Extremal sequences of polynomial complexity

The joint spectral radius of a bounded set of $d \times d$ real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called \emph{extremal} if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. Wang asked whether every finite set of matrices admits an extremal sequence which is periodic. This is equivalent to the assertion that every finite set of matrices admits an extremal sequence with bounded subword complexity. Counterexamples were subsequently constructed which have the property that every extremal sequence has at least linear subword complexity. In this paper we extend this result to show that for each integer $p \geq 1$, there exists a pair of square matrices of dimension $2^p(2^{p+1}-1)$ for which every extremal sequence has subword complexity at least $2^{-p^2}n^p$.Comment: 15 page

Elemental bio-imaging : in situ analysis of trace elements in tissue by laser ablation-inductively coupled plasma-mass spectrometry

University of Technology, Sydney. Faculty of Science.Elemental bio-imaging is a new application of laser ablation - inductively coupled plasma - mass spectrometry (LA-ICP-MS) that determines in situ trace element concentrations in thin sections of biological tissues. This project developed a LA-ICP-MS method for creating colour images of the regional distribution of both metals and non-metals in a variety of biological samples. The developed method was capable of producing images with a lateral resolution as low as 30 µm. This was achieved using a 30 µm laser spot that was rastered across the sample at a rate of 30 µm s⁻¹. It was found that a laser fluence of 0.22-0.28 J cm⁻² was best suited for soft tissue sections with minimal particle redeposition and fracturing of the surrounding sample. Evaluation of the octopole reaction system (ORS) fitted to the Agilent 7500 ce instrument found that the use of a collision gas for interference removal was inadvisable for imaging experiments. Experiments were carried out to determine the significance of potential polyatomic interferences in the ablation of tissue. It was found that the 'dry' nature of the plasma in LA-ICP-MS significantly reduced the occurrence of O and H based polyatomics, and the small sample load in each ablation reduced the effect of other matrix-based interferences on elements in the mg kg⁻¹ concentration range. Application of the method to sections of human lymph nodes impregnated with malignant melanoma found imaging of ³¹P was able to accurately discern healthy cells from cancerous tissue. Measurement of the ratio between ³¹P and other elements improved the contrast between the two types of cells. Three-dimensional models of imaged lymph nodes further improved the distinction between the melanoma cells and normal tissue. A method was developed for producing matrix-matched tissue standards by homogenising chicken tissue with known amounts of added elemental standards. Digestion of the tissue standards was performed and analysis by solution nebulisation ICP-MS confirmed the concentration of each added element. The standards were frozen and cut to the desired thickness for ablation and construction of multi-point calibration curves. An Fe-fed mouse model for Parkinson's disease (PD) was used to demonstrate the characteristics of the technique. C57BL6 mice were fed a diet high in Fe during development and were treated with both clioquinol (CQ) and L-DOPA, both of which are thought to chelate Fe in the brain. Results showed a decrease in Fe in the treated animals within the region of the brain called the substantia nigra (SN), which is adversely affected in PD. A 6-hydroxydopamine (6-OHDA) model for PD was also examined. 6-OHDA is directly injected into the rodent brain, stimulating the loss of cells within the SN. Imaging of sections taken from 6-OHDA lesioned animals showed a significant increase in Fe within the SN bilaterally when compared to control animals. In summary, elemental bio-imaging is a new method that can be applied to tissue sections from many sources, including humans. The technique has the potential to assist biologists in identifying possible new biomarkers for disease, related specifically to trace elements

An explicit counterexample to the Lagarias-Wang finiteness conjecture

The joint spectral radius of a finite set of real $d \times d$ matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the \emph{finiteness property} if there exists a periodic product which achieves this maximal rate of growth. J.C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real $d \times d$ matrices satisfies the finiteness property. However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the finiteness conjecture exist, showing in particular that there exists a family of pairs of $2 \times 2$ matrices which contains a counterexample. Similar results were subsequently given by V.D. Blondel, J. Theys and A.A. Vladimirov and by V.S. Kozyakin, but no explicit counterexample to the finiteness conjecture has so far been given. The purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the Lagarias-Wang finiteness conjecture. Namely, for the set \mathsf{A}_{\alpha_*}:= \{({cc}1&1\\0&1), \alpha_*({cc}1&0\\1&1)\} we give an explicit value of \alpha_* \simeq 0.749326546330367557943961948091344672091327370236064317358024...] such that $\mathsf{A}_{\alpha_*}$ does not satisfy the finiteness property.Comment: 27 pages, 2 figure

