Alternating direction implicit numerical method for the space and time fractional Bloch-Torrey equation in 3-D

Recently, some authors have considered a new diffusion model–space and time fractional Bloch-Torrey equation (ST-FBTE). Magin et al. (2008) have derived analytical solutions with fractional order dynamics in space (i.e., _ = 1, β an arbitrary real number, 1 < β ≤ 2) and time (i.e., 0 < α < 1, and β = 2), respectively. Yu et al. (2011) have derived an analytical solution and an effective implicit numerical method for solving ST-FBTEs, and also discussed the stability and convergence of the implicit numerical method. However, due to the computational overheads necessary to perform the simulations for nuclear magnetic resonance (NMR) in three dimensions, they present a study based on a two-dimensional example to confirm their theoretical analysis. Alternating direction implicit (ADI) schemes have been proposed for the numerical simulations of classic differential equations. The ADI schemes will reduce a multidimensional problem to a series of independent one-dimensional problems and are thus computationally efficient. In this paper, we consider the numerical solution of a ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. A fractional alternating direction implicit scheme (FADIS) for the ST-FBTE in 3-D is proposed. Stability and convergence properties of the FADIS are discussed. Finally, some numerical results for ST-FBTE are given.

A new implicit numerical method for the space and time fractional Bloch-Torrey equation

In recent years, it has been found that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from fractional calculus. Recently, noted a new space and time fractional Bloch-Torrey equation (ST-FBTE) has been proposed (see Magin et al. (2008)), and successfully applied to analyse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we propose a new effective implicit numerical method (INM) for the STFBTE whereby we discretize the Riesz fractional derivative using a fractional centered difference. Secondly, we prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent, and the order of convergence of the implicit numerical method is ( T2 - α + h2 x + h2 y + h2 z ). Finally, some numerical results are presented to support our theoretical analysis.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Whe' yu' from?

"Whe' yu' from?" The question was put to me as I wandered, camera in hand, in the old square of Spanish Town, Jamaica's former capital. The local man, lounging in the shade of one of the colonial Georgian buildings that enclose the square, was mildly curious about what he took to be a typical white tourish photgraphing the sights of the decayed historic town. At that time, my home was in Kingston where i lived with my wife and baby son. I was then working in the Jamaican Government Town Planning Department in a job that took me all over the island. Turning to my questioner, I replied, "Kingston". There was a brief pause, and then the man spoke again: "No Man! Whe' yu' really from?" I still have difficulties when asked this question. Where am I from? What does this question mean? Does it refer to where I was born, where I spent my previous life or where I live now? Does it have a broader meaning, an enquiry about my origins in terms of background and previous experience? The following chapters are my attempt to answer these questions for my own satisfaction and, I hope, for the amusement of others who may be interested in the life of an ordinary English boy whose dream to travel and see the world was realized in ways he could not possibly have imagined. Finding an appropriate title for this book was difficult. Thursday's Child, North and South and War and Peace all came to mind but, unfortunately for me, those titles had been appropriated by other writers. Thursdays's Child is quite a popular book title, presumably because people who were born on that day and, in the words of the nursery rhyme, had 'far to go', are especially likely to have travellers' tales to tell or life stories of the rags-to-riches variety. Born on a Thursday, I have travelled a lot and I suppose that I have gone far in life. Coming from a working class family, I 'got on' by 'getting a good education' and a 'good job'. I decided against adding to the list of Thursday's Children. North and South would have reflected my life in Britain, spent in both the North and South of England, and my later years, divided between the Northern and Southern Hemispheres of the globe, as well as in countries commonly referred to as the 'advanced' North and the 'underdeveloped' South. North and South has already been appropriated by Mrs Gaskell, something that did not deter one popular American writer from using the title for a book of his. My memories of World War Two and the years afterwards made War and Peace a possible candidate, but readers expectnig an epic tale of Tolstoyan proportions may have been disappointed. To my knowledge, no other book has the title "Whe' Yu' From?". I am grateful to the Jamaican man whose question lingered in my memory and provided the title of this memoir, written decades later. This book is a word picture. It is, in a sense, a self-portrait, and like all portraits, it captures something of the character, it attempts to tell the truth, but it is not the whole truth. This is because it is not my intention to write my entire life story; rather I wish to tell about some of the things in my experience of life that have seemed important or interesting to me. Unlike a painted portrait, the picture I have created is intended to suggest the passage of time. While, for most of us in Western society, time is linear and unidirectional, like the flight of an arrov or the trajectory of a bullet, memory rearranges things, calling up images of the past in no particular order, making connections that may link events in various patterns, circular, web-like, superimposed. The stream of consciousness is very unlike that of streams we encounter in the physical world. Connections are made in all directions; thoughts hop back and forth in time and space, from topic to topic. My book is a composition drawn from periods, events and thoughts as I remember them. Like life itself, it is made up of patches, some good, some bad, but in my experience, always fascinating. In recording my memories, I have been as accurate as possible. Little of what I have written is about spectacular sights and strange customs. Much of it focuses on my more modest explorations includng observations of everyday things that have attracted my attention. Reading through the chapters, I am struck by my childhood freedom to roam and engage in 'dangerous' activities like climbing trees and playing beside streams, things that many children today are no longer allowed to enjoy. Also noticeable is the survival of traditions and superstitions from the distant past. Obvious too, is my preoccupation with place names, both official ones that appear on maps and sign boards and those used by locals and children, names rarely seen in print. If there is any uniting theme to be found in what I have written, it must be my education in the fields, woods and streets of my English homeland, in the various other countries in which I have lived and travelled, as well as more formally from books and in classrooms. Much of my book is concerned with people and places. Many of the people I mention are among those who have been, and often have remained, important and close to me. Others I remember from only the briefest of encounters, but they remain in my memory because of some specific incident or circumstance that fixed a lasting image in my mind. Some of my closest friends and relatives, however, appear nowhere in these pages or they receive only the slightest mention. This is not because they played an unimportant roles in my life. It is because this book is not the whole story. Among those whe receive little or no mention are some who are especially close to me, with whom I have shared happy and sad times and who have shown me and my family much kindness, giving support when this was needed. Some I have known since childhood and have popped up at various times in my life, often in different parts of the world. Although years may pass without me seeing them, in an important sense they are always with me. These people know who they are. I hope that they know how much I love and appreciate them. When writing my memoir, I consulted a few of the people mentioned in this book, but in the main, I have relied on my own memory, asided by daiary and notebook entries and old correspondence. In the preparation of this manuscript, I benefited greatly from the expert advice and encouragement of Neil Marr of BeWrite Books. My wife Anne, inspiration for this book, also contributed in the valuable role of critic. She has my undying gratitude.

