Fractional Calculus: Quo Vadimus? (Where Are We Going?)

Discussions under this title were held during a special session in frames of the International Conference “Fractional Differentiation and Applications” (ICFDA ’14) held in Catania (Italy), 23-25 June 2014, see details at http://www.icfda14.dieei.unict.it/. Along with the presentations made during this session, we include here some contributions by the participants sent afterwards and also by few colleagues planning but failed to attend. The intention of this special session was to continue the useful traditions from the first conferences on the Fractional Calculus (FC) topics, to pose open problems, challenging hypotheses and questions “where to go”, to discuss them and try to find ways to resolve.

Efficient Legendre spectral tau algorithm for solving two-sided space-time Caputo fractional advection-dispersion equation

In this paper we present the operational matrices of the left Caputo fractional derivative, right Caputo fractional derivative and Riemann–Liouville fractional integral for shifted Legendre polynomials. We develop an accurate numerical algorithm to solve the two-sided space–time fractional advection–dispersion equation (FADE) based on a spectral shifted Legendre tau (SLT) method in combination with the derived shifted Legendre operational matrices. The fractional derivatives are described in the Caputo sense. We propose a spectral SLT method, both in temporal and spatial discretizations for the two-sided space–time FADE. This technique reduces the two-sided space–time FADE to a system of algebraic equations that simplifies the problem. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithm. By selecting relatively few Legendre polynomial degrees, we are able to get very accurate approximations, demonstrating the utility of the new approach over other numerical methods.

Entropy Analysis of Industrial Accident Data Series

Complex industrial plants exhibit multiple interactions among smaller parts and with human operators. Failure in one part can propagate across subsystem boundaries causing a serious disaster. This paper analyzes the industrial accident data series in the perspective of dynamical systems. First, we process real world data and show that the statistics of the number of fatalities reveal features that are well described by power law (PL) distributions. For early years, the data reveal double PL behavior, while, for more recent time periods, a single PL fits better into the experimental data. Second, we analyze the entropy of the data series statistics over time. Third, we use the Kullback–Leibler divergence to compare the empirical data and multidimensional scaling (MDS) techniques for data analysis and visualization. Entropy-based analysis is adopted to assess complexity, having the advantage of yielding a single parameter to express relationships between the data. The classical and the generalized (fractional) entropy and Kullback–Leibler divergence are used. The generalized measures allow a clear identification of patterns embedded in the data.

Short-Circuit Impedance of Power Transformers: the Fractional Order Calculus Approach

First IFAC Workshop on Fractional Differentiation and Its Application - 19-21 July 2004, Enseirb, Bordeaux, France - FDA'04

Matrix fractional systems

This paper addresses the matrix representation of dynamical systems in the perspective of fractional calculus. Fractional elements and fractional systems are interpreted under the light of the classical Cole–Cole, Davidson–Cole, and Havriliak–Negami heuristic models. Numerical simulations for an electrical circuit enlighten the results for matrix based models and high fractional orders. The conclusions clarify the distinction between fractional elements and fractional systems.

Fractional State Space Analysis of Temperature Time Series

Atmospheric temperatures characterize Earth as a slow dynamics spatiotemporal system, revealing long-memory and complex behavior. Temperature time series of 54 worldwide geographic locations are considered as representative of the Earth weather dynamics. These data are then interpreted as the time evolution of a set of state space variables describing a complex system. The data are analyzed by means of multidimensional scaling (MDS), and the fractional state space portrait (fSSP). A centennial perspective covering the period from 1910 to 2012 allows MDS to identify similarities among different Earth’s locations. The multivariate mutual information is proposed to determine the “optimal” order of the time derivative for the fSSP representation. The fSSP emerges as a valuable alternative for visualizing system dynamics.

Visualizing control systems performance: A fractional perspective

This article presents a novel method for visualizing the control systems behavior. The proposed scheme uses the tools of fractional calculus and computes the signals propagating within the system structure as a time/frequency-space wave. Linear and nonlinear closed-loop control systems are analyzed, for both the time and frequency responses, under the action of a reference step input signal. Several nonlinearities, namely, Coulomb friction and backlash, are also tested. The numerical experiments demonstrate the feasibility of the proposed methodology as a visualization tool and motivate its extension for other systems and classes of nonlinearities.

