Diffusion tensor of water in model articular cartilage

We used Monte Carlo simulations of Brownian dynamics of water to study anisotropic water diffusion in an idealised model of articular cartilage. The main aim was to use the simulations as a tool for translation of the fractional anisotropy of the water diffusion tensor in cartilage into quantitative characteristics of its collagen fibre network. The key finding was a linear empirical relationship between the collagen volume fraction and the fractional anisotropy of the diffusion tensor. Fractional anisotropy of the diffusion tensor is potentially a robust indicator of the microstructure of the tissue because, in the first approximation, it is invariant to the inclusion of proteoglycans or chemical exchange between free and collagen-bound water in the model. We discuss potential applications of Monte Carlo diffusion-tensor simulations for quantitative biophysical interpretation of MRI diffusion-tensor images of cartilage. Extension of the model to include collagen fibre disorder is also discussed.

An enhanced explicit rate algorithm for ABR traffic control in ATM networks

A high performance, low computational complexity rate-based flow control algorithm which can avoid congestion and achieve fairness is important to ATM available bit rate service. The explicit rate allocation algorithm proposed by Kalampoukas et al. is designed to achieve max–min fairness in ATM networks. It has several attractive features, such as a fixed computational complexity of O(1) and the guaranteed convergence to max–min fairness. In this paper, certain drawbacks of the algorithm, such as the severe overload of an outgoing link during transient period and the non-conforming use of the current cell rate field in a resource management cell, have been identified and analysed; a new algorithm which overcomes these drawbacks is proposed. The proposed algorithm simplifies the rate computation as well. Compared with Kalampoukas's algorithm, it has better performance in terms of congestion avoidance and smoothness of rate allocation.

New insights into Crustal Contributions to Large-volume Rhyolite Generation in the Mid-Tertiary Sierra Madre Occidental Province, Mexico, revealed by U Pb Geochronology

Voluminous (≥3·9 × 105 km3), prolonged (∼18 Myr) explosive silicic volcanism makes the mid-Tertiary Sierra Madre Occidental province of Mexico one of the largest intact silicic volcanic provinces known. Previous models have proposed an assimilation–fractional crystallization origin for the rhyolites involving closed-system fractional crystallization from crustally contaminated andesitic parental magmas, with <20% crustal contributions. The lack of isotopic variation among the lower crustal xenoliths inferred to represent the crustal contaminants and coeval Sierra Madre Occidental rhyolite and basaltic andesite to andesite volcanic rocks has constrained interpretations for larger crustal contributions. Here, we use zircon age populations as probes to assess crustal involvement in Sierra Madre Occidental silicic magmatism. Laser ablation-inductively coupled plasma-mass spectrometry analyses of zircons from rhyolitic ignimbrites from the northeastern and southwestern sectors of the province yield U–Pb ages that show significant age discrepancies of 1–4 Myr compared with previously determined K/Ar and 40Ar/39Ar ages from the same ignimbrites; the age differences are greater than the errors attributable to analytical uncertainty. Zircon xenocrysts with new overgrowths in the Late Eocene to earliest Oligocene rhyolite ignimbrites from the northeastern sector provide direct evidence for some involvement of Proterozoic crustal materials, and, potentially more importantly, the derivation of zircon from Mesozoic and Eocene age, isotopically primitive, subduction-related igneous basement. The youngest rhyolitic ignimbrites from the southwestern sector show even stronger evidence for inheritance in the age spectra, but lack old inherited zircon (i.e. Eocene or older). Instead, these Early Miocene ignimbrites are dominated by antecrystic zircons, representing >33 to ∼100% of the dated population; most antecrysts range in age between ∼20 and 32 Ma. A sub-population of the antecrystic zircons is chemically distinct in terms of their high U (>1000 ppm to 1·3 wt %) and heavy REE contents; these are not present in the Oligocene ignimbrites in the northeastern sector of the Sierra Madre Occidental. The combination of antecryst zircon U–Pb ages and chemistry suggests that much of the zircon in the youngest rhyolites was derived by remelting of partially molten to solidified igneous rocks formed during preceding phases of Sierra Madre Occidental volcanism. Strong Zr undersaturation, and estimations for very rapid dissolution rates of entrained zircons, preclude coeval mafic magmas being parental to the rhyolite magmas by a process of lower crustal assimilation followed by closed-system crystal fractionation as interpreted in previous studies of the Sierra Madre Occidental rhyolites. Mafic magmas were more probably important in providing a long-lived heat and material flux into the crust, resulting in the remelting and recycling of older crust and newly formed igneous materials related to Sierra Madre Occidental magmatism.

Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

In this paper, we consider the variable-order Galilei advection diffusion equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis. Keywords: The variable-order Galilei invariant advection diffusion equation with a nonlinear source term; The variable-order Riemann–Liouville fractional partial derivative; Stability; Convergence; Numerical scheme improving temporal accuracy

Numerical analysis for a variable-order nonlinear cable equation

In this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.

