Identifying component modules

A computer-based system for modelling component dependencies and identifying component modules is presented. A variation of the Dependency Structure Matrix (DSM) representation was used to model component dependencies. The system utilises a two-stage approach towards facilitating the identification of a hierarchical modular structure. The first stage calculates a value for a clustering criterion that may be used to group component dependencies together. A Genetic Algorithm is described to optimise the order of the components within the DSM with the focus of minimising the value of the clustering criterion to identify the most significant component groupings (modules) within the product structure. The second stage utilises a 'Module Strength Indicator' (MSI) function to determine a value representative of the degree of modularity of the component groupings. The application of this function to the DSM produces a 'Module Structure Matrix' (MSM) depicting the relative modularity of available component groupings within it. The approach enabled the identification of hierarchical modularity in the product structure without the requirement for any additional domain specific knowledge within the system. The system supports design by providing mechanisms to explicitly represent and utilise component and dependency knowledge to facilitate the nontrivial task of determining near-optimal component modules and representing product modularity

Phasefield theory for fractional diffusion-reaction equations and applications

This paper is concerned with diffusion-reaction equations where the classical diffusion term, such as the Laplacian operator, is replaced with a singular integral term, such as the fractional Laplacian operator. As far as the reaction term is concerned, we consider bistable non-linearities. After properly rescaling (in time and space) these integro-differential evolution equations, we show that the limits of their solutions as the scaling parameter goes to zero exhibit interfaces moving by anisotropic mean curvature. The singularity and the unbounded support of the potential at stake are both the novelty and the challenging difficulty of this work.Comment: 41 page

Chromospheric explosions

Three issues relative to chromospheric explosions were debated. (1) Resolved: The blue-shifted components of x-ray spectral lines are signatures of chromospheric evaporation. It was concluded that the plasma rising with the corona is indeed the primary source of thermal plasma observed in the corona during flares. (2) Resolved: The excess line broading of UV and X-ray lines is accounted for by a convective velocity distribution in evaporation. It is concluded that the hypothesis that convective evaporation produces the observed X-ray line widths in flares is no more than a hypothesis. It is not supported by any self-consistent physical theory. (3) Resolved: Most chromospheric heating is driven by electron beams. Although it is possible to cast doubt on many lines of evidence for electron beams in the chromosphere, a balanced view that debaters on both sides of the question might agree to is that electron beams probably heat the low corona and upper chromosphere, but their direct impact on evaporating the chromosphere is energetically unimportant when compared to conduction. This represents a major departure from the thick-target flare models that were popular before the Workshop

Power sums and Homfly skein theory

The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements, whose symmetric functions are central in H_n. In [Skein theory and the Murphy operators, J. Knot Theory Ramif. 11 (2002), 475-492] I defined geometrically a homomorphism from the Homfly skein C of the annulus to the centre of each algebra H_n, and found an element P_m in C, independent of n, whose image, up to an explicit linear combination with the identity of H_n, is the m-th power sum of the Murphy operators. The aim of this paper is to give simple geometric representatives for the elements P_m, and to discuss their role in a similar construction for central elements of an extended family of algebras H_{n,p}.Comment: Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper15.abs.htm

The Theory Behind TheoryMine

Abstract. We describe the technology behind the TheoryMine novelty gift company, which sells the rights to name novel mathematical theorems. A tower of four computer systems is used to generate recursive theories, then to speculate conjectures in those theories and then to prove these conjectures. All stages of the process are entirely automatic. The process guarantees large numbers of sound, novel theorems of some intrinsic merit.

Part 3: James Clerk Maxwell\u27s Zoetrope

James Clerk Maxwell attempted to understand how three rings of ether would pass through each other. On October 6, 1868, he wrote to Kelvin: [Helmholtz\u27s] 3 rings do as 2 rings in his own paper that is those in front expand and go slower those behind contract and when small go faster and thread through the others. I drew 3 to make the motion more slow and visible, not that I have solved the case of 3 rings more than to get a rough notion about this case and to make the sum of the three areas [constant] I have made them fat when small and thin when big. [Maxwell\u27s lack of punctuation was not typical of his writing, which was generally excellent.] Maxwell drew the figures to be viewed on a zoetrope ( wheel of life ) of his own design. The Cavendish Museum of Cambridge University owns the device. This PDF contains a photograph of the Zoetrope created by James Clerk Maxwell reproduced from The Scientific Letters and Papers of James Clerk Maxwell (Harman, P. M. Book Review: The scientific letters and papers of James Clerk Maxwell. Volume II, 1862-1873/Cambridge U Press, 1995. The Observatory 116 (1996): 44.), and a photograph of Professor Gordon Squires and Daniel Silver in 1996, during a visit to the Cavendish Museum of Cambridge. At that time, Professor Squires was the museum curator. Photo of Professor Squires and Dr. Silver was taken by Susan Williams

Part 5: A Colorful Approach to Knot Theory (or: “How Happy I Could Be With Ether”)

Ralph Fox’s p-colorings offer the simplest yet effective invariants of knots. This exposition, based on the 2020 Lewis-Parker Lecture delivered by the author, introduces the method of knot coloring as well as some historical background