Constant Curvature Coefficients and Exact Solutions in Fractional Gravity and Geometric Mechanics

We study fractional configurations in gravity theories and Lagrange mechanics. The approach is based on Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists in a proof that for corresponding classes of nonholonomic distributions a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dynamics of interactions and constraints into the geometry of curve flows and solitonic hierarchies.Comment: latex2e, 11pt, 27 pages, the variant accepted to CEJP; added and up-dated reference

Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

Recently, the fractional Bloch-Torrey model has been used to study anomalous diffusion in the human brain. In this paper, we consider three types of space and time fractional Bloch-Torrey equations in two dimensions: Model-1 with the Riesz fractional derivative; Model-2 with the one-dimensional fractional Laplacian operator; and Model-3 with the two-dimensional fractional Laplacian operator. Firstly, we propose a spatially second-order accurate implicit numerical method for Model-1 whereby we discretize the Riesz fractional derivative using a fractional centered difference. We consider a finite domain where the time and space derivatives are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Secondly, we utilize the matrix transfer technique for solving Model-2 and Model-3. Finally, some numerical results are given to show the behaviours of these three models especially on varying domain sizes with zero Dirichlet boundary conditions

Nonequilibrium radiation measurements and modelling relevant to Titan entry

An update to a collisional-radiative model developed by Magin1 for Huygens Titan atmospheric entry is proposed. The model is designed to predict the nonequilibrium populations and the radiation emitted from cyanogen and nitrogen during the entry of the Huygens probe into the Titan atmosphere. Radiation during Titan entry is important at lower speeds (around 5 – 6 km/s) more so than other planetary entries due to the formation of cyanogen in the shock layer, which is a highly radiative species. The model has been tested against measurements obtained with the EAST shock tube of NASA Ames Research Centre.1,2 The motivation for the update is due to the large discrepancies shown in the postshock fall-off rates of the radiation when compared to the experimental EAST shock tube test results. Modifications were made to the reaction rates used to calculate the species concentrations in the flow field. The reaction that was deemed most influential for the radiation fall off rate was the dissociation of molecular nitrogen. The model with modified reaction rates showed significantly better agreement with the EAST data. This paper also includes experimental results for radiation and spectra for Titan entry. Experiments were performed on the University of Queensland's X2 expansion tube. Spectra were recorded at various positions behind the shock. This enabled the construction of radiation profiles for Titan entry, as well as wavelength plots to identify various radiating species, in this case, predominately CN violet. This paper includes radiation profiles to compare with experiments performed at NASA Ames. It is planned that further experiments will be performed to cover a larger pressure range than NASA Ames. Good qualitative agreement has so far been obtained between our data and NASA Ames, however, it should be noted at the time of printing, the experimental spectrum have not been calibrated absolutely

Fractional Euler-Lagrange differential equations via Caputo derivatives

We review some recent results of the fractional variational calculus. Necessary optimality conditions of Euler-Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are considered: with fixed or free boundary conditions, and in presence of integral constraints that also depend on Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form will appear as Chapter 9 of the book Fractional Dynamics and Control, D. Baleanu et al. (eds.), Springer New York, 2012, DOI:10.1007/978-1-4614-0457-6_9, in pres

Fractional Hamilton formalism within Caputo's derivative

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canonical Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler-Lagrange formulations lead to the same set of equations.Comment: 8 page

Australian general practice trainees’ exposure to ophthalmic problems and implications for training: A cross-sectional analysis

INTRODUCTION: Eye conditions are common presentations in Australian general practice, with the potential for serious sequelae. Pre-vocational ophthalmology training for General Practitioner (GP) trainees is limited. AIM: To describe the rate, nature and associations of ophthalmic problems managed by Australian GP trainees, and derive implications for education and training. METHODS: Cross-sectional analysis from an ongoing cohort study of GP trainees’ clinical consultations. Trainees recorded demographic, clinical and educational details of consecutive patient consultations. Descriptive analyses report trainee, patient and practice demographics. Proportions of all problems managed in these consultations that were ophthalmology-related were calculated with 95% confidence intervals (CI). Associations were tested using simple logistic regression within the generalised estimating equations (GEE) framework. RESULTS: In total, 884 trainees returned data on 184,476 individual problems or diagnoses from 118,541 encounters. There were 2649 ophthalmology-related problems, equating to 1.4% (95% CI: 1.38-1.49) of all problems managed. The most common eye presentations were conjunctivitis (32.5% of total problems), eyelid problems (14.9%), foreign body (5.3%) and dry eye (4.7%). Statistically significant associations were male trainee; male patient and patient aged 14 years or under; the problem being new and the patient being new to both trainee and practice; urban and of higher socioeconomic status practice location; the practice nurse not being involved; planned follow up not arranged; referral made; in-consultation information sought; and learning goals generated. DISCUSSION: Trainees have comparable ophthalmology exposure to established GPs. However, associations with referral and information-seeking suggest GP trainees find ophthalmic problems challenging, reinforcing the critical importance of appropriate training