Public enterprise reforms and efficiency in regulated environments: the case of the postal sector

Purpose - This paper focuses on analyzing the effect that public reforms have on the efficiency of state-owned enterprises in regulated environments. Design/methodology/approach - The research is focused in the postal sector where public and private companies must obey a legal framework related to provide a universal service. The analysis is carried out using a panel of 7 European postal operators for the period 1997-2003. The activity analyzed was the letter mail division; we take as key variable the unit cost of a letter and use a translog cost function where as independent variables we include traffic levels, labor cost per employee, quality and availability of the service as well as the type of ownership (public or private). Additionally, in a second stage the geographical differences among countries are considered. Findings - Results indicate that postal operators that experienced organizational changes without being privatized, such as the Spanish and Greek operators, do not have higher unit costs than privatized postal operators like the one of Germany and the Netherlands. Moreover, we find that in all cases the operator of Ireland appear to be the most efficient. In this case restructuring process has been carried out giving an important leadership role to workers. This suggests us that labor culture could be a key variable to study when analyzing reform processes in public enterprises. Originality/value - Our findings show that in a regulated environment like in the postal sector, public and private companies can obtain similar efficiency levels in term of unit costs.

Investigation of field and diffusion time dependence of the diffusion-weighted signal at ultrahigh magnetic fields.

Over the last decade, there has been a significant increase in the number of high-magnetic-field MRI magnets. However, the exact effect of a high magnetic field strength (B0 ) on diffusion-weighted MR signals is not yet fully understood. The goal of this study was to investigate the influence of different high magnetic field strengths (9.4 T and 14.1 T) and diffusion times (9, 11, 13, 15, 17 and 24 ms) on the diffusion-weighted signal in rat brain white matter. At a short diffusion time (9 ms), fractional anisotropy values were found to be lower at 14.1 T than at 9.4 T, but this difference disappeared at longer diffusion times. A simple two-pool model was used to explain these findings. The model describes the white matter as a first hindered compartment (often associated with the extra-axonal space), characterized by a faster orthogonal diffusion and a lower fractional anisotropy, and a second restricted compartment (often associated with the intra-axonal space), characterized by a slower orthogonal diffusion (i.e. orthogonal to the axon direction) and a higher fractional anisotropy. Apparent T2 relaxation time measurements of the hindered and restricted pools were performed. The shortening of the pseudo-T2 value from the restricted compartment with B0 is likely to be more pronounced than the apparent T2 changes in the hindered compartment. This study suggests that the observed differences in diffusion tensor imaging parameters between the two magnetic field strengths at short diffusion time may be related to differences in the apparent T2 values between the pools. Copyright © 2013 John Wiley & Sons, Ltd.

Hours worked - Productivity puzzle: identification in fractional integration settings

A recent finding of the structural VAR literature is that the response of hours worked to a technology shock depends on the assumption on the order of integration of the hours. In this work we relax this assumption, allowing for fractional integration and long memory in the process for hours and productivity. We find that the sign and magnitude of the estimated impulse responses of hours to a positive technology shock depend crucially on the assumptions applied to identify them. Responses estimated with short-run identification are positive and statistically significant in all datasets analyzed. Long-run identification results in negative often not statistically significant responses. We check validity of these assumptions with the Sims (1989) procedure, concluding that both types of assumptions are appropriate to recover the impulse responses of hours in a fractionally integrated VAR. However, the application of longrun identification results in a substantial increase of the sampling uncertainty. JEL Classification numbers: C22, E32. Keywords: technology shock, fractional integration, hours worked, structural VAR, identification

Work and health conditions of sugar cane workers in Brazil

This is an exploratory research, with a quantitative approach, developed with the objective of analyzing the work and of life situations that can offer risks to the workers' health involved in the manual and automated cut of the sugar cane. The sample was composed by 39 sugar cane cutters and 16 operators of harvesters. The data collection occurred during the months of July and August of 2006, by the technique of direct observation of work situations and workers' homes and through interviews semi-structured. The interviews were recorded and later transcribed. Data were analyzed according to Social Ecological Theory. It was observed that the workers deal with multiple health risk situations, predominantly to the risks of occurrence of respiratory, musculoskeletal and psychological problems and work-related accidents due to the work activities. The interaction of individual, social and environmental factors can determine the workers' tendency to falling ill.

