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This paper presents the continuous-time fractional linear systems and their main properties. Two particular classes of models are introduced: the fractional autoregressive-moving average type and the tempered linear system. For both classes, the computations of the impulse response, transfer function, and frequency response are discussed. It is shown that such systems can have integer and fractional components. From the integer component we deduce the stability. The fractional order component is always stable. The initial-condition problem is analyzed and it is verified that it depends on the structure of the system. For a correct definition and backward compatibility with classic systems, suitable fractional derivatives are also introduced. The Grünwald-Letnikov and Liouville derivatives, as well as the corresponding tempered versions, are formulated. [ABSTRACT FROM AUTHOR]

The substantial, tempered, and shifted fractional derivatives, useful in medium range systems, are reviewed and highlighted in a unified framework. Their historical evolution is described and their properties studied. Moreover, they are characterized and assessed under the light of the strict sense criterion for the definition of derivatives. The relationship between these derivatives and several formulations developed in fractional calculus are explored. Additionally, the interpretations in the time and frequency domains are included. In the scope of the framework, new tempered linear systems and transfer functions are introduced. As particular applications, several systems for compensation and control purposes are revisited. [ABSTRACT FROM AUTHOR]

The distributed-order fractional diffusion equation is a generalization of the standard fractional diffusion equation that can model processes lacking power-law scaling over the whole time-domain. An important application of distributed-order diffusions is to model ultraslow diffusion where a plume of particles spreads at a logarithmic rate. To broaden the range of applicability of distributed-order fractional diffusion models, efficient numerical methods are needed to solve the model equation. In this work, we develop spectral tau schemes to discretize the fractional diffusion equation with distributed-order fractional derivative in time and Dirichlet boundary conditions. The model solution is expanded in multi-dimensions in terms of Legendre polynomials and the discrete equations are obtained with the tau method. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. The proposed spectral tau methods yield an exponential rate of convergence when the solution is smooth. Our results confirm that nonlocal numerical methods are best suited to discretize distributed-order fractional differential equations as they naturally take the global behavior of the solution into account. [ABSTRACT FROM AUTHOR]

In this paper the numerical solution of the fractional Navier–Stokes equations (FNSE) is derived by means of a new non-standard finite difference method (NSFDM). The fractional differential operators are taken in sense of the Riesz fractional derivative. The pressure driven by the flow between two parallel plates is solved with the NSFDM and a modified trapezoidal quadrature rule of fractional order. The stability and convergence of the proposed scheme are proved. Numerical examples illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the DNA code of eleven mammals from the perspective of fractional dynamics. The application of Fourier transform and power law trendlines leads to a categorical representation of species and chromosomes. The DNA information reveals long range memory characteristics. [ABSTRACT FROM AUTHOR]

Power law PL and fractional calculus are two faces of phenomena with long memory behavior. This paper applies PL description to analyze different periods of the business cycle. With such purpose the evolution of ten important stock market indices (DAX, Dow Jones, NASDAQ), Nikkei, NYSE, S&P500, SSEC, HSI, TWII, and BSE) over time is studied. An evolutionary algorithm is used for the fitting of the PL parameters. It is observed that the PL curve fitting constitutes a good tool for revealing the signal main characteristics leading to the emergence of the global financial dynamic evolution. [ABSTRACT FROM AUTHOR]

Stock market indices (SMIs) are important measures of financial and economical performance. Considerable research efforts during the last years demonstrated that these signals have a chaotic nature and require sophisticated mathematical tools for analyzing their characteristics. Classical methods, such as the Fourier transform, reveal considerable limitations in discriminating different periods of time. This paper studies the dynamics of SMI by combining the wavelet transform and the multidimensional scaling (MDS). Six continuous wavelets are tested for analyzing the information content of the stock signals. In a first phase, the real Shannon wavelet is adopted for performing the evaluation of the SMI dynamics, while their comparison is visualized by means of the MDS. In a second phase, the other wavelets are also tested, and the corresponding MDS plots are analyzed. [ABSTRACT FROM AUTHOR]

Statistical significance measures the reliability of a result obtained from a random experiment. We investigate the number of repetitions needed for a statistical result to have a certain significance. In the first step, we consider binomially distributed variables in the example of medication testing with fixed placebo efficacy, asking how many experiments are needed in order to achieve a significance of 95%. In the next step, we take the probability distribution of the placebo efficacy into account, which to the best of our knowledge has not been done so far. Depending on the specifics, we show that in order to obtain identical significance, it may be necessary to perform twice as many experiments than in a setting where the placebo distribution is neglected. We proceed by considering more general probability distributions and close with comments on some erroneous assumptions on probability distributions which lead, for instance, to a trivial explanation of the fat tail. [ABSTRACT FROM AUTHOR]

The study of the assessment and reflection of biological assets in the economic processes of agricultural enterprises can be represented as a chain of phenomena in which scientists and practitioners try to study and understand the nature and essence of biological assets in various aspects. This article discusses the principles of accounting for biological assets in the agricultural enterprises' economic life of the Republic of Tajikistan and identifies the reasons and mechanisms for their reflection based on the principles of the International Financial Reporting System (IFRS) in refraction to the specifics and national features of accounting. The author gives his own interpretation of these approaches and constructs the architecture of the information module for accounting for biological assets and the results of biotransformation, based on the web services architecture (WSA) within the framework of the development trend of these and new accounting models in connection with the transition of the world economy to a digital format. The article provides specific authors' approaches to implementing this architecture in PHP (Hypertext Preprocessor) and also implements one of the approaches to assessing the value of biological assets, based on the theory of fuzzy sets, taking into account the risk of investment inefficiency. [ABSTRACT FROM AUTHOR]