Modelling travelling waves in spatially constrained environment solutions

In this study, we consider how Fractional Differential Equations (FDEs) can be used to study the travelling wave phenomena in parabolic equations. As our method is conducted under intracellular environments that are highly crowded, it was discovered that there is a simple relationship between the travelling wave speed and obstacle density.

Multifractal characterisation and analysis of complex networks

Complex networks have been studied extensively due to their relevance to many real-world systems such as the world-wide web, the internet, biological and social systems. During the past two decades, studies of such networks in different fields have produced many significant results concerning their structures, topological properties, and dynamics. Three well-known properties of complex networks are scale-free degree distribution, small-world effect and self-similarity. The search for additional meaningful properties and the relationships among these properties is an active area of current research. This thesis investigates a newer aspect of complex networks, namely their multifractality, which is an extension of the concept of selfsimilarity. The first part of the thesis aims to confirm that the study of properties of complex networks can be expanded to a wider field including more complex weighted networks. Those real networks that have been shown to possess the self-similarity property in the existing literature are all unweighted networks. We use the proteinprotein interaction (PPI) networks as a key example to show that their weighted networks inherit the self-similarity from the original unweighted networks. Firstly, we confirm that the random sequential box-covering algorithm is an effective tool to compute the fractal dimension of complex networks. This is demonstrated on the Homo sapiens and E. coli PPI networks as well as their skeletons. Our results verify that the fractal dimension of the skeleton is smaller than that of the original network due to the shortest distance between nodes is larger in the skeleton, hence for a fixed box-size more boxes will be needed to cover the skeleton. Then we adopt the iterative scoring method to generate weighted PPI networks of five species, namely Homo sapiens, E. coli, yeast, C. elegans and Arabidopsis Thaliana. By using the random sequential box-covering algorithm, we calculate the fractal dimensions for both the original unweighted PPI networks and the generated weighted networks. The results show that self-similarity is still present in generated weighted PPI networks. This implication will be useful for our treatment of the networks in the third part of the thesis. The second part of the thesis aims to explore the multifractal behavior of different complex networks. Fractals such as the Cantor set, the Koch curve and the Sierspinski gasket are homogeneous since these fractals consist of a geometrical figure which repeats on an ever-reduced scale. Fractal analysis is a useful method for their study. However, real-world fractals are not homogeneous; there is rarely an identical motif repeated on all scales. Their singularity may vary on different subsets; implying that these objects are multifractal. Multifractal analysis is a useful way to systematically characterize the spatial heterogeneity of both theoretical and experimental fractal patterns. However, the tools for multifractal analysis of objects in Euclidean space are not suitable for complex networks. In this thesis, we propose a new box covering algorithm for multifractal analysis of complex networks. This algorithm is demonstrated in the computation of the generalized fractal dimensions of some theoretical networks, namely scale-free networks, small-world networks, random networks, and a kind of real networks, namely PPI networks of different species. Our main finding is the existence of multifractality in scale-free networks and PPI networks, while the multifractal behaviour is not confirmed for small-world networks and random networks. As another application, we generate gene interactions networks for patients and healthy people using the correlation coefficients between microarrays of different genes. Our results confirm the existence of multifractality in gene interactions networks. This multifractal analysis then provides a potentially useful tool for gene clustering and identification. The third part of the thesis aims to investigate the topological properties of networks constructed from time series. Characterizing complicated dynamics from time series is a fundamental problem of continuing interest in a wide variety of fields. Recent works indicate that complex network theory can be a powerful tool to analyse time series. Many existing methods for transforming time series into complex networks share a common feature: they define the connectivity of a complex network by the mutual proximity of different parts (e.g., individual states, state vectors, or cycles) of a single trajectory. In this thesis, we propose a new method to construct networks of time series: we define nodes by vectors of a certain length in the time series, and weight of edges between any two nodes by the Euclidean distance between the corresponding two vectors. We apply this method to build networks for fractional Brownian motions, whose long-range dependence is characterised by their Hurst exponent. We verify the validity of this method by showing that time series with stronger correlation, hence larger Hurst exponent, tend to have smaller fractal dimension, hence smoother sample paths. We then construct networks via the technique of horizontal visibility graph (HVG), which has been widely used recently. We confirm a known linear relationship between the Hurst exponent of fractional Brownian motion and the fractal dimension of the corresponding HVG network. In the first application, we apply our newly developed box-covering algorithm to calculate the generalized fractal dimensions of the HVG networks of fractional Brownian motions as well as those for binomial cascades and five bacterial genomes. The results confirm the monoscaling of fractional Brownian motion and the multifractality of the rest. As an additional application, we discuss the resilience of networks constructed from time series via two different approaches: visibility graph and horizontal visibility graph. Our finding is that the degree distribution of VG networks of fractional Brownian motions is scale-free (i.e., having a power law) meaning that one needs to destroy a large percentage of nodes before the network collapses into isolated parts; while for HVG networks of fractional Brownian motions, the degree distribution has exponential tails, implying that HVG networks would not survive the same kind of attack.