Difficulties of familes in caring for children and adolescents with mental disorders: an integrative review

Objective To identify the difficulties of families with children and/or adolescents with mental disorder. Method This is an integrative review. In December 2013, an electronic search was performed on Latin American Caribbean Literature on Health Sciences databases (LILACS) and on Electronic Medicus Index of the National Library of Medicine (MEDLINE) indexed in the Health Virtual Library (BVS) using a combination of descriptors and boolean operators as follows: mental disorders and child or adolescent and caregivers and/not health staff. Results 557 studies were identified, of which 15 were selected for this study. The findings indicated difficulties related to the care for or to interaction with children/adolescents with mental disorder. Conclusion The studies revealed difficulties related to everyday practices of care and feelings expressed during care practices, as well as in relationships with children or adolescents with mental disorder.



Combined analysis of grey matter voxel-based morphometry and white matter tract-based spatial statistics in late-life bipolar disorder.

Background: Previous magnetic resonance imaging (MRI) studies in young patients with bipolar disorder indicated the presence of grey matter concentration changes as well as microstructural alterations in white matter in various neocortical areas and the corpus callosum. Whether these structural changes are also present in elderly patients with bipolar disorder with long-lasting clinical evolution remains unclear. Methods: We performed a prospective MRI study of consecutive elderly, euthymic patients with bipolar disorder and healthy, elderly controls. We conducted a voxel-based morphometry (VBM) analysis and a tract-based spatial statistics (TBSS) analysis to assess fractional anisotropy and longitudinal, radial and mean diffusivity derived by diffusion tensor imaging (DTI). Results: We included 19 patients with bipolar disorder and 47 controls in our study. Fractional anisotropy was the most sensitive DTI marker and decreased significantly in the ventral part of the corpus callosum in patients with bipolar disorder. Longitudinal, radial and mean diffusivity showed no significant between-group differences. Grey matter concentration was reduced in patients with bipolar disorder in the right anterior insula, head of the caudate nucleus, nucleus accumbens, ventral putamen and frontal orbital cortex. Conversely, there was no grey matter concentration or fractional anisotropy increase in any brain region in patients with bipolar disorder compared with controls. Limitations: The major limitation of our study is the small number of patients with bipolar disorder. Conclusion: Our data document the concomitant presence of grey matter concentration decreases in the anterior limbic areas and the reduced fibre tract coherence in the corpus callosum of elderly patients with long-lasting bipolar disorder.

Dispersal probability distributions and the wave-front speed problem

The speed and width of front solutions to reaction-dispersal models are analyzed both analytically and numerically. We perform our analysis for Laplace and Gaussian distribution kernels, both for delayed and nondelayed models. The results are discussed in terms of the characteristic parameters of the models

Effect of fructose overfeeding and fish oil administration on hepatic de novo lipogenesis and insulin sensitivity in healthy men.

High-fructose diet stimulates hepatic de novo lipogenesis (DNL) and causes hypertriglyceridemia and insulin resistance in rodents. Fructose-induced insulin resistance may be secondary to alterations of lipid metabolism. In contrast, fish oil supplementation decreases triglycerides and may improve insulin resistance. Therefore, we studied the effect of high-fructose diet and fish oil on DNL and VLDL triglycerides and their impact on insulin resistance. Seven normal men were studied on four occasions: after fish oil (7.2 g/day) for 28 days; a 6-day high-fructose diet (corresponding to an extra 25% of total calories); fish oil plus high-fructose diet; and control conditions. Following each condition, fasting fractional DNL and endogenous glucose production (EGP) were evaluated using [1-13C]sodium acetate and 6,6-2H2 glucose and a two-step hyperinsulinemic-euglycemic clamp was performed to assess insulin sensitivity. High-fructose diet significantly increased fasting glycemia (7 +/- 2%), triglycerides (79 +/- 22%), fractional DNL (sixfold), and EGP (14 +/- 3%, all P < 0.05). It also impaired insulin-induced suppression of adipose tissue lipolysis and EGP (P < 0.05) but had no effect on whole- body insulin-mediated glucose disposal. Fish oil significantly decreased triglycerides (37%, P < 0.05) after high-fructose diet compared with high-fructose diet without fish oil and tended to reduce DNL but had no other significant effect. In conclusion, high-fructose diet induced dyslipidemia and hepatic and adipose tissue insulin resistance. Fish oil reversed dyslipidemia but not insulin resistance.