Paired 2-disjoint path covers of burnt pancake graphs with faulty elements

The burnt pancake graph $BP_n$ is the Cayley graph of the hyperoctahedral group using prefix reversals as generators. Let $\{u,v\}$ and $\{x,y\}$ be any two pairs of distinct vertices of $BP_n$ for $n\geq 4$. We show that there are $u-v$ and $x-y$ paths whose vertices partition the vertex set of $BP_n$ even if $BP_n$ has up to $n-4$ faulty elements. On the other hand, for every $n\ge3$ there is a set of $n-2$ faulty edges or faulty vertices for which such a fault-free disjoint path cover does not exist.Comment: 14 pages, 4 figure

Cycles in the burnt pancake graphs

The pancake graph $P_n$ is the Cayley graph of the symmetric group $S_n$ on $n$ elements generated by prefix reversals. $P_n$ has been shown to have properties that makes it a useful network scheme for parallel processors. For example, it is $(n-1)$-regular, vertex-transitive, and one can embed cycles in it of length $\ell$ with $6\leq\ell\leq n!$. The burnt pancake graph $BP_n$, which is the Cayley graph of the group of signed permutations $B_n$ using prefix reversals as generators, has similar properties. Indeed, $BP_n$ is $n$-regular and vertex-transitive. In this paper, we show that $BP_n$ has every cycle of length $\ell$ with $8\leq\ell\leq 2^n n!$. The proof given is a constructive one that utilizes the recursive structure of $BP_n$. We also present a complete characterization of all the $8$-cycles in $BP_n$ for $n \geq 2$, which are the smallest cycles embeddable in $BP_n$, by presenting their canonical forms as products of the prefix reversal generators.Comment: Added a reference, clarified some definitions, fixed some typos. 42 pages, 9 figures, 20 pages of appendice

Modelling tools and methodologies for rapid protocell prototyping

The field of unconventional computing considers the possibility of implementing computational devices using novel paradigms and materials to produce computers which may be more efficient, adaptable and robust than their silicon based counterparts. The integration of computation into the realms of chemistry and biology will allow the embedding of engineered logic into living systems and could produce truly ubiquitous computing devices. Recently, advances in synthetic biology have resulted in the modification of microorganism genomes to create computational behaviour in living cells, so called “cellular computing”. The cellular computing paradigm offers the possibility of intelligent bacterial agents which may respond and communicate with one another according to chemical signals received from the environment. However, the high levels of complexity when altering an organism which has been well adapted to certain environments over millions of years of evolution suggests an alternative approach in which chemical computational devices can be constructed completely from the bottom up, allowing the designer exquisite control and knowledge about the system being created. This thesis presents the development of a simulation and modelling framework to aid the study and design of bottom-up chemical computers, involving the encapsulation of computational re-actions within vesicles. The new “vesicle computing” paradigm is investigated using a sophisticated multi-scale simulation framework, developed from mesoscale, macroscale and executable biology techniques

Teacher roles during amusement park visits – insights from observations, interviews and questionnaires

Amusement parks offer rich possibilities for physics learning, through observations and experiments that illustrate important physical principles and often involve the whole body. Amusement parks are also among the most popular school excursions, but very often the learning possibilities are underused. In this work we have studied different teacher roles and discuss how universities, parks or event managers can encourage and support teachers and schools in their efforts to make amusement park visits true learning experiences for their students