Real, complex, and binary semantic vectors

This paper presents a combined structure for using real, complex, and binary valued vectors for semantic representation. The theory, implementation, and application of this structure are all significant. For the theory underlying quantum interaction, it is important to develop a core set of mathematical operators that describe systems of information, just as core mathematical operators in quantum mechanics are used to describe the behavior of physical systems. The system described in this paper enables us to compare more traditional quantum mechanical models (which use complex state vectors), alongside more generalized quantum models that use real and binary vectors. The implementation of such a system presents fundamental computational challenges. For large and sometimes sparse datasets, the demands on time and space are different for real, complex, and binary vectors. To accommodate these demands, the Semantic Vectors package has been carefully adapted and can now switch between different number types comparatively seamlessly. This paper describes the key abstract operations in our semantic vector models, and describes the implementations for real, complex, and binary vectors. We also discuss some of the key questions that arise in the field of quantum interaction and informatics, explaining how the wide availability of modelling options for different number fields will help to investigate some of these questions.

Untanglings: A novel approach to analyzing concurrent systems

Substantial research efforts have been expended to deal with the complexity of concurrent systems that is inherent to their analysis, e.g., works that tackle the well-known state space explosion problem. Approaches differ in the classes of properties that they are able to suitably check and this is largely a result of the way they balance the trade-off between analysis time and space employed to describe a concurrent system. One interesting class of properties is concerned with behavioral characteristics. These properties are conveniently expressed in terms of computations, or runs, in concurrent systems. This article introduces the theory of untanglings that exploits a particular representation of a collection of runs in a concurrent system. It is shown that a representative untangling of a bounded concurrent system can be constructed that captures all and only the behavior of the system. Representative untanglings strike a unique balance between time and space, yet provide a single model for the convenient extraction of various behavioral properties. Performance measurements in terms of construction time and size of representative untanglings with respect to the original specifications of concurrent systems, conducted on a collection of models from practice, confirm the scalability of the approach. Finally, this article demonstrates practical benefits of using representative untanglings when checking various behavioral properties of concurrent systems.

