Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

Homoclinic Splitting without Trees

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations. When the coupling takes place through an even trigonometric polynomial in the angle variables, we extend analytically the solutions of the equations of motion, order by order in the perturbation parameter, to a large neighbourhood of the real line representing time. Subsequently, we devise an asymptotic expansion for the splitting (matrix) associated with a homoclinic point. This expansion consists of contributions that are manifestly exponentially small in the limit of vanishing gravity, by a shift-of-countour argument. Hence, we infer a similar upper bound for the splitting itself. In particular, the derivation of the result does not call for a tree expansion with explicit cancellation mechanisms.

Distortion of Dimension under Quasiconformal Mappings

Quasiconformal mappings are natural generalizations of conformal mappings. They are homeomorphisms with 'bounded distortion' of which there exist several approaches. In this work we study dimension distortion properties of quasiconformal mappings both in the plane and in higher dimensional Euclidean setting. The thesis consists of a summary and three research articles. A basic property of quasiconformal mappings is the local Hölder continuity. It has long been conjectured that this regularity holds at the Sobolev level (Gehring's higher integrabilty conjecture). Optimal regularity would also provide sharp bounds for the distortion of Hausdorff dimension. The higher integrability conjecture was solved in the plane by Astala in 1994 and it is still open in higher dimensions. Thus in the plane we have a precise description how Hausdorff dimension changes under quasiconformal deformations for general sets. The first two articles contribute to two remaining issues in the planar theory. The first one concerns distortion of more special sets, for rectifiable sets we expect improved bounds to hold. The second issue consists of understanding distortion of dimension on a finer level, namely on the level of Hausdorff measures. In the third article we study flatness properties of quasiconformal images of spheres in a quantitative way. These also lead to nontrivial bounds for their Hausdorff dimension even in the n-dimensional case.

Flying-Fox Roost Disturbance and Hendra Virus Spillover Risk

Bats of the genus Pteropus (flying-foxes) are the natural host of Hendra virus (HeV) which periodically causes fatal disease in horses and humans in Australia. The increased urban presence of flying-foxes often provokes negative community sentiments because of reduced social amenity and concerns of HeV exposure risk, and has resulted in calls for the dispersal of urban flying-fox roosts. However, it has been hypothesised that disturbance of urban roosts may result in a stress-mediated increase in HeV infection in flying-foxes, and an increased spillover risk. We sought to examine the impact of roost modification and dispersal on HeV infection dynamics and cortisol concentration dynamics in flying-foxes. The data were analysed in generalised linear mixed models using restricted maximum likelihood (REML). The difference in mean HeV prevalence in samples collected before (4.9%), during (4.7%) and after (3.4%) roost disturbance was small and non-significant (P = 0.440). Similarly, the difference in mean urine specific gravity-corrected urinary cortisol concentrations was small and non-significant (before = 22.71 ng/mL, during = 27.17, after = 18.39) (P= 0.550). We did find an underlying association between cortisol concentration and season, and cortisol concentration and region, suggesting that other (plausibly biological or environmental) variables play a role in cortisol concentration dynamics. The effect of roost disturbance on cortisol concentration approached statistical significance for region, suggesting that the relationship is not fixed, and plausibly reflecting the nature and timing of disturbance. We also found a small positive statistical association between HeV excretion status and urinary cortisol concentration. Finally, we found that the level of flying-fox distress associated with roost disturbance reflected the nature and timing of the activity, highlighting the need for a ‘best practice’ approach to dispersal or roost modification activities. The findings usefully inform public discussion and policy development in relation to Hendra virus and flying-fox management.

On the Rademacher maximal function

Tools known as maximal functions are frequently used in harmonic analysis when studying local behaviour of functions. Typically they measure the suprema of local averages of non-negative functions. It is essential that the size (more precisely, the L^p-norm) of the maximal function is comparable to the size of the original function. When dealing with families of operators between Banach spaces we are often forced to replace the uniform bound with the larger R-bound. Hence such a replacement is also needed in the maximal function for functions taking values in spaces of operators. More specifically, the suprema of norms of local averages (i.e. their uniform bound in the operator norm) has to be replaced by their R-bound. This procedure gives us the Rademacher maximal function, which was introduced by Hytönen, McIntosh and Portal in order to prove a certain vector-valued Carleson's embedding theorem. They noticed that the sizes of an operator-valued function and its Rademacher maximal function are comparable for many common range spaces, but not for all. Certain requirements on the type and cotype of the spaces involved are necessary for this comparability, henceforth referred to as the “RMF-property”. It was shown, that other objects and parameters appearing in the definition, such as the domain of functions and the exponent p of the norm, make no difference to this. After a short introduction to randomized norms and geometry in Banach spaces we study the Rademacher maximal function on Euclidean spaces. The requirements on the type and cotype are considered, providing examples of spaces without RMF. L^p-spaces are shown to have RMF not only for p greater or equal to 2 (when it is trivial) but also for 1 < p < 2. A dyadic version of Carleson's embedding theorem is proven for scalar- and operator-valued functions. As the analysis with dyadic cubes can be generalized to filtrations on sigma-finite measure spaces, we consider the Rademacher maximal function in this case as well. It turns out that the RMF-property is independent of the filtration and the underlying measure space and that it is enough to consider very simple ones known as Haar filtrations. Scalar- and operator-valued analogues of Carleson's embedding theorem are also provided. With the RMF-property proven independent of the underlying measure space, we can use probabilistic notions and formulate it for martingales. Following a similar result for UMD-spaces, a weak type inequality is shown to be (necessary and) sufficient for the RMF-property. The RMF-property is also studied using concave functions giving yet another proof of its independence from various parameters.

Gene flow from Corymbia hybrids in northern New South Wales

Some of the most productive taxa for forestry are interspecific F1 hybrids grown as exotics in the tropics and subtropics. Attributes of resilience, adaptability and vigour which engender the hybrids for wood production, may also exacerbate the risk they present from gene flow to native species gene pools or to local ecologies as weeds. To determine the biological and genetic factors that influence the extent of hybridisation, we examine the distribution and genealogy of wildlings surrounding plantings of locally-exotic Corymbia torelliana (Section Cadageria) near native C. henryi (Section Maculatae) in northern New South Wales. Our study showed pre-mating and pre- and post-zygotic barriers were incomplete, with in situ generation and natural establishment of both F1 hybrids (n = 3) and advanced generation hybrids under the disturbed conditions bordering native forest. As hybrids were located on alluvial flats exposed to frost, they also likely have an extended ecological range relative to native C. henryi. Despite the likely generation of large viable seed crops on F1 trees at the site over many years, establishment success and survival of advanced generation hybrids may be low, as only 5 immature and no mature advanced generation hybrids were identified. Propagation and genetic analysis of a seed crop from one F1 wildling showed early survival and vigour of seedlings in cultivation was high, and that at least for some F1 in some seasons, backcrossing to the recurrent native C. henryi parent is favoured (60%), whereas selfing (10%) and crossing with other F1 (30%) was less frequent. Transport of seed by stingless bees probably accounted for long distance dispersal from C. torelliana, but this mechanism does not appear to supplement gravity-dispersal of seed from the F1. Coupled with other evidence from studies of bee behaviour, controlled pollination in Corymbia sp., and long-term fitness in second generation eucalypt hybrids, we anticipate gene flow via pollen rather than seed will be the greater challenge for managing the risk of introgression of C. torelliana ancestry into native species from the planted F1 hybrid. If large sources of F1 pollen become available to compete with native pollen, gene flow will probably be frequent and hybrids may establish in disturbed conditions and in habitats beyond the ecological range of their native parent. Further study is needed to determine the degree to which outbreeding depression and poor survival inhibits on-going gene flow.

Superhydrophobic fabrics for oil-water separation based on the metal organic charge transfer complex CuTCNAQ

This work reports on the fabrication of a superhydrophobic nylon textile based on the organic charge transfer complex CuTCNAQ (TCNAQ = 11,11,12,12-tetracyanoanthraquinodimethane). The nylon fabric that is metallized with copper undergoes a spontaneous chemical reaction with TCNAQ dissolved in acetonitrile to form nanorods of CuTCNAQ that are intertwined over the entire surface of the fabric. This creates the necessary micro and nanoscale roughness that is required for the Cassie-Baxter state thereby achieving a superhydrophobic/superoleophilic surface without the need for a fluorinated surface. The material is characterised with SEM, FT-IR and XPS spectroscopy and investigated for its ability to separate oil and water in two modes, namely under gravity and as an absorbent. It is found that the fabric can separate dichloromethane, olive oil and crude oil from water and in fact reduce the water content of the oil during the separation process. The fabric is reusable and tolerant to conditions such as seawater, hydrochloric acid and extensive time periods on the shelf. Given that CuTCNAQ is a copper based semiconductor may also open up the possibility of other applications in areas such as photocatalysis and antibacterial applications.

A note on the movement of single large liquid bubbles in vertical tubes

The flow of single large liquid bubbles under gravity in closed tubes is studied here for the case when the liquid bubble exhibits micropolar behaviour. The film thickness, velocity profile in the bubble and film, and nonNewtonian effects are studied and compared with those for the correspondingNewtonian fluid. The investigation is restricted to the case where the bubble length is far greater than the tube radius.

The height limit of a siphon

The maximum height of a siphon is generally assumed to be dependent on barometric pressure—about 10 m at sea level. This limit arises because the pressure in a siphon above the upper reservoir level is below the ambient pressure, and when the height of a siphon approaches 10 m, the pressure at the crown of the siphon falls below the vapour pressure of water causing water to boil breaking the column. After breaking, the columns on either side are supported by differential pressure between ambient and the low-pressure region at the top of the siphon. Here we report an experiment of a siphon operating at sea level at a height of 15 m, well above 10 m. Prior degassing of the water prevented cavitation. This experiment provides conclusive evidence that siphons operate through gravity and molecular cohesion.

Comparison of Prediction Based LSF Quantization Methods using Split VQ

Further improvement in performance, to achieve near transparent quality LSF quantization, is shown to be possible by using a higher order two dimensional (2-D) prediction in the coefficient domain. The prediction is performed in a closed-loop manner so that the LSF reconstruction error is the same as the quantization error of the prediction residual. We show that an optimum 2-D predictor, exploiting both inter-frame and intra-frame correlations, performs better than existing predictive methods. Computationally efficient split vector quantization technique is used to implement the proposed 2-D prediction based method. We show further improvement in performance by using weighted Euclidean distance.

Tools for compositional data with a total

Compositional data analysis usually deals with relative information between parts where the total (abundances, mass, amount, etc.) is unknown or uninformative. This article addresses the question of what to do when the total is known and is of interest. Tools used in this case are reviewed and analysed, in particular the relationship between the positive orthant of D-dimensional real space, the product space of the real line times the D-part simplex, and their Euclidean space structures. The first alternative corresponds to data analysis taking logarithms on each component, and the second one to treat a log-transformed total jointly with a composition describing the distribution of component amounts. Real data about total abundances of phytoplankton in an Australian river motivated the present study and are used for illustration.

A simple method for large-scale preparation of chick embryo tRNA

A simple method for preparing bulk quantities of tRNA from chick embryo has been developed. In this method chick embryos were homogenized in a buffer of pH 4.5, followed by deproteinization with phenol. The aqueous layer was allowed to separate under gravity. The resulting aqueous layer, after two more phenol treatments, was directly passed through a DEAE-cellulose column and the tRNA eluted therefrom with 1 Image NaCl. The tRNA prepared by this method was as active as the one prepared at neutral pH.

On Multi-Dimensional Packets of Surface Waves

By using the method of operators of multiple scales, two coupled nonlinear equations are derived, which govern the slow amplitude modulation of surface gravity waves in two space dimensions. The equations of Davey and Stewartson, which also govern the two-dimensional modulation of the amplitude of gravity waves, are derived as a special case of our equations. For a fully dispersed wave, symmetric about a point which moves with the group velocity, the coupled equations reduce to a nonlinear Schrödinger equation with extra terms representing the effect of the curvature of the wavefront.

Maltose and maltotriose transport into ale and lager brewer's yeast strains

Maltose and maltotriose are the two most abundant sugars in brewer s wort, and thus brewer s yeast s ability to utilize them efficiently is of major importance in the brewing process. The increasing tendency to utilize high and very-high-gravity worts containing increased concentrations of maltose and maltotriose renders the need for efficient transport of these sugars even more pronounced. Residual maltose and especially maltotriose are quite often present especially after high and very-high-gravity fermentations. Sugar uptake capacity has been shown to be the rate limiting factor for maltose and maltotriose utilization. The main aim of the present study was to find novel ways to improve maltose and maltotriose utilization during the main fermentation. Maltose and maltotriose uptake characteristics of several ale and lager strains were studied. Genotype determination of the genes needed for maltose and maltotriose utilization was performed. Maltose uptake inhibition studies were performed to reveal the dominant transporter types actually functioning in each of the strains. Temperature-dependence of maltose transport was studied for ale and for lager strains as well as for each of the single sugar transporter proteins Agt1p, Malx1p and Mtt1p. The AGT1 promoter regions of one ale and two lager strains were sequenced by chromosome walking and the promoter elements were searched for using computational methods. The results showed that ale and lager strains predominantly use different maltose and maltotriose transporter types for maltose and maltotriose uptake. Agt1 transporter was found to be the dominant maltose/maltotriose transporter in the ale strains whereas Malx1 and Mtt1- type transporters dominated in the lager strains. All lager strains studied were found to possess a non-functional Agt1 transporter. The ale strains were observed to be more sensitive to temperature decrease in their maltose uptake compared to the lager strains. Single transporters were observed to differ in their sensitivity to temperature decrease and their temperature-dependence was shown to decrease in the order Agt1≥Malx1>Mtt1. The different temperature-dependence between the ale and lager strains was observed to be due to the different dominant maltose/maltotriose transporters ale and lager strains possessed. The AGT1 promoter regions of ale and lager strains were found to differ markedly from the corresponding regions of laboratory strains. The ale strain was found to possess an extra MAL-activator binding site compared to the lager strains. Improved maltose and maltotriose uptake capacity was obtained with a modified lager strain where the AGT1 gene was repaired and put under the control of a strong promoter. Modified strains fermented wort faster and more completely, producing beers containing more ethanol and less residual maltose and maltotriose. Significant savings in the main fermentation time were obtained when modified strains were used. In high-gravity wort fermentations 8 20% and in very-high-gravity wort fermentations even 11 37% time savings were obtained. These are economically significant changes and would cause a marked increase in annual output from the same-size of brewhouse and fermentor facilities.

Nonlinear secondary whirl of an overhung rotor

Gravity critical speeds of rotors have hitherto been studied using linear analysis, and ascribed to rotor stiffness asymmetry. Here, we study an idealized asymmetric nonlinear overhung rotor model of Crandall and Brosens, spinning close to its gravity critical speed.Nonlinearities arise from finite displacements, and the rotor's staticlateral deflection under gravity is taken as small. Assuming small asymmetry and damping, slow modulations of whirl amplitudes are studied using the method of multiple scales. Inertia asymmetry appears only at second order. More interestingly, even without stiffness asymmetry, the gravity-induced resonance survives through geometric nonlinearities. The gravity resonant forcing does not influence the resonant mode at leading order, unlike the typical resonant oscillations. Nevertheless,the usual phenomena of resonances, namely saddle-node bifurcations, jump phenomena and hysteresis, are all observed. An unanticipated periodic solution branch is found. In the three-dimensional space oftwo modal coefficients and a detuning parameter, the full set of periodic solutions is found to be an imperfect version of three mutually intersecting curves: a straight line,a parabola and an ellipse.