Some Remarks on the Approximation of Transitional Density in Stochastic Differential Equations

Aijt-Sahalia (2002) introduced a method to estimate transitional probability densities of di®usion processes by means of Hermite expansions with coe±cients determined by means of Taylor series. This note describes a numerical procedure to ¯nd these coe±cients based on the calculation of moments. One advantage of this procedure is that it can be used e®ectively when the mathematical operations required to ¯nd closed-form expressions for these coe±cients are otherwise infeasible.

Non-alcoholic fatty liver disease : can ultrasound assist in early diagnosis of prediabetes and delay progression to type2 diabetes mellitus

Non Alcoholic Fatty Liver Disease (NAFLD) is a condition that is frequently seen but seldom investigated. Until recently, NAFLD was considered benign, self-limiting and unworthy of further investigation. This opinion is based on retrospective studies with relatively small numbers and scant follow-up of histology data. (1) The prevalence for adults, in the USA is, 30%, and NAFLD is recognized as a common and increasing form of liver disease in the paediatric population (1). Australian data, from New South Wales, suggests the prevalence of NAFLD in “healthy” 15 year olds as being 10%.(2) Non-alcoholic fatty liver disease is a condition where fat progressively invades the liver parenchyma. The degree of infiltration ranges from simple steatosis (fat only) to steatohepatitis (fat and inflammation) steatohepatitis plus fibrosis (fat, inflammation and fibrosis) to cirrhosis (replacement of liver texture by scarred, fibrotic and non functioning tissue).Non-alcoholic fatty liver is diagnosed by exclusion rather than inclusion. None of the currently available diagnostic techniques -liver biopsy, liver function tests (LFT) or Imaging; ultrasound, Computerised tomography (CT) or Magnetic Resonance Imaging (MRI) are specific for non-alcoholic fatty liver. An association exists between NAFLD, Non Alcoholic Steatosis Hepatitis (NASH) and irreversible liver damage, cirrhosis and hepatoma. However, a more pervasive aspect of NAFLD is the association with Metabolic Syndrome. This Syndrome is categorised by increased insulin resistance (IR) and NAFLD is thought to be the hepatic representation. Those with NAFLD have an increased risk of death (3) and it is an independent predictor of atherosclerosis and cardiovascular disease (1). Liver biopsy is considered the gold standard for diagnosis, (4), and grading and staging, of non-alcoholic fatty liver disease. Fatty-liver is diagnosed when there is macrovesicular steatosis with displacement of the nucleus to the edge of the cell and at least 5% of the hepatocytes are seen to contain fat (4).Steatosis represents fat accumulation in liver tissue without inflammation. However, it is only called non-alcoholic fatty liver disease when alcohol - >20gms-30gms per day (5), has been excluded from the diet. Both non-alcoholic and alcoholic fatty liver are identical on histology. (4).LFT’s are indicative, not diagnostic. They indicate that a condition may be present but they are unable to diagnosis what the condition is. When a patient presents with raised fasting blood glucose, low HDL (high density lipoprotein), and elevated fasting triacylglycerols they are likely to have NAFLD. (6) Of the imaging techniques MRI is the least variable and the most reproducible. With CT scanning liver fat content can be semi quantitatively estimated. With increasing hepatic steatosis, liver attenuation values decrease by 1.6 Hounsfield units for every milligram of triglyceride deposited per gram of liver tissue (7). Ultrasound permits early detection of fatty liver, often in the preclinical stages before symptoms are present and serum alterations occur. Earlier, accurate reporting of this condition will allow appropriate intervention resulting in better patient health outcomes. References 1. Chalasami N. Does fat alone cause significant liver disease: It remains unclear whether simple steatosis is truly benign. American Gastroenterological Association Perspectives, February/March 2008 www.gastro.org/wmspage.cfm?parm1=5097 Viewed 20th October, 2008 2. Booth, M. George, J.Denney-Wilson, E: The population prevalence of adverse concentrations with adiposity of liver tests among Australian adolescents. Journal of Paediatrics and Child Health.2008 November 3. Catalano, D, Trovato, GM, Martines, GF, Randazzo, M, Tonzuso, A. Bright liver, body composition and insulin resistance changes with nutritional intervention: a follow-up study .Liver Int.2008; February 1280-9 4. Choudhury, J, Sanysl, A. Clinical aspects of Fatty Liver Disease. Semin in Liver Dis. 2004:24 (4):349-62 5. Dionysus Study Group. Drinking factors as cofactors of risk for alcohol induced liver change. Gut. 1997; 41 845-50 6. Preiss, D, Sattar, N. Non-alcoholic fatty liver disease: an overview of prevalence, diagnosis, pathogenesis and treatment considerations. Clin Sci.2008; 115 141-50 7. American Gastroenterological Association. Technical review on nonalcoholic fatty liver disease. Gastroenterology.2002; 123: 1705-25

Delay-dependent robust H-infinity control for uncertain systems with time-varying delay

This paper proposes a new approach for delay-dependent robust H-infinity stability analysis and control synthesis of uncertain systems with time-varying delay. The key features of the approach include the introduction of a new Lyapunov–Krasovskii functional, the construction of an augmented matrix with uncorrelated terms, and the employment of a tighter bounding technique. As a result, significant performance improvement is achieved in system analysis and synthesis without using either free weighting matrices or model transformation. Examples are given to demonstrate the effectiveness of the proposed approach.

Constructive development of the solutions of linear equations in introductory ordinary differential equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognising what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced studies in mathematics. In this study we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery based approach where students employ their existing skills as a framework for constructing the solutions of first and second order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first year undergraduate class. Finally we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

New Approach on Robust Delay-Dependent H∞ Control for Uncertain T-S Fuzzy Systems With Interval Time-Varying Delay

This paper investigates the robust H∞ control for Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delay. By employing a new and tighter integral inequality and constructing an appropriate type of Lyapunov functional, delay-dependent stability criteria are derived for the control problem. Because neither any model transformation nor free weighting matrices are employed in our theoretical derivation, the developed stability criteria significantly improve and simplify the existing stability conditions. Also, the maximum allowable upper delay bound and controller feedback gains can be obtained simultaneously from the developed approach by solving a constrained convex optimization problem. Numerical examples are given to demonstrate the effectiveness of the proposed methods.

Closed-form design equations for decoupling networks of small arrays

Small element spacing in compact arrays results in strong mutual coupling between the array elements. A decoupling network consisting of reactive cross-coupling elements can alleviate problems associated with the coupling. Closed-form design equations for the decoupling networks of symmetrical arrays with two or three elements are presented.