Atmosphere-Ocean Ozone Exchange – A Global Modeling Study of Biogeochemical, Atmospheric and Water-Side Turbulence Dependencies

The significance of the removal of tropospheric ozone by the oceans, covering ~2/3 of the Earth's surface, has only been addressed in a few studies involving water tank, aircraft, and tower flux measurements. On the basis of results from these few observations of the ozone dry deposition velocity (VdO3), atmospheric chemistry models generally apply an empirical, constant ocean uptake rate of 0.05 cm s-1. This value is substantially smaller than the atmospheric turbulent transport velocity for ozone. On the other hand, the uptake is higher than expected from the solubility of ozone in clean water alone, suggesting that there is an enhancement in oceanic ozone uptake, e.g., through a chemical destruction mechanism. We present an evaluation of a global-scale analysis with a new mechanistic representation of atmosphere-ocean ozone exchange. The applied atmosphere chemistry-climate model includes not only atmospheric but also waterside turbulence and the role of waterside chemical loss processes as a function of oceanic biogeochemistry. The simulations suggest a larger role of biogeochemistry in tropical and subtropical ozone oceanic uptake with a relative small temporal variability, whereas in midlatitude and high-latitude regions, highly variable ozone uptake rates are expected because of the stronger influence of waterside turbulence. Despite a relatively large range in the explicitly calculated ocean uptake rate, there is a surprisingly small sensitivity of simulated Marine Boundary Layer ozone concentrations compared to the sensitivity for the commonly applied constant ocean uptake approach. This small sensitivity points at compensating effects through inclusion of the process-based ocean uptake mechanisms to consider variability in oceanic O3 deposition consistent with that in atmospheric and oceanic physical, chemical, and biological processe

A Study of the Actuarial Profession

The aim of this thesis is to educate the reader about the career of an actuary and will put into practice some of the rigorous mathematical applications that are used. There will also be an emphasis on the financial and business applications of the previous information, which encompasses the well roundedness of the profession. The actuarial profession is ranked one of the best in the country and rightfully so because of the skill and time it takes to perform it well. This thesis will guide the reader in understanding more clearly the high rank of the actuarial profession and the details of the statistical mathematics that are used

Design, Fabrication and Analysis of a Utility Strip for Light Utility Trucks

This senior project discusses the design and fabrication of a utility strip for a light utility truck. The utility strip is desirable for off road enthusiasts with its double use as a dirt bike ramp as well as a traction strip. This report will show the cost of materials as well as labor, and if it is feasible for mass production. Tests were conducted which indicated that the Utility Strip was able to support the weight of a dirt bike as a ramp, as well as regain traction to a two-wheel drive truck stuck in the sand. This product can be put on the market to appeal to off-road enthusiasts

Smallholder Farmer Seed Production of Mulato II Hybrid Brachiaria in Thailand and Laos - Bringing Quality Forage Seeds Home to Africa

Seeds of Mulato II hybrid brachiaria [Urochloa ruziziensis (syn. Brachiaria ruziziensis) x U. brizantha (syn. B. brizantha) x U. decumbens (syn, B.decumbens)] produced by smallholder farmers in villages in northeast Thailand and northern Laos, under contract to Ubon Forage Seeds Co. Ltd, a tropical forage seed company in Thailand, have come home to Africa. Since 2012, over 20,000 kg of Mulato II seed have been exported from Thailand to Kenya. The seed is hand-harvested from seed-heads in Laos and by hand-sweeping the seed from the ground in Thailand. The farmers semi-clean the seed in the field and then further winnow the seed back at their houses in cane trays or through small motorized seed cleaners. Ubon Forage Seeds Co. Ltd purchases the seed and then treats the seed with sulphuric acid to improve seed germination from below 10% to above 80%. Since 2008, 300,000 kg of Mulato II seed have been produced by the smallholder farmers and exported to over 20 countries. The pathway of smallholder farmers producing Mulato II seeds in Thailand and Laos and then exporting this seed to Kenya and other countries in Africa to be used by smallholder farmers to improve livestock production, is discussed