Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.

The relationship between chronic disease self-efficacy and nutritional status, functional ability and quality of life in older adults at risk of hospital readmission

Background and significance: Older adults with chronic diseases are at increasing risk of hospital admission and readmission. Approximately 75% of adults have at least one chronic condition, and the odds of developing a chronic condition increases with age. Chronic diseases consume about 70% of the total Australian health expenditure, and about 59% of hospital events for chronic conditions are potentially preventable. These figures have brought to light the importance of the management of chronic disease among the growing older population. Many studies have endeavoured to develop effective chronic disease management programs by applying social cognitive theory. However, limited studies have focused on chronic disease self-management in older adults at high risk of hospital readmission. Moreover, although the majority of studies have covered wide and valuable outcome measures, there is scant evidence on examining the fundamental health outcomes such as nutritional status, functional status and health-related quality of life. Aim: The aim of this research was to test social cognitive theory in relation to self-efficacy in managing chronic disease and three health outcomes, namely nutritional status, functional status, and health-related quality of life, in older adults at high risk of hospital readmission. Methods: A cross-sectional study design was employed for this research. Three studies were undertaken. Study One examined the nutritional status and validation of a nutritional screening tool; Study Two explored the relationships between participants. characteristics, self-efficacy beliefs, and health outcomes based on the study.s hypothesized model; Study Three tested a theoretical model based on social cognitive theory, which examines potential mechanisms of the mediation effects of social support and self-efficacy beliefs. One hundred and fifty-seven patients aged 65 years and older with a medical admission and at least one risk factor for readmission were recruited. Data were collected from medical records on demographics, medical history, and from self-report questionnaires. The nutrition data were collected by two registered nurses. For Study One, a contingency table and the kappa statistic was used to determine the validity of the Malnutrition Screening Tool. In Study Two, standard multiple regression, hierarchical multiple regression and logistic regression were undertaken to determine the significant influential predictors for the three health outcome measures. For Study Three, a structural equation modelling approach was taken to test the hypothesized self-efficacy model. Results: The findings of Study One suggested that a high prevalence of malnutrition continues to be a concern in older adults as the prevalence of malnutrition was 20.6% according to the Subjective Global Assessment. Additionally, the findings confirmed that the Malnutrition Screening Tool is a valid nutritional screening tool for hospitalized older adults at risk of readmission when compared to the Subjective Global Assessment with high sensitivity (94%), and specificity (89%) and substantial agreement between these two methods (k = .74, p < .001; 95% CI .62-.86). Analysis data for Study Two found that depressive symptoms and perceived social support were the two strongest influential factors for self-efficacy in managing chronic disease in a hierarchical multiple regression. Results of multivariable regression models suggested advancing age, depressive symptoms and less tangible support were three important predictors for malnutrition. In terms of functional status, a standard regression model found that social support was the strongest predictor for the Instrumental Activities of Daily Living, followed by self-efficacy in managing chronic disease. The results of standard multiple regression revealed that the number of hospital readmission risk factors adversely affected the physical component score, while depressive symptoms and self-efficacy beliefs were two significant predictors for the mental component score. In Study Three, the results of the structural equation modelling found that self-efficacy partially mediated the effect of health characteristics and depression on health-related quality of life. The health characteristics had strong direct effects on functional status and body mass index. The results also indicated that social support partially mediated the relationship between health characteristics and functional status. With regard to the joint effects of social support and self-efficacy, social support fully mediated the effect of health characteristics on self-efficacy, and self-efficacy partially mediated the effect of social support on functional status and health-related quality of life. The results also demonstrated that the models fitted the data well with relative high variance explained by the models, implying the hypothesized constructs under discussion were highly relevant, and hence the application for social cognitive theory in this context was supported. Conclusion: This thesis highlights the applicability of social cognitive theory on chronic disease self-management in older adults at risk of hospital readmission. Further studies are recommended to validate and continue to extend the development of social cognitive theory on chronic disease self-management in older adults to improve their nutritional and functional status, and health-related quality of life.

Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise

In this paper we extend the ideas of Brugnano, Iavernaro and Trigiante in their development of HBVM($s,r$) methods to construct symplectic Runge-Kutta methods for all values of $s$ and $r$ with $s\geq r$. However, these methods do not see the dramatic performance improvement that HBVMs can attain. Nevertheless, in the case of additive stochastic Hamiltonian problems an extension of these ideas, which requires the simulation of an independent Wiener process at each stage of a Runge-Kutta method, leads to methods that have very favourable properties. These ideas are illustrated by some simple numerical tests for the modified midpoint rule.

Practical physical activity measurement in youth : a review of contemporary approaches

Background: The accurate evaluation of physical activity levels amongst youth is critical for quantifying physical activity behaviors and evaluating the effect of physical activity interventions. The purpose of this review is to evaluate contemporary approaches to physical activity evaluation amongst youth. Data sources: The literature from a range of sources was reviewed and synthesized to provide an overview of contemporary approaches for measuring youth physical activity. Results: Five broad categories are described: self-report, instrumental movement detection, biological approaches, direct observation, and combined methods. Emerging technologies and priorities for future research are also identified. Conclusions: There will always be a trade-off between accuracy and available resources when choosing the best approach for measuring physical activity amongst youth. Unfortunately, cost and logistical challenges may prohibit the use of "gold standard" physical activity measurement approaches such as doubly labelled water. Other objective methods such as heart rate monitoring, accelerometry, pedometry, indirect calorimetry, or a combination of measures have the potential to better capture the duration and intensity of physical activity, while self-reported measures are useful for capturing the type and context of activity.

Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation

Anomalous subdiffusion equations have in recent years received much attention. In this paper, we consider a two-dimensional variable-order anomalous subdiffusion equation. Two numerical methods (the implicit and explicit methods) are developed to solve the equation. Their stability, convergence and solvability are investigated by the Fourier method. Moreover, the effectiveness of our theoretical analysis is demonstrated by some numerical examples. © 2011 American Mathematical Society.

Numerical methods and analysis for a class of fractional advection-dispersion models

In this paper, a class of fractional advection–dispersion models (FADMs) is considered. These models include five fractional advection–dispersion models, i.e., the time FADM, the mobile/immobile time FADM with a time Caputo fractional derivative 0 < γ < 1, the space FADM with two sides Riemann–Liouville derivatives, the time–space FADM and the time fractional advection–diffusion-wave model with damping with index 1 < γ < 2. These equations can be used to simulate the regional-scale anomalous dispersion with heavy tails. We propose computationally effective implicit numerical methods for these FADMs. The stability and convergence of the implicit numerical methods are analysed and compared systematically. Finally, some results are given to demonstrate the effectiveness of theoretical analysis.

The SAGE Handbook of Criminological Research Methods

Conducting research into crime and criminal justice carries unique challenges. This Handbook focuses on the application of 'methods' to address the core substantive questions that currently motivate contemporary criminological research. It maps a canon of methods that are more elaborated than in most other fields of social science, and the intellectual terrain of research problems with which criminologists are routinely confronted. Drawing on exemplary studies, chapters in each section illustrate the techniques (qualitative and quantitative) that are commonly applied in empirical studies, as well as the logic of criminological enquiry. Organized into five sections, each prefaced by an editorial introduction, the Handbook covers: • Crime and Criminals • Contextualizing Crimes in Space and Time: Networks, Communities and Culture • Perceptual Dimensions of Crime • Criminal Justice Systems: Organizations and Institutions • Preventing Crime and Improving Justice Edited by leaders in the field of criminological research, and with contributions from internationally renowned experts, The SAGE Handbook of Criminological Research Methods is set to become the definitive resource for postgraduates, researchers and academics in criminology, criminal justice, policing, law, and sociology.

A comparison of methods for classifying clinical samples based on proteomics data : a case study for statistical and machine learning approaches

The discovery of protein variation is an important strategy in disease diagnosis within the biological sciences. The current benchmark for elucidating information from multiple biological variables is the so called “omics” disciplines of the biological sciences. Such variability is uncovered by implementation of multivariable data mining techniques which come under two primary categories, machine learning strategies and statistical based approaches. Typically proteomic studies can produce hundreds or thousands of variables, p, per observation, n, depending on the analytical platform or method employed to generate the data. Many classification methods are limited by an n≪p constraint, and as such, require pre-treatment to reduce the dimensionality prior to classification. Recently machine learning techniques have gained popularity in the field for their ability to successfully classify unknown samples. One limitation of such methods is the lack of a functional model allowing meaningful interpretation of results in terms of the features used for classification. This is a problem that might be solved using a statistical model-based approach where not only is the importance of the individual protein explicit, they are combined into a readily interpretable classification rule without relying on a black box approach. Here we incorporate statistical dimension reduction techniques Partial Least Squares (PLS) and Principal Components Analysis (PCA) followed by both statistical and machine learning classification methods, and compared them to a popular machine learning technique, Support Vector Machines (SVM). Both PLS and SVM demonstrate strong utility for proteomic classification problems.