Rhapsody in fractional

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

A Review of Definitions for Fractional Derivatives and Integral

This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering.

Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.

Delay approximation of fractional integrals

This paper explores the calculation of fractional integrals by means of the time delay operator. The study starts by reviewing the memory properties of fractional operators and their relationship with time delay. Based on the time response of the Mittag-Leffler function an approximation of fractional integrals consisting of time delayed samples is proposed. The tuning of the approximation is optimized by means of a genetic algorithm. The results demonstrate the feasibility of the new perspective and the limits of their application.

Fractional calculus analysis of the cosmic microwave background

Cosmic microwave background (CMB) radiation is the imprint from an early stage of the Universe and investigation of its properties is crucial for understanding the fundamental laws governing the structure and evolution of the Universe. Measurements of the CMB anisotropies are decisive to cosmology, since any cosmological model must explain it. The brightness, strongest at the microwave frequencies, is almost uniform in all directions, but tiny variations reveal a spatial pattern of small anisotropies. Active research is being developed seeking better interpretations of the phenomenon. This paper analyses the recent data in the perspective of fractional calculus. By taking advantage of the inherent memory of fractional operators some hidden properties are captured and described.

Fractional order modelling of dynamic backlash

This paper studies the dynamical properties of systems with backlash and impact phenomena. This type of non-linearity can be tackled in the perspective of the fractional calculus theory. Fractional and integer order models are compared and their influence upon the emerging dynamics is analysed. It is demonstrated that fractional models can memorize dynamical effects due to multiple micro-collisions.

Evaluation of the double diffusion, enzyme immunoassay and immunoblotting techniques, for the diagnosis of human hydatid disease in tropical areas

Hydatid disease in tropical areas poses a serious diagnostic problem due to the high frequence of cross-reactivity with other endemic helminthic infections. The enzyme-linked-immunosorbent assay (ELISA) and the double diffusion arc 5 showed respectively a sensitivity of 73% and 57% and a specificity of 84-95% and 100%. However, the specificity of ELISA was greatly increased by using ovine serum and phosphorylcholine in the diluent buffer. The hydatic antigen obtained from ovine cyst fluid showed three main protein bands of 64,58 and 30 KDa using SDS PAGE and immunoblotting. Sera from patients with onchocerciasis, cysticercosis, toxocariasis and Strongyloides infection cross-reacted with the 64 and 58 KDa bands by immunoblotting. However, none of the analyzed sera recognized the 30 KDa band, that seems to be specific in this assay. The immunoblotting showed a sensitivity of 80% and a specificity of 100% when used to recognize the 30 KDa band.

Viscosity and self-diffusion coefficients of dialkyl adipates: a correlation scheme with predictive capabilities

A correlation and predictive scheme for the viscosity and self-diffusivity of liquid dialkyl adipates is presented. The scheme is based on the kinetic theory for dense hard-sphere fluids, applied to the van der Waals model of a liquid to predict the transport properties. A "universal" curve for a dimensionless viscosity of dialkyl adipates was obtained using recently published experimental viscosity and density data of compressed liquid dimethyl (DMA), dipropyl (DPA), and dibutyl (DBA) adipates. The experimental data are described by the correlation scheme with a root-mean-square deviation of +/- 0.34 %. The parameters describing the temperature dependence of the characteristic volume, V-0, and the roughness parameter, R-eta, for each adipate are well correlated with one single molecular parameter. Recently published experimental self-diffusion coefficients of the same set of liquid dialkyl adipates at atmospheric pressure were correlated using the characteristic volumes obtained from the viscosity data. The roughness factors, R-D, are well correlated with the same single molecular parameter found for viscosity. The root-mean-square deviation of the data from the correlation is less than 1.07 %. Tests are presented in order to assess the capability of the correlation scheme to estimate the viscosity of compressed liquid diethyl adipate (DEA) in a range of temperatures and pressures by comparison with literature data and of its self-diffusivity at atmospheric pressure in a range of temperatures. It is noteworthy that no data for DEA were used to build the correlation scheme. The deviations encountered between predicted and experimental data for the viscosity and self-diffusivity do not exceed 2.0 % and 2.2 %, respectively, which are commensurate with the estimated experimental measurement uncertainty, in both cases.

Evaluation of the enzyme-linked-immuno-electro-diffusion-assay (ELIEDA) for the diagnosis of Schistosoma mansoni infection with low worm burden

An immunoprecipitation technique, ELIEDA (enzyme-linked-immuno-electro-diffusion assay), was evaluated for the diagnosis of Schistosoma mansoni infection with low worm burden. One hundred of serum samples from patients excreting less than 600 eggs per gram of feces (epg), with unrelated diseases and clinically healthy subjects were studied. In patients with egg counts higher than 200 epg, the sensitivities of IgM and IgG ELIEDA were 1.000 and 0.923, respectively, not differing from other Serologic techniques, such as indirect hemaglutination (IHAT), immunofluorescence (IFT) tests and immuno-electrodiffusion assay (IEDA). However in patients with low egg counts (< 100 epg), the IgG ELIEDA provided better results (0.821) than IgM ELIEDA (0.679), showing sensitivity that did not differ from that of IgG IFT (0.929), but lower than that of IgM IFT (0.964). However, its sensivity was higher than that found with IHAT (0.607) and IEDA (0.536). The specificity of IgG ELIEDA was comparable to that of other techniques. The data indicate that IgG ELIEDA might be useful for the diagnosis of slight S. mansoni infections, and the cellulose acetate membrane strips can be stored for further retrospective studies.

Numerical analysis of the initial conditions in fractional systems

Fractional dynamics is a growing topic in theoretical and experimental scientific research. A classical problem is the initialization required by fractional operators. While the problem is clear from the mathematical point of view, it constitutes a challenge in applied sciences. This paper addresses the problem of initialization and its effect upon dynamical system simulation when adopting numerical approximations. The results are compatible with system dynamics and clarify the formulation of adequate values for the initial conditions in numerical simulations.

Local Fractional Variational Iteration Method for Local Fractional Poisson Equations in Two Independent Variables

The local fractional Poisson equations in two independent variables that appear in mathematical physics involving the local fractional derivatives are investigated in this paper. The approximate solutions with the nondifferentiable functions are obtained by using the local fractional variational iteration method.

Fractional Order Generalized Information

This paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow an high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The concepts are also extended to relative distances and tested with several sets of data, confirming the goodness of the generalization.

Rhapsody in Fractional

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

Fractional Dynamics of Computer Virus Propagation

We propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.

Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.