A Caputo fractional derivative of a function with respect to another function

In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor’s Theorem, Fermat’s Theorem, etc., are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.

Fractional differential equations with dependence on the Caputo-Katugampola derivative

In this paper we present a new type of fractional operator, the Caputo–Katugampola derivative. The Caputo and the Caputo–Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy type problem, with dependence on the Caputo–Katugampola derivative, is proven. A decomposition formula for the Caputo–Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation.

Modeling some real phenomena by fractional differential equations

This paper deals with fractional differential equations, with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional differential equations may model more efficiently certain problems than ordinary differential equations. A numerical optimization approach based on least squares approximation is used to determine the order of the fractional operator that better describes real data, as well as other related parameters.

Finite time extinction for nonlinear fractional evolution equations and related properties

The finite time extinction phenomenon (the solution reaches an equilibrium after a finite time) is peculiar to certain nonlinear problems whose solutions exhibit an asymptotic behavior entirely different from the typical behavior of solutions associated to linear problems. The main goal of this work is twofold. Firstly, we extend some of the results known in the literature to the case in which the ordinary time derivative is considered jointly with a fractional time differentiation. Secondly, we consider the limit case when only the fractional derivative remains. The latter is the most extraordinary case, since we prove that the finite time extinction phenomenon still appears, even with a non-smooth profile near the extinction time. Some concrete examples of quasi-linear partial differential operators are proposed. Our results can also be applied in the framework of suitable nonlinear Volterra integro-differential equations.

Finite time extinction for nonlinear fractional evolution equations and related properties.

The finite time extinction phenomenon (the solution reaches an equilibrium after a finite time) is peculiar to certain nonlinear problems whose solutions exhibit an asymptotic behavior entirely different from the typical behavior of solutions associated to linear problems. The main goal of this work is twofold. Firstly, we extend some of the results known in the literature to the case in which the ordinary time derivative is considered jointly with a fractional time differentiation. Secondly, we consider the limit case when only the fractional derivative remains. The latter is the most extraordinary case, since we prove that the finite time extinction phenomenon still appears, even with a non-smooth profile near the extinction time. Some concrete examples of quasi-linear partial differential operators are proposed. Our results can also be applied in the framework of suitable nonlinear Volterra integro-differential equations.

Nature of the spin-glass phase in dense packings of Ising dipoles with random anisotropy axes

By Monte Carlo simulations, we study the character of the spinglass (SG) phase in dense disordered packings of magnetic nanoparticles (NPs). We focus on NPs which have large uniaxial anisotropies and can be well represented as Ising dipoles. Dipoles are placed on SC lattices and point along randomly oriented axes. From the behaviour of a SG correlation length we determine the transition temperature Tc between the paramagnetic and a SG phase. For temperatures well below Tc we find distributions of the SG overlap parameter q that are strongly sample-dependent and exhibit several spikes. We find that the average width of spikes, and the fraction of samples with spikes higher than a certain threshold does not vary appreciably with the system sizes studied. We compare these results with the ones found previously for 3D site-diluted systems of parallel Ising dipoles and with the behaviour of the Sherrington-Kirkpatrick model.

Part-time and fractional staff in the learning and skills sector

This Universities and College Union Launch Event presentation reported on the findings of Learning and Skills Research Network (LSRN) London and South East (LSE) Regional Research Project. The presentation reflected on research carried out during 2002-06 on the development and deployment of part-time staff in the Learning and Skills Sector. Although the lifelong learning sector is the largest UK education sector, little attention has as yet been paid to the role of LSC sector part-time staff. Worrying trends of an increasing casualisation of staffing have been reported. The role of part-timers as highly committed (philanthropic) but generally underpaid and exploited staff (ragged-trousered) emerged from the data collected by this investigation, which examined the role of part-timers in several colleges and adult education institutions in London and the South East. The metaphor of the 'ragged-trousered philanthropist' was consciously selected to investigate the interactivity between philantrophy, employment practices for PT staff, and education as social action, in addressing the need for good practice to achieve quality outcomes in learning and teaching. The results are to some extent transferable to other education and training sectors employing part-time staff, e.g. higher education institutions and work-based training organisations.

Hydrogeophysical Characterization of Anisotropy in the Biscayne Aquifer Using Geophysical Methods

