the conditioning of LCNO problems. Our work may also explain the performance of the code SLEIGN2 recently reported by Bailey et al. [1] on a number of problems. Received January 22, 1993 / Revised version received April 22, 1994     

The computational beauty of flocking: boids revisited

Artificial-life research was founded in the mid-1980s. It promotes the idea of the bottom-up research approach, where only the basic units of a situation and their local interaction are modelled, and then the system is left to evolve. However, the notable progress of the processing power of personal computers, evident in the last two decades, has had little influence on the ways the basic units (artificial animals or animats) are constructed. This impacts largely on the applicability of the methods in other research fields. Our field of choice is the modelling of bird flocks. This area was at its peak in the late 1980s when Craig W. Reynolds presented the first and most influential model – the boids. In spite of his many following works no formal definition has ever been presented. This might be the reason why a second generation of flocking models is still awaited. In this article we make a step forward, all in view of allowing for the development of the second-generation models. We present an artificial animal construction framework that has been obtained as a generalization of the existing bird flocking models, but is not limited to them. The article thus presents a formal definition of the framework and gives an example of its use. In the latter the framework is employed to present a formalization of Reynolds's boids.     

Threshold concepts in finance: conceptualizing the curriculum

Graduates with well-developed capabilities in finance are invaluable to our society and in increasing demand. Universities face the challenge of designing finance programmes to develop these capabilities and the essential knowledge that underpins them. Our research responds to this challenge by identifying threshold concepts that are central to the mastery of finance and by exploring their potential for informing curriculum design and pedagogical practices to improve student outcomes. In this paper, we report the results of an online survey of finance academics at multiple institutions in Australia, Canada, New Zealand, South Africa and the United Kingdom. The outcomes of our research are recommendations for threshold concepts in finance endorsed by quantitative evidence, as well as a model of the finance curriculum incorporating finance, modelling and statistics threshold concepts. In addition, we draw conclusions about the application of threshold concept theory supported by both quantitative and qualitative evidence. Our methodology and findings have general relevance to the application of threshold concept theory as a means to investigate and inform curriculum design and delivery in higher education.     

Analytic psychology: Source concepts for artificial intelligence

As intelligence technology advances from problem specific to adaptive designs it is conceivable that design concepts will be found in analytic psychology. This psychology features a mechanistic and structured model of human adaptivity from which such concepts may derive. This paper develops this position and illustrates it with three examples: the von Neumann machine, digital simulation of dreams, and digital simulation of the mechanism of the creative process. The simulating algorithms of the last two examples are implemented in Fortran.