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Multifractals
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed. Multifractal systems are common in nature. They include the length of coastlines, mountain topography, fully developed turbulence, real-world scenes, heartbeat dynamics, human gait and activity, human brain activity, and natural luminosity time series. Models have been proposed in various contexts ranging from turbulence in fluid dynamics to internet traffic, finance, image modeling, texture synthesis, meteorology, geophysics and