Effect of aging time and Al substitution on the morphology of aluminous goethite

Goethite and Al-substituted goethite were synthesized from the reaction between ferric nitrate and/or aluminum nitrate and potassium hydroxide. XRF, XRD, TEM with EDS were used to characterize the chemical composition, phase and lattice parameters, and morphology of the synthesized products. The results show that d(020) decreases from 4.953 to 4.949 Å and the b dimension decreases from 9.951 Å to 9.906 Å when the aging time increases from 6 days to 42 days for 9.09 mol% Al-substituted goethite. A sample with 9.09 mol% Al substitution in Al-substituted goethite was prepared by a rapid co-precipitation method. In the sample, 13.45 mol%, 12.31 mol% and 5.85 mol% Al substitution with a crystal size of 163, 131, and 45 nm are observed as shown in the TEM images and EDS. The crystal size of goethite is positively related to the degree of Al substitution according to the TEM images and EDS results. Thus, this methodology is proved to be effective to distinguish the morphology of goethite and Al substituted goethite.

Groundwater visualisation system (GVS) : a software framework for integrated display and interrogation of conceptual hydrogeological models, data and time-series animation

Management of groundwater systems requires realistic conceptual hydrogeological models as a framework for numerical simulation modelling, but also for system understanding and communicating this to stakeholders and the broader community. To help overcome these challenges we developed GVS (Groundwater Visualisation System), a stand-alone desktop software package that uses interactive 3D visualisation and animation techniques. The goal was a user-friendly groundwater management tool that could support a range of existing real-world and pre-processed data, both surface and subsurface, including geology and various types of temporal hydrological information. GVS allows these data to be integrated into a single conceptual hydrogeological model. In addition, 3D geological models produced externally using other software packages, can readily be imported into GVS models, as can outputs of simulations (e.g. piezometric surfaces) produced by software such as MODFLOW or FEFLOW. Boreholes can be integrated, showing any down-hole data and properties, including screen information, intersected geology, water level data and water chemistry. Animation is used to display spatial and temporal changes, with time-series data such as rainfall, standing water levels and electrical conductivity, displaying dynamic processes. Time and space variations can be presented using a range of contouring and colour mapping techniques, in addition to interactive plots of time-series parameters. Other types of data, for example, demographics and cultural information, can also be readily incorporated. The GVS software can execute on a standard Windows or Linux-based PC with a minimum of 2 GB RAM, and the model output is easy and inexpensive to distribute, by download or via USB/DVD/CD. Example models are described here for three groundwater systems in Queensland, northeastern Australia: two unconfined alluvial groundwater systems with intensive irrigation, the Lockyer Valley and the upper Condamine Valley, and the Surat Basin, a large sedimentary basin of confined artesian aquifers. This latter example required more detail in the hydrostratigraphy, correlation of formations with drillholes and visualisation of simulation piezometric surfaces. Both alluvial system GVS models were developed during drought conditions to support government strategies to implement groundwater management. The Surat Basin model was industry sponsored research, for coal seam gas groundwater management and community information and consultation. The “virtual” groundwater systems in these 3D GVS models can be interactively interrogated by standard functions, plus production of 2D cross-sections, data selection from the 3D scene, rear end database and plot displays. A unique feature is that GVS allows investigation of time-series data across different display modes, both 2D and 3D. GVS has been used successfully as a tool to enhance community/stakeholder understanding and knowledge of groundwater systems and is of value for training and educational purposes. Projects completed confirm that GVS provides a powerful support to management and decision making, and as a tool for interpretation of groundwater system hydrological processes. A highly effective visualisation output is the production of short videos (e.g. 2–5 min) based on sequences of camera ‘fly-throughs’ and screen images. Further work involves developing support for multi-screen displays and touch-screen technologies, distributed rendering, gestural interaction systems. To highlight the visualisation and animation capability of the GVS software, links to related multimedia hosted online sites are included in the references.

