scholarly journals And I say to myself: “What a fractional world!”

2011 ◽  
Vol 14 (4) ◽  
Author(s):  
J. Tenreiro Machado
Keyword(s):  
Global Warming ◽  
Fourier Transform ◽  
Fractional Calculus ◽  
Complex Systems ◽  
Power Law ◽  
Deoxyribonucleic Acid ◽  
Warming Trend ◽  
Fractional Dynamics ◽  
The Fourier Transform ◽  
Musical Rhythms

AbstractThis paper discusses several complex systems in the perspective of fractional dynamics. For prototype systems are considered the cases of deoxyribonucleic acid decoding, financial evolution, earthquakes events, global warming trend, and musical rhythms. The application of the Fourier transform and of the power law trendlines leads to an assertive representation of the dynamics and to a simple comparison of their characteristics. Moreover, the gallery of different systems, both natural and man made, demonstrates the richness of phenomena that can be described and studied with the tools of fractional calculus.

scholarly journals ON THE DNA OF ELEVEN MAMMALS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250074 ◽  
Author(s):  
J. TENREIRO MACHADO ◽  
ANTÓNIO C. COSTA ◽  
MARIA DULCE QUELHAS
Keyword(s):  
Fourier Transform ◽  
Long Range ◽  
Power Law ◽  
Fractional Dynamics ◽  
Categorical Representation ◽  
Memory Characteristics

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.

scholarly journals Analysis of Natural and Artificial Phenomena Using Signal Processing and Fractional Calculus

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
J. A. Tenreiro Machado ◽  
António M. Lopes
Keyword(s):  
Time Series ◽  
Fourier Transform ◽  
Vector Field ◽  
System Dynamics ◽  
Power Law ◽  
System Behavior ◽  
Amplitude Spectra ◽  
The Fourier Transform ◽  
Complex Phenomena ◽  
Power Law Parameters

AbstractIn this paper we study several natural and man-made complex phenomena in the perspective of dynamical systems. For each class of phenomena, the system outputs are time-series records obtained in identical conditions. The time-series are viewed as manifestations of the system behavior and are processed for analyzing the system dynamics. First, we use the Fourier transform to process the data and we approximate the amplitude spectra by means of power law functions. We interpret the power law parameters as a phenomenological signature of the system dynamics. Second, we adopt the techniques of non-hierarchical clustering and multidimensional scaling to visualize hidden relationships between the complex phenomena. Third, we propose a vector field based analogy to interpret the patterns unveiled by the PL parameters.

scholarly journals Quantum Szilard engine for the fractional power-law potentials

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Ekrem Aydiner
Keyword(s):  
Quantum Information ◽  
Fractional Calculus ◽  
Power Law ◽  
Fractional Power ◽  
Thermodynamic Cycle ◽  
Partition Functions ◽  
Fractional Dynamics ◽  
Energy Eigenvalues ◽  
Positive Work ◽  
Degenerate Cases

AbstractIn this study, we consider the quantum Szilárd engine with a single particle under the fractional power-law potential. We suggest that such kind of the Szilárd engine works a Stirling-like cycle. We obtain energy eigenvalues and canonical partition functions for the degenerate and non-degenerate cases in this cycle process. By using these quantities we numerically compute work and efficiency for this thermodynamic cycle for various power-law potentials with integer and non-integer exponents. We show that the presented simple engine also yields positive work and efficiency. We discuss the importance of fractional dynamics in physics and finally, we conclude that fractional calculus should be included in the fields of quantum information and thermodynamics.

scholarly journals On the Numerical Computation of the Mittag–Leffler Function

2019 ◽  
Vol 20 (6) ◽  
pp. 725-736 ◽  
Author(s):  
Manuel D. Ortigueira ◽  
António M. Lopes ◽  
José Tenreiro Machado
Keyword(s):  
Fourier Transform ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Fast Fourier Transform ◽  
Numerical Computation ◽  
Fractional Order ◽  
Power Law ◽  
Computational Time ◽  
Integral Formulas

AbstractThe Mittag–Leffler function (MLF) plays an important role in many applications of fractional calculus, establishing a connection between exponential and power law behaviors that characterize integer and fractional order phenomena, respectively. Nevertheless, the numerical computation of the MLF poses problems both of accuracy and convergence. In this paper, we study the calculation of the 2-parameter MLF by using polynomial computation and integral formulas. For the particular cases having Laplace transform (LT) the method relies on the inversion of the LT using the fast Fourier transform. Experiments with two other available methods compare also the computational time and accuracy. The 3-parameter MLF and its calculation are also considered.

scholarly journals REVIEW OF SOME PROMISING FRACTIONAL PHYSICAL MODELS

2013 ◽  
Vol 27 (09) ◽  
pp. 1330005 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fractional Calculus ◽  
Power Law ◽  
Particle Interaction ◽  
Discrete Systems ◽  
Physical Models ◽  
Long Term Memory ◽  
Fractional Dynamics ◽  
Fractal Media ◽  
Systems With Memory ◽  
With Memory

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law nonlocality, power-law long-term memory or fractal properties by using integrations and differentiation of non-integer orders, i.e., by methods in the fractional calculus. This paper is a review of physical models that look very promising for future development of fractional dynamics. We suggest a short introduction to fractional calculus as a theory of integration and differentiation of noninteger order. Some applications of integro-differentiations of fractional orders in physics are discussed. Models of discrete systems with memory, lattice with long-range inter-particle interaction, dynamics of fractal media are presented. Quantum analogs of fractional derivatives and model of open nano-system systems with memory are also discussed.

