Comment on “A computational technique to classify several fractional Brownian motion processes” [Chaos Solitons and Fractals 150(2021) 111152]
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Fractional Brownian motion
Fractional Brownian motion
Brownian dynamics
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Brownian trajectories, obtained from molecular dynamics simulations of two-dimensional systems with inverse power-law potentials, exhibit fractal behaviour. The fractal dimensionality D goes to two asymptotically. The approach to the asymptotic value depends on the thermodynamic state of the system. Experiments are proposed to verify the authors' predictions for the behaviour of D.
Fractional Brownian motion
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The record statistics in stationary and non-stationary fractal time series is studied extensively. By calculating various concepts in record dynamics, we find some interesting results. In stationary fractional Gaussian noises, we observe a universal behavior for the whole range of Hurst exponents. However, for non-stationary fractional Brownian motions, the record dynamics is crucially dependent on the memory, which plays the role of a non-stationarity index, here. Indeed, the deviation from the results of the stationary case increases by increasing the Hurst exponent in fractional Brownian motions. We demonstrate that the memory governs the dynamics of the records as long as it causes non-stationarity in fractal stochastic processes; otherwise, it has no impact on the record statistics.
Hurst exponent
Fractional Brownian motion
Rescaled range
Stationary process
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Fractional Brownian motion
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In this thesis we study Brownian motion, a random process named by Robert Brown who
discovered that the pollen particles dispersed in the liquid move constantly and randomly.
The basic characteristics of this random process are independent and normally distributed
increments. Even though the applications and implications of Brownian motion are numerous
and significant, it is not suitable for modeling a phenomenon with an unimaginable
probability of extreme events. This is the motivation for introducing fractal Brownian motion,
as a generalization of Brownian motion and model for stable processes. In the thesis we
discuss the concepts of fractal and fractal dimensions. In examples, we show self similarity as
one of the basic characteristics of a fractal. We consider the application of fractal Brownian
motion to the financial market. First of all we consider the historical problem of cotton
prices which is characterized by self similarity in time series of prices. Benoit Mandelbrot
noticed that cotton prices do not follow the normal distribution but one of the distributions
with the heavy tail. We also define terms multifractal and multifractal time, on the basis of
which we are conducting a multifractal model of the market.
Multifractal system
Mandelbrot set
Fractional Brownian motion
Self-similarity
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Fractional Brownian motion
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This paper discusses to establish the characteristic model of particles' Brownian motion by fractal theory. Particles' Brownian motion is a kind of moving at random and particles' position oscillates with extreme irregularity. After having analyzed this process, authors present a fractal model to describe the physical process and prove theoretically the model is valid. The relationship between fractal characteristic information and particle size parameters is deduced. Combining image processing technique, it can provide a new method and technique of submicron particle sizing.
Fractional Brownian motion
Position (finance)
Particle (ecology)
Fractal derivative
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Each and every particle in this world is moving in random and it is impossible to find the motion of such particles. In short, motion of such particles is uncertain. This uncertain movement can be seen in particles ranging from simple atoms to the complex biological cells and tissues in human brain and other human organs. This random motion has been analyzed by various researchers and this is being considered very important in the field of science and technology. This paper aims at providing the concept of Brownian motion and fractional Brownian motion. And also, fractal analysis of brain MRI images is also discussed since the brain MRI images are easily affected by this fractional Brownian motion.
Fractional Brownian motion
Particle (ecology)
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Dynamics
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One kind of fractal signals are signals that are derived from the theory of Brownian motion. Brownian motion is a random process, which has the properties of self-similarity. Mandelbrott obtained analytical expression describing the amplitude of fractal signal [1].
Fractional Brownian motion
Self-similarity
Basis (linear algebra)
SIGNAL (programming language)
Similarity (geometry)
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