The anisotropy of the Biscayne Aquifer which serves as the source of potable water for Miami-Dade County was investigated by applying geophysical methods. Electrical resistivity imaging, self potential and ground penetration radar techniques were employed in both regional and site specific studies. In the regional study, electrical anisotropy and resistivity variation with depth were investigated with azimuthal square array measurements at 13 sites. The observed coefficient of electrical anisotropy ranged from 1.01 to 1.36. The general direction of measured anisotropy is uniform for most sites and trends W-E or SE-NW irrespective of depth. Measured electrical properties were used to estimate anisotropic component of the secondary porosity and hydraulic anisotropy which ranged from 1 to 11% and 1.18 to 2.83 respectively. 1-D sounding analysis was used to models the variation of formation resistivity with depth. Resistivities decreased from NW (close to the margins of the everglades) to SE on the shores of Biscayne Bay. Porosity calculated from Archie's law, ranged from 18 to 61% with higher values found along the ridge. Higher anisotropy, porosities and hydraulic conductivities were on the Atlantic Coastal Ridge and lower values at low lying areas west of the ridge. The cause of higher anisotropy and porosity is attributed to higher dissolution rates of the oolitic facies of the Miami Formation composing the ridge. The direction of minimum resistivity from this study is similar to the predevelopment groundwater flow direction indicated in published modeling studies. Detailed investigations were carried out to evaluate higher anisotropy at West Perrine Park located on the ridge and Snapper Creek Municipal well field where the anisotropy trend changes with depth. The higher anisotropy is attributed to the presence of solution cavities oriented in the E-SE direction on the ridge. Similarly, the change in hydraulic anisotropy at the well field might be related to solution cavities, the surface canal and groundwater extraction wells.

Utility and safety of invasive (fractional flow reserve) and non-invasive (cardiac magnetic resonance imaging) diagnostic tests in patients with NSTEMI

A prospective randomised controlled clinical trial of treatment decisions informed by invasive functional testing of coronary artery disease severity compared with standard angiography-guided management was implemented in 350 patients with a recent non-ST elevation myocardial infarction (NSTEMI) admitted to 6 hospitals in the National Health Service. The main aims of this study were to examine the utility of both invasive fractional flow reserve (FFR) and non-invasive cardiac magnetic resonance imaging (MRI) amongst patients with a recent diagnosis of NSTEMI. In summary, the findings of this thesis are: (1) the use of FFR combined with intravenous adenosine was feasible and safe amongst patients with NSTEMI and has clinical utility; (2) there was discordance between the visual, angiographic estimation of lesion significance and FFR; (3). The use of FFR led to changes in treatment strategy and an increase in prescription of medical therapy in the short term compared with an angiographically guided strategy; (4) in the incidence of major adverse cardiac events (MACE) at 12 months follow up was similar in the two groups. Cardiac MRI was used in a subset of patients enrolled in two hospitals in the West of Scotland. T1 and T2 mapping methods were used to delineate territories of acute myocardial injury. T1 and T2 mapping were superior when compared with conventional T2-weighted dark blood imaging for estimation of the ischaemic area-at-risk (AAR) with less artifact in NSTEMI. There was poor correlation between the angiographic AAR and MRI methods of AAR estimation in patients with NSTEMI. FFR had a high accuracy at predicting inducible perfusion defects demonstrated on stress perfusion MRI. This thesis describes the largest randomized trial published to date specifically looking at the clinical utility of FFR in the NSTEMI population. We have provided evidence of the diagnostic and clinical utility of FFR in this group of patients and provide evidence to inform larger studies. This thesis also describes the largest ever MRI cohort, including with myocardial stress perfusion assessments, specifically looking at the NSTEMI population. We have demonstrated the diagnostic accuracy of FFR to predict reversible ischaemia as referenced to a non-invasive gold standard with MRI. This thesis has also shown the futility of using dark blood oedema imaging amongst all comer NSTEMI patients when compared to novel T1 and T2 mapping methods.

Does Final Energy Demand in Portugal Exhibit Long Memory? A Fractional Integration Analysis

In this paper, we measure the degree of fractional integration in final energy demand in Portugal using an ARFIMA model with and without adjustments for seasonality. We consider aggregate energy demand as well as final demand for petroleum, electricity, coal, and natural gas. Our findings suggest the presence of long memory in all of the components of energy demand. All fractional-difference parameters are positive and lower than 0.5 indicating that the series are stationary, although with mean reversion patterns slower than in the typical short-run processes. These results have important implications for the design of energy policies. As a result of the long-memory in final energy demand, the effects of temporary policy shocks will tend to disappear slowly. This means that even transitory shocks have long lasting effects. Given the temporary nature of these effects, however, permanent effects on final energy demand require permanent policies. This is unlike what would be suggested by the more standard, but much more limited, unit root approach, which would incorrectly indicate that even transitory policies would have permanent effects

Do Global CO2 Emissions from Fossil-Fuel Consumption Exhibit Long Memory? A Fractional Integration Analysis

In this article we use an autoregressive fractionally integrated moving average approach to measure the degree of fractional integration of aggregate world CO2 emissions and its five components – coal, oil, gas, cement, and gas flaring. We find that all variables are stationary and mean reverting, but exhibit long-term memory. Our results suggest that both coal and oil combustion emissions have the weakest degree of long-range dependence, while emissions from gas and gas flaring have the strongest. With evidence of long memory, we conclude that transitory policy shocks are likely to have long-lasting effects, but not permanent effects. Accordingly, permanent effects on CO2 emissions require a more permanent policy stance. In this context, if one were to rely only on testing for stationarity and non-stationarity, one would likely conclude in favour of non-stationarity, and therefore that even transitory policy shocks