Search results for: Mandelbrot set

Authors: Ana Barbir, S. Ivelic Bradanovic, D. Pecaric, J. Pecaric

Abstract:

We proved a converse to Sherman's inequality. Using the concept of f-divergence we obtained some inequalities for the well-known entropies, such as Shannon entropies that have many applications in many applied sciences, for example, in information theory, biology and economics Zipf-Mandelbrot law gave improvement in account for the low-rankwords in corpus. Applications of Zipf-Mandelbrot law can be found in linguistics, information sciences and also mostly applicable in ecological eld studies. We also introduced an entropy by applying the Zipf-Mandelbrot law and derived some related inequalities.

Keywords: f-divergence, majorization inequality, Sherman inequality, Zipf-Mandelbrot entropy

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Authors: Neda Lovričević, Đilda Pečarić, Josip Pečarić

Abstract:

The Jensen functional in its discrete form is brought in relation to the Csiszar divergence functional, this time via its monotonicity property. This approach presents a generalization of the previously obtained results that made use of interpolating Jensen-type inequalities. Thus the monotonicity property is integrated with the Zipf-Mandelbrot law and applied to f-divergences for probability distributions that originate from the Csiszar divergence functional: Kullback-Leibler divergence, Hellinger distance, Bhattacharyya distance, chi-square divergence, total variation distance. The Zipf-Mandelbrot and the Zipf law are widely used in various scientific fields and interdisciplinary and here the focus is on the aspect of the mathematical inequalities.

Keywords: Jensen functional, monotonicity, Csiszar divergence functional, f-divergences, Zipf-Mandelbrot law

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Authors: O. Boussoufi, K. Lamrini Uahabi, M. Atounti

Abstract:

The main purpose of this paper is to calculate the fractal dimension of some Julia Sets and Mandelbrot Set in the Misiurewicz Points. Using Matlab to generate the Julia Sets images that match the Misiurewicz points and using a Fractal software, we were able to find different measures that characterize those fractals in textures and other features. We are actually focusing on fractal dimension and the error calculated by the software. When executing the given equation of regression or the log-log slope of image a Box Counting method is applied to the entire image, and chosen settings are available in a FracLAc Program. Finally, a comparison is done for each image corresponding to the area (boundary) where Misiurewicz Point is located.

Keywords: box counting, FracLac, fractal dimension, Julia Sets, Mandelbrot Set, Misiurewicz Points

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Authors: Adnan H. M. Al-Helali, Hamza A. Ali

Abstract:

Growth of censorship and pervasive monitoring on the Internet, Steganography arises as a new means of achieving secret communication. Steganography is the art and science of embedding information within electronic media used by common applications and systems. Generally, hiding information of multimedia within images will change some of their properties that may introduce few degradation or unusual characteristics. This paper presents a new image steganography approach for hiding information of multimedia (images, text, and audio) using generated Mandelbrot Fractal image as a cover. The proposed technique has been extensively tested with different images. The results show that the method is a very secure means of hiding and retrieving steganographic information. Experimental results demonstrate that an effective improvement in the values of the Peak Signal to Noise Ratio (PSNR), Mean Square Error (MSE), Normalized Cross Correlation (NCC) and Image Fidelity (IF) over the previous techniques.

Keywords: fractal image, information hiding, Mandelbrot et fractal, steganography

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Authors: Adnan H. M. Al-Helali, Hamza A. Ali

Abstract:

The growth of censorship and pervasive monitoring on the Internet, Steganography arises as a new means of achieving secret communication. Steganography is the art and science of embedding information within electronic media used by common applications and systems. Generally, hiding information of multimedia within images will change some of their properties that may introduce few degradation or unusual characteristics. This paper presents a new image steganography approach for hiding information of multimedia (images, text, and audio) using generated Mandelbrot Fractal image as a cover. The proposed technique has been extensively tested with different images. The results show that the method is a very secure means of hiding and retrieving steganographic information. Experimental results demonstrate that an effective improvement in the values of the Peak Signal to Noise Ratio (PSNR), Mean Square Error (MSE), Normalized Cross Correlation (NCC), and Image Fidelity (IF) over the pervious techniques.

Keywords: fractal image, information hiding, Mandelbrot set fractal, steganography

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Authors: Anca Radulescu, Danae Evans

Abstract:

The relationship between a network’s hardwiring and its emergent dynamics are central to neuroscience. We study the principles of this correspondence in a canonical setup (in which network nodes exhibit well-studied complex quadratic dynamics), then test their universality in biological networks. By extending methods from discrete dynamics, we study the effects of network connectivity on temporal patterns, encapsulating long-term behavior into the rich topology of network Mandelbrot sets. Then elements of fractal geometry can be used to predict and classify network behavior.

Keywords: canonical model, complex dynamics, dynamic networks, fractals, Mandelbrot set, network connectivity

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Authors: Marina Benito-Parejo, Raul Perez-Lopez, Miguel Herraiz, Carolina Guardiola-Albert, Cesar Martinez

Abstract:

Persistency, long-term memory and randomness are intrinsic properties of time-series of earthquakes. The Rescaled Range Analysis (RS-Analysis) was introduced by Hurst in 1956 and modified by Mandelbrot and Wallis in 1964. This method represents a simple and elegant analysis which determines the range of variation of one natural property (the seismic energy released in this case) in a time interval. Despite the simplicity, there is complexity inherent in the property measured. The cumulative curve of the energy released in time is the well-known fractal geometry of a devil’s staircase. This geometry is used for determining the maximum and minimum value of the range, which is normalized by the standard deviation. The rescaled range obtained obeys a power-law with the time, and the exponent is the Hurst value. Depending on this value, time-series can be classified in long-term or short-term memory. Hence, an algorithm has been developed for compiling the RS-Analysis for time series of earthquakes by days. Completeness time distribution and locally stationarity of the time series are required. The interest of this analysis is their application for a complex seismic crisis where different earthquakes take place in clusters in a short period. Therefore, the Hurst exponent has been obtained for the seismic crisis of Alhoceima (Mediterranean Sea) of January-March, 2016, where at least five medium-sized earthquakes were triggered. According to the values obtained from the Hurst exponent for each cluster, a different mechanical origin can be detected, corroborated by the focal mechanisms calculated by the official institutions. Therefore, this type of analysis not only allows an approach to a greater understanding of a seismic series but also makes possible to discern different types of seismic origins.

Keywords: Alhoceima crisis, earthquake time series, Hurst exponent, rescaled range analysis

Procedia PDF Downloads 287