Characterization of Pyrolysis and Combustion of Complex Systems Using Fourier Transform Infrared Spectroscopy

1980 ◽  
Vol 34 (2) ◽  
pp. 174-185 ◽  
Author(s):  
John O. Lephardt ◽  
Robert A. Fenner
Keyword(s):  
Fourier Transform ◽  
Complex Systems ◽  
Evolved Gas Analysis ◽  
Analytical Techniques ◽  
Fourier Transform Infrared ◽  
Gas Analysis ◽  
Before And After ◽  
Infrared Spectral ◽  
Ft Ir ◽  
The Fourier Transform

Thermal analysis plays an important role in the evaluation and development of many complex chemical systems. Often combustion data as well as pyrolytic data on these systems may be important. Analytical techniques are required which maximize the amount of useful information which can be obtained per thermal experiment. Fourier transform infrared spectral analysis of gases evolving from materials vs temperature can provide information on both pyrolysis and combustion processes, and an apparatus for performing such analysis routinely is described. While the Fourier transform infrared evolved gas analysis (FT-IR-EGA) system has numerous applications, three applications using tobacco as a sample are presented for illustration. These include comparison of different materials, comparison of materials before and after modification and examination of oxygen-induced effects relative to pyrolysis effects. The FT-IR-EGA technique should prove to be a useful tool in elucidating thermal processes in complex systems.

scholarly journals Long and Short Memory in Economics: Fractional-Order Difference and Differentiation

2016 ◽  
Vol 5 (2) ◽  
pp. 327 ◽  
Author(s):  
Vasily E. Tarasov ◽  
Valentina V. Tarasova
Keyword(s):  
Fourier Transform ◽  
Fractional Order ◽  
Power Law ◽  
Fractional Integration ◽  
Order Difference ◽  
Fractional Differencing ◽  
Discrete Operators ◽  
Short Memory ◽  
Fractional Differential ◽  
The Fourier Transform

<div><p><em>Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the </em><em>Grunwald-Letnikov fractional differences of non-integer order d. Equations of ARIMA(p,d,q) and ARFIMA(p,d,q) models are the fractional-order difference equations with the Grunwald-Letnikov differences of order d. We prove that the long and short memory with power law should be described by the exact fractional-order differences, for which </em><em>the Fourier transform </em><em>demonstrates the power law exactly. The fractional differencing and the Grunwald-Letnikov fractional differences cannot give exact results for the long and short memory with power law, since </em><em>the Fourier transform</em><em> of these discrete operators satisfy the </em><em>power law in the neighborhood of zero only. We prove that the economic processes with t</em><em>he continuous time long and short memory, which is characterized by the power law, should be described by the fractional differential equations.</em></p></div>

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

scholarly journals Strain Sensitivity Enhancement of Broadband Ultrasonic Signals in Plates Using Spectral Phase Filtering

2021 ◽  
Vol 11 (6) ◽  
pp. 2582
Author(s):  
Lucas M. Martinho ◽  
Alan C. Kubrusly ◽  
Nicolás Pérez ◽  
Jean Pierre von der Weid
Keyword(s):  
Fourier Transform ◽  
Guided Waves ◽  
Strain Level ◽  
Energy Concentration ◽  
Short Time Fourier Transform ◽  
Time Frequency ◽  
Frequency Representation ◽  
The Fourier Transform ◽  
Short Time ◽  
Sensitivity Gain

The focused signal obtained by the time-reversal or the cross-correlation techniques of ultrasonic guided waves in plates changes when the medium is subject to strain, which can be used to monitor the medium strain level. In this paper, the sensitivity to strain of cross-correlated signals is enhanced by a post-processing filtering procedure aiming to preserve only strain-sensitive spectrum components. Two different strategies were adopted, based on the phase of either the Fourier transform or the short-time Fourier transform. Both use prior knowledge of the system impulse response at some strain level. The technique was evaluated in an aluminum plate, effectively providing up to twice higher sensitivity to strain. The sensitivity increase depends on a phase threshold parameter used in the filtering process. Its performance was assessed based on the sensitivity gain, the loss of energy concentration capability, and the value of the foreknown strain. Signals synthesized with the time–frequency representation, through the short-time Fourier transform, provided a better tradeoff between sensitivity gain and loss of energy concentration.

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