A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

We develop an elementary formula for certain non-trivial elements of upper cluster algebras. These elements have positive coefficients. We show that when the cluster algebra is acyclic these elements form a basis. Using this formula, we show that each non-acyclic skew-symmetric cluster algebra of rank 3 is properly contained in its upper cluster algebra

Resin transfer moulding: Novel fabrics and tow placement techniques in highly loaded carbon fibre composite aircraft spars

A BAE Systems /UK EPSRC funded project Flaviir, is investigating the design and manufacture of low cost carbon fibre composite airframe structures. Novel binder coated unidirectional fibre tapes and tows were developed to enable the design of optimised primary structures. The RTM technique was applied to mould net shape sections of spar components. Various designs of wing attachment lugs were manufactured with a range of reinforcement materials, including non crimp fabric, novel binder coated tapes and conventional unidirectional prepreg. Alongside these, a novel technique termed optimised tow lay up (OTL) was used to reduce the weight. Binder coated carbon fibre tow is placed around the structure in the principal stress directions to increase both bearing strength and overall component stiffness. The novel materials, manufacturing technique and initial element test results are presented

Optical Propagation of Self-sustaining Wavefronts and Nonlinear Dynamics in Parabolic Multimode Fibers

The aim of this thesis is to introduce my work which has generally been focused on optical wavefronts that have the unusual property of resisting commonplace phenomena such as diffraction and dispersion. Interestingly, these special beams are found both in linear and nonlinear situations. For example, in the linear regime, localized spatio-temporal waves which resemble the spherical harmonic symmetries of the hydrogen quantum orbitals can simultaneously negotiate both diffractive and dispersive effects. In the nonlinear regime, dressed optical filaments can be arranged to propagate multi-photon produced plasma channels orders of magnitude longer than expected. The first portion of this dissertation will begin by surveying the history of diffraction-free beams and introducing some of their mathematical treatments. Interjected throughout this discussion will be several relevant concepts which I explored during my first years a CREOL. The discussion will then be steered into a detailed account of diffraction/dispersion free wavefronts which display hydrogen-like symmetries. The second segment of the document will cover the highly nonlinear process of optical filamentation. This chapter will almost entirely investigate the idea of the dressed filament, an entity which allows for substantial prolongation of this light string. I will then conclude by delving into the topic of supercontinuum generation in parabolic multimode fibers which, in the upcoming years, has great potential of becoming important in optics

Geometric description of C-vectors and real L\"osungen

We introduce real Loesungen as an analogue of real roots. For each mutation sequence of an arbitrary skew-symmetrizable matrix, we define a family of reflections along with associated vectors which are real Loesungen and a set of curves on a Riemann surface. The matrix consisting of these vectors is called L-matrix. We explain how the L-matrix naturally arises in connection with the C-matrix. Then we conjecture that the L-matrix depends (up to signs of row vectors) only on the seed, and that the curves can be drawn without self-intersections, providing a new combinatorial/geometric description of c-vectors

The Bunsen gas solubility coefficient of ethylene as a function of temperature and salinity and its importance for nitrogen fixation assays

The acetylene reduction assay is a common method for assessing nitrogen fixation in a variety of marine and freshwater systems. The method measures ethylene, the product of the conversion of the gas acetylene to its reduced form by nitrogenase. Knowledge of the solubility of ethylene in aqueous solution is crucial to the calculation of nitrogen fixation rates and depends on the temperature and salinity of the assay conditions. Despite the increasing interest in marine nitrogen fixation, no gas solubility (Bunsen) coefficients for ethylene in seawater are published to date. Here, we provide a set of equations and present semiempirically derived Bunsen coefficients for ethylene in water (ranging from 0.069 to 0.226) for a range of temperatures and salinities that are relevant for aquatic nitrogen fixation. We apply these data to nitrogen fixation scenarios at different temperatures and salinities and stress the importance of using accurate Bunsen coefficients in nitrogen fixation assays