Lingodroids : learning terms for time

For humans and robots to communicate using natural language it is necessary for the robots to develop concepts and associated terms that correspond to the human use of words. Time and space are foundational concepts in human language, and to develop a set of words that correspond to human notions of time and space, it is necessary to take into account the way that they are used in natural human conversations, where terms and phrases such as `soon', `in a while', or `near' are often used. We present language learning robots called Lingodroids that can learn and use simple terms for time and space. In previous work, the Lingodroids were able to learn terms for space. In this work we extend their abilities by adding temporal variables which allow them to learn terms for time. The robots build their own maps of the world and interact socially to form a shared lexicon for location and duration terms. The robots successfully use the shared lexicons to communicate places and times to meet again.

A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D

The space and time fractional Bloch–Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo–Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.

“Would you; could you; on a boat?” Wait-time and travel-time activities during river ferry journeys and their relationship to overall satisfaction in Brisbane Queensland

This study investigates travel behaviour and wait-time activities as a component of passenger satisfaction with public transport in Brisbane, Australia. Australian transport planners recognise a variety of benefits to encouraging a mode shift away from automobile travel in favour of active and public transport use. Efforts to increase public transport ridership have included introducing state of the art passenger information systems, improving physical station access, and integrating system pricing, routes and scheduling for train, bus and ferry. Previous research regarding satisfaction with public transport emphasizes technical dimensions of service quality, including the timing and reliability of service. Those factors might be especially significant for frequent (commuting) travellers who look to balance the cost and efficiency of their travel options. In contrast, infrequent (leisure) passengers may be more concerned with way finding and the sensory experience of the journey. Perhaps due to the small relative proportion of trips made by river ferry compared to bus and rail, this mode of public transport has not received as much attention in travel-behaviour research. This case study of Brisbane’s river ferry system examines ferry passengers at selected terminals during peak and off-peak travel times to find out how travel behaviours and activities correlate to satisfaction with ferry travel. Data include 416 questionnaires completed by passengers intercepted during wait times at seven CityCat terminals in Brisbane. Descriptive statistical analysis revealed associations between specific wait time activities and satisfaction levels that could inform planners seeking to increase ridership and quality of life through ferry-oriented development.

Forward: On the taming of space

An editorial commentary on applications of critical social geography, communications theory and Indigenous studies to the analysis of spatialization in literacy education research.

Making mangoes move

A landscape of mangoes most likely brings to mind a place in a tropical location. By the end of the nineteenth century that place could have been located on any continent in the world. Mangoes were found in geographic locations; in scientific institutions; as crop plants; and as a backyard trees. Here I trace the movement of mangoes Mangifera indica Linn., focusing on the transnational links formed through colonial botanic gardens in Australia. Botanic gardens were largely understood through their work in establishing economically successful plantation crops, such as sugar and tea. Mangoes were not a success crop of the age of botanic imperialism. Instead, mangoes were simply one species among the millions of plants that botanic gardens moved in addition to these well known commercial crops. Colonial science moved plants for a myriad of other types of reasons, for ornament, for curiosity, for lesser commercial purposes and for pure science. In each site the mango emerged, the discourses and technologies that traveled with it changed according to local needs. Indeed, rather than finding mangoes located in one place, tracing their movement demonstrates that this was an extended landscape connecting these things across time and space...

Plasmodium vivax malaria incidence over time and its association with temperature and rainfall in four counties of Yunnan Province, China

Background Transmission of Plasmodium vivax malaria is dependent on vector availability, biting rates and parasite development. In turn, each of these is influenced by climatic conditions. Correlations have previously been detected between seasonal rainfall, temperature and malaria incidence patterns in various settings. An understanding of seasonal patterns of malaria, and their weather drivers, can provide vital information for control and elimination activities. This research aimed to describe temporal patterns in malaria, rainfall and temperature, and to examine the relationships between these variables within four counties of Yunnan Province, China. Methods Plasmodium vivax malaria surveillance data (1991–2006), and average monthly temperature and rainfall were acquired. Seasonal trend decomposition was used to examine secular trends and seasonal patterns in malaria. Distributed lag non-linear models were used to estimate the weather drivers of malaria seasonality, including the lag periods between weather conditions and malaria incidence. Results There was a declining trend in malaria incidence in all four counties. Increasing temperature resulted in increased malaria risk in all four areas and increasing rainfall resulted in increased malaria risk in one area and decreased malaria risk in one area. The lag times for these associations varied between areas. Conclusions The differences detected between the four counties highlight the need for local understanding of seasonal patterns of malaria and its climatic drivers.