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.

Multimodal and monomodal discourses of marketization in higher education : power, ideology, and the absence of the image

The ubiquity of multimodality in hypermedia environments is undeniable. Bezemer and Kress (2008) have argued that writing has been displaced by image as the central mode for representation. Given the current technical affordances of digital technology and user-friendly interfaces that enable the ease of multimodal design, the conspicuous absence of images in certain domains of cyberspace is deserving of critical analysis. In this presentation, I examine the politics of discourses implicit within hypertextual spaces, drawing textual examples from a higher education website. I critically examine the role of writing and other modes of production used in what Fairclough (1993) refers to as discourses of marketisation in higher education, tracing four pervasive discourses of teaching and learning in the current economy: i) materialization, ii) personalization, iii) technologisation, and iv) commodification (Fairclough, 1999). Each of these arguments is supported by the critical analysis of multimodal texts. The first is a podcast highlighting the new architectonic features of a university learning space. The second is a podcast and transcript of a university Open Day interview with prospective students. The third is a time-lapse video showing the construction of a new science and engineering precinct. These three multimodal texts contrast a final web-based text that exhibits a predominance of writing and the powerful absence or silencing of the image. I connect the weightiness of words and the function of monomodality in the commodification of discourses, and its resistance to the multimodal affordances of web-based technologies, and how this is used to establish particular sets of subject positions and ideologies through which readers are constrained to occupy. Applying principles of critical language study by theorists that include Fairclough, Kress, Lemke, and others whose semiotic analysis of texts focuses on the connections between language, power, and ideology, I demonstrate how the denial of image and the privileging of written words in the multimodality of cyberspace is an ideological effect to accentuate the dominance of the institution.

Corneal nerve structure and function as markers of diabetic neuropathy

Diabetic neuropathy is a significant clinical problem that currently has no effective therapy, and in advanced cases, leads to foot ulceration and lower limb amputation. The accurate detection, characterisation and quantification of this condition are important in order to define at-risk patients, anticipate deterioration, monitor progression and assess new therapies. This thesis evaluates novel corneal methods of assessing diabetic neuropathy. Over the past several years two new non-invasive corneal markers have emerged, and in cross-sectional studies have demonstrated their ability to stratify the severity of this disease. Corneal confocal microscopy (CCM) allows quantification of corneal nerve parameters and non-contact corneal aesthesiometry (NCCA), the presumed functional correlate of corneal structure, assesses the sensitivity of the cornea. Both these techniques are quick to perform, produce little or no discomfort for the patient, and with automatic analysis paradigms developed, are suitable for clinical settings. Each has advantages and disadvantages over established techniques for assessing diabetic neuropathy. New information is presented regarding measurement bias of CCM images, and a unique sampling paradigm and associated accuracy determination method of combinations is described. A novel high-speed corneal nerve mapping procedure has been developed and application of this procedure in individuals with neuropathy has revealed regions of sub-basal nerve plexus that dictate further evaluation, as they appear to show earlier signs of damage than the central region of the cornea that has to date been examined. The discriminative capacity of corneal sensitivity measured by NCCA is revealed to have reasonable potential as a marker of diabetic neuropathy. Application of these new corneal markers for longitudinal evaluation of diabetic neuropathy has the potential to reduce dependence on more invasive, costly, and time-consuming assessments, such as skin biopsy.

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.

Stability and convergence of implicit numerical methods for a class of fractional advection-dispersion models

In this paper, a class of fractional advection-dispersion models (FADM) is investigated. These models include five fractional advection-dispersion models: the immobile, mobile/immobile time FADM with a temporal fractional derivative 0 < γ < 1, the space FADM with skewness, both the time and space FADM and the time fractional advection-diffusion-wave model with damping with index 1 < γ < 2. They describe nonlocal dependence on either time or space, or both, to explain the development of anomalous dispersion. These equations can be used to simulate regional-scale anomalous dispersion with heavy tails, for example, the solute transport in watershed catchments and rivers. We propose computationally effective implicit numerical methods for these FADM. The stability and convergence of the implicit numerical methods are analyzed and compared systematically. Finally, some results are given to demonstrate the effectiveness of our theoretical analysis.

Human proximal tubule epithelial cells modulate autologous dendritic cell function

Background We have previously demonstrated that human kidney proximal tubule epithelial cells (PTEC) are able to modulate autologous T and B lymphocyte responses. It is well established that dendritic cells (DC) are responsible for the initiation and direction of adaptive immune responses and that these cells occur in the renal interstitium in close apposition to PTEC under inflammatory disease settings. However, there is no information regarding the interaction of PTEC with DC in an autologous human context. Methods Human monocytes were differentiated into monocyte-derived DC (MoDC) in the absence or presence of primary autologous activated PTEC and matured with polyinosinic:polycytidylic acid [poly(I:C)], while purified, pre-formed myeloid blood DC (CD1c+ BDC) were cultured with autologous activated PTEC in the absence or presence of poly(I:C) stimulation. DC responses were monitored by surface antigen expression, cytokine secretion, antigen uptake capacity and allogeneic T-cell-stimulatory ability. Results The presence of autologous activated PTEC inhibited the differentiation of monocytes to MoDC. Furthermore, MoDC differentiated in the presence of PTEC displayed an immature surface phenotype, efficient phagocytic capacity and, upon poly(I:C) stimulation, secreted low levels of pro-inflammatory cytokine interleukin (IL)-12p70, high levels of anti-inflammatory cytokine IL-10 and induced weak Th1 responses. Similarly, pre-formed CD1c+ BDC matured in the presence of PTEC exhibited an immature tolerogenic surface phenotype, strong endocytic and phagocytic ability and stimulated significantly attenuated T-cell proliferative responses. Conclusions Our data suggest that activated PTEC regulate human autologous immunity via complex interactions with DC. The ability of PTEC to modulate autologous DC function has important implications for the dampening of pro-inflammatory immune responses within the tubulointerstitium in renal injuries. Further dissection of the mechanisms of PTEC modulation of autologous immune responses may offer targets for therapeutic intervention in renal medicine.