Navigating Between Packings of Graphic Sequences

Let $\pi_1=(d_1^{(1)}, \ldots,d_n^{(1)})$ and $\pi_2=(d_1^{(2)},\ldots,d_n^{(2)})$ be graphic sequences. We say they \emph{pack} if there exist edge-disjoint realizations $G_1$ and $G_2$ of $\pi_1$ and $\pi_2$, respectively, on vertex set $\{v_1,\dots,v_n\}$ such that for $j\in\{1,2\}$, $d_{G_j}(v_i)=d_i^{(j)}$ for all $i\in\{1,\ldots,n\}$. In this case, we say that $(G_1,G_2)$ is a $(\pi_1,\pi_2)$-\textit{packing}. A clear necessary condition for graphic sequences $\pi_1$ and $\pi_2$ to pack is that $\pi_1+\pi_2$, their componentwise sum, is also graphic. It is known, however, that this condition is not sufficient, and furthermore that the general problem of determining if two sequences pack is $NP$- complete. S.~Kundu proved in 1973 that if $\pi_2$ is almost regular, that is each element is from $\{k-1, k\}$, then $\pi_1$ and $\pi_2$ pack if and only if $\pi_1+\pi_2$ is graphic. In this paper we will consider graphic sequences $\pi$ with the property that $\pi+\mathbf{1}$ is graphic. By Kundu's theorem, the sequences $\pi$ and $\mathbf{1}$ pack, and there exist edge-disjoint realizations $G$ and $\mathcal{I}$, where $\mathcal{I}$ is a 1-factor. We call such a $(\pi,\mathbf{1})$ packing a {\em Kundu realization}. Assume that $\pi$ is a graphic sequence, in which each term is at most $n/24$, that packs with $\mathbf{1}$. This paper contains two results. On one hand, any two Kundu realizations of the degree sequence $\pi+\mathbf{1}$ can be transformed into each other through a sequence of other Kundu realizations by swap operations. On the other hand, the same conditions ensure that any particular 1-factor can be part of a Kundu realization of $\pi+\mathbf{1}$

Improving Chemical Plant Safety Training Using Virtual Reality

The chemical engineering industry often requires people to work in hazardous environments and to operate complicated equipment which often limits the type of training that be carried out on site. The daily job of chemical plant operators is becoming more demanding due to the increasing plant complexity together with increasing requirements on plant safety, production capacity, product quality and cost effectiveness. The importance of designing systems and environments that are as safe as possible to educate and train personnel is vital for the chemical process industries. Virtual reality offers the potential to expose personnel to hazardous situations in a safe, highly visual and interactive manner. Virtual reality has been proposed as a technological breakthrough that holds the power to facilitate learning. The ability to visualise complex and dynamic systems involving personnel, equipment and layouts during any real operation is a potential advantage of such an approach. Virtual reality and multimedia training is commonly used in many industries, aiding understanding and memory retention and creating a more interactive learning experience. Four desktop virtual reality training environments were developed for this research which highlighted issues related to chemical process dynamic simulation and plant safety. The pump training system is a virtual reality environment, which was built using the SAFE-VR virtual engine, to train personnel to operate two centrifugal pumps. The virtual hazard spotting exercise focuses on improving the users' safety awareness of electrical and occupational hygiene hazards. The virtual boiler plant is a complicated and high detailed virtual training environment, which is characterised by its flexibility and by a real time dynamic simulation of the steam generation chemical process. The virtual flooding and gas absorption experiment was based on an undergraduate laboratory experiment for the Chemical Engineering degree course at the University of Nottingham, focusing primarily on training and safety issues of students using the equipment. The dynamic features of the virtual absorption column simulation give high level of realism in the virtual environment

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Improving Chemical Plant Safety Training Using Virtual Reality

The chemical engineering industry often requires people to work in hazardous environments and to operate complicated equipment which often limits the type of training that be carried out on site. The daily job of chemical plant operators is becoming more demanding due to the increasing plant complexity together with increasing requirements on plant safety, production capacity, product quality and cost effectiveness. The importance of designing systems and environments that are as safe as possible to educate and train personnel is vital for the chemical process industries. Virtual reality offers the potential to expose personnel to hazardous situations in a safe, highly visual and interactive manner. Virtual reality has been proposed as a technological breakthrough that holds the power to facilitate learning. The ability to visualise complex and dynamic systems involving personnel, equipment and layouts during any real operation is a potential advantage of such an approach. Virtual reality and multimedia training is commonly used in many industries, aiding understanding and memory retention and creating a more interactive learning experience. Four desktop virtual reality training environments were developed for this research which highlighted issues related to chemical process dynamic simulation and plant safety. The pump training system is a virtual reality environment, which was built using the SAFE-VR virtual engine, to train personnel to operate two centrifugal pumps. The virtual hazard spotting exercise focuses on improving the users' safety awareness of electrical and occupational hygiene hazards. The virtual boiler plant is a complicated and high detailed virtual training environment, which is characterised by its flexibility and by a real time dynamic simulation of the steam generation chemical process. The virtual flooding and gas absorption experiment was based on an undergraduate laboratory experiment for the Chemical Engineering degree course at the University of Nottingham, focusing primarily on training and safety issues of students using the equipment. The dynamic features of the virtual absorption column simulation give high level of realism in the virtual environment

Towards a unified method to synthesising scenarios and solvers in combinatorial optimisation via graph-based approaches

Hyper-heuristics is a collection of search methods for selecting, combining and generating heuristics used to solve combinatorial optimisation problems. The primary objective of hyper-heuristics research is to develop more generally applicable search procedures that can be easily applied to a wide variety of problems. However, current hyper-heuristic architectures assume the existence of a domain barrier that does not allow low-level heuristics or operators to be applied outside their designed problem domain. Additionally the representation used to encode solvers differs from the one used to encode solutions. This means that hyper-heuristic internal components can not be optimised by the system itself. In this thesis we address these issues by using graph reformulations of selected problems and search in the space of operators by using Grammatical Evolution techniques to evolve new perturbative and constructive heuristics. The low-level heuristics (representing graph transformations) are evolved using a single grammar which is capable of adapting to multiple domains. We test our generators of heuristics on instances of the Travelling Salesman Problem, Knapsack Problem and Load Balancing Problem and show that the best evolved heuristics can compete with human written heuristics and representations designed for each problem domain. Further we propose a conceptual framework for the production and combination of graph structures. We show how these concepts can be used to describe and provide a representation for problems in combinatorics and the inner mechanics of hyper-heuristic systems. The final contribution is a new benchmark that can generate problem instances for multiple problem domains that can be used for the assessment of multi-domain problem solvers

Proceedings of the tenth international conference Models in developing mathematics education: September 11 - 17, 2009, Dresden, Saxony, Germany

This volume contains the papers presented at the International Conference on “Models in Developing Mathematics Education” held from September 11-17, 2009 at The University of Applied Sciences, Dresden, Germany. The Conference was organized jointly by The University of Applied Sciences and The Mathematics Education into the 21st Century Project - a non-commercial international educational project founded in 1986. The Mathematics Education into the 21st Century Project is dedicated to the improvement of mathematics education world-wide through the publication and dissemination of innovative ideas. Many prominent mathematics educators have supported and contributed to the project, including the late Hans Freudental, Andrejs Dunkels and Hilary Shuard, as well as Bruce Meserve and Marilyn Suydam, Alan Osborne and Margaret Kasten, Mogens Niss, Tibor Nemetz, Ubi D’Ambrosio, Brian Wilson, Tatsuro Miwa, Henry Pollack, Werner Blum, Roberto Baldino, Waclaw Zawadowski, and many others throughout the world. Information on our project and its future work can be found on Our Project Home Page http://math.unipa.it/~grim/21project.htm It has been our pleasure to edit all of the papers for these Proceedings. Not all papers are about research in mathematics education, a number of them report on innovative experiences in the classroom and on new technology. We believe that “mathematics education” is fundamentally a “practicum” and in order to be “successful” all new materials, new ideas and new research must be tested and implemented in the classroom, the real “chalk face” of our discipline, and of our profession as mathematics educators. These Proceedings begin with a Plenary Paper and then the contributions of the Principal Authors in alphabetical name order. We sincerely thank all of the contributors for their time and creative effort. It is clear from the variety and quality of the papers that the conference has attracted many innovative mathematics educators from around the world. These Proceedings will therefore be useful in reviewing past work and looking ahead to the future