Diameter of Cayley graphs of permutation groups generated by transposition trees

Let $\Gamma$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the Cayley graph $\Gamma$ is bounded as \diam(\Gamma) \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}, where the maximization is over all permutations $\pi$, $c(\pi)$ denotes the number of cycles in $\pi$, and \dist_T is the distance function in $T$. In this work, we first assess the performance (the sharpness and strictness) of this upper bound. We show that the upper bound is sharp for all trees of maximum diameter and also for all trees of minimum diameter, and we exhibit some families of trees for which the bound is strict. We then show that for every $n$, there exists a tree on $n$ vertices, such that the difference between the upper bound and the true diameter value is at least $n-4$. Observe that evaluating this upper bound requires on the order of $n!$ (times a polynomial) computations. We provide an algorithm that obtains an estimate of the diameter, but which requires only on the order of (polynomial in) $n$ computations; furthermore, the value obtained by our algorithm is less than or equal to the previously known diameter upper bound. This result is possible because our algorithm works directly with the transposition tree on $n$ vertices and does not require examining any of the permutations (only the proof requires examining the permutations). For all families of trees examined so far, the value $\beta$ computed by our algorithm happens to also be an upper bound on the diameter, i.e. \diam(\Gamma) \le \beta \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}.Comment: This is an extension of arXiv:1106.535

STRATIGRAPHIC DISTRIBUTION AND PALEOENVIRONMENTAL IMPLICATIONS OF PEDOGENIC FEATURES AND TRACE FOSSILS FROM THE LOWER PERMIAN OF KANSAS

ABSTRACT Paleosols in the Lower Permian (Asselian-Sakmarian) Council Grove Group from western Kansas, USA, contain numerous pedogenic features, including carbonate nodules (caliche), redoximorphic coloration, rhizoliths, and trace fossils, which can be used to reconstruct the paleoenvironmental and paleoclimatic conditions during the interval of soil formation. Due to the paleogeographic setting of the Hugoton Basin, 5th-order cycles recorded in distant Council Grove Group outcrops are condensed in the study area into cumulic paleosols that contain pedogenic features signifying both wet and dry climates. Wet/dry cycles are interpreted to be the product of eustatically driven climate change, as well as seasonal changes related to monsoonal weather variability. Statistical analyses of the distribution of pedogenic features indicate a change from drier to wetter conditions over the upper Council Grove Group interval. Detailed study of trace fossils revealed unique morphologies that warrant the erection of a new ichnospecies, proposed here as Naktodemasis krausi. These assemblages are the oldest known occurrence of the genus Naktodemasis. Trace fossils were interpreted as the burrows of soil dwelling insects or insect larvae that colonized semiarid coastal plain environments during periods of subaerial exposure, creating topography and reducing accommodation, thereby preventing the formation of discrete 5th-order lithologic cycles. In order to provide further evidence for the origin of trace fossils, neoichnological experiments were conducted to compare ancient burrows to those of modern fossorial insects. Experimental results showed that the burrows of modern coleopterans belonging to the masked chafer beetles (Cyclocephala sp.) are convincing analogs for Permian trace fossils, and provide some general constraints on moisture tolerances for potential Permian tracemakers. The presence of these traces in Permian sediments, therefore, indicate that soil moisture levels during sea-level lowstands were high enough to support sustained populations of similar organisms. This study also conclusively demonstrates that burrows of this type could be constructed in subaerial settings, and documents the unique burrowing mechanism of the masked chafer beetle

Analytical techniques and instrumentation: A compilation

Technical information is presented covering the areas of: (1) analytical instrumentation useful in the analysis of physical phenomena; (2) analytical techniques used to determine the performance of materials; and (3) systems and component analyses for design and quality control

On The 2-Spanning Cyclability Of Honeycomb Toroidal Graphs

A graph $X$ is 2-spanning cyclable if for any pair of distinct vertices $u$ and $v$ there is a 2-factor of $X$ consisting of two cycles such that $u$ and $v$ belong to distinct cycles. In this paper we examine the 2-spanning cyclability of honeycomb toroidal graphs.Comment: 17 pages, 2 figure

Characterizing the mechanical properties of fused deposition modelling natural fiber recycled polypropylene composites

The objective of this investigation was to characterize the performance of natural fiber reinforced polypropylene composites in fused deposition modelling (FDM). Composite filaments comprising of pre-consumer recycled polypropylene with varying contents of hemp or harakeke fibers were extruded from which tensile test specimens were made using FDM. Filament and test specimens were tensile tested and properties were compared with plain polypropylene samples; the ultimate tensile strength and Young’s modulus of reinforced filament increased by more than 50% and 143%, respectively, for both 30 wt % hemp or harakeke compared to polypropylene filament. However, the same degree of improvement was not seen with the FDM test specimens, with several compositions having properties lower than for unfilled polypropylene. SEM analysis of fracture surfaces revealed uniform fiber dispersion and reasonable fiber alignment, but porosity and fiber pull-out were also observed. Fiber reinforcement was found to give benefit regarding dimensional stability during extrusion and FDM, which is of major importance for its implementation in FDM. Recommendations for optimization of processing in order to enhance build quality and improve mechanical properties are provided

Financial Market Models for the Grid

The existing network of computing devices around the world created by the Internet gives the possibility of establishing a global market for computing power, where anybody connected to this network can acquire computing power or sell his own spare computing resources in exchange for real money. This potential global market for computing power, which does not exist yet, is what we study in this thesis. Specifically, we study the market with both analytic and simulated models. This thesis predicts how a future global market for Grid computing will behave. We give arguments that such a large market, together with its potential indefinite growth, would not be able to scale if it were organized with a central server, and therefore we study a peer-to-peer market model in our simulations. We create a high-level model with the most relevant characteristics of the market, where buyers and sellers trade a single commodity. In our simulations, the parameters of the volume of contracts, proportion of satisfied agents and number of messages in the network achieve stable values in the long run. We also derive analytically the conditions that make the price get stable over time; we then implement these conditions in the simulation as local mechanisms of the market participants, which make the whole system achieve a stable price evolution. We are also confident that, as soon as the Grid market emerges, a parallel market of derivatives will be created as well. This market of derivatives will be important due to the non-storability nature of computing power. We develop a futures market for computing power based on Markov chains, where we initially model the behaviour of each participant with a particular Markov chain, and then we derive a global transition probability matrix that models the market as a whole. Furthermore, we analyse the performance of a futures trader operating in such a market, and we obtain an optimal trading strategy with the use of Markov Decision Processes. We finally develop a stochastic differential equation model that captures the essence of the spot price evolution of computing power observed in our market simulations. This model is based on a previously one proposed for the electricity market, and consists of the use of a Markov regime-switching mechanism in order to model the existence of spikes in the spot price. We then estimate the parameters in the model with the output data of our simulation program; the estimation is carried out both by maximum likelihood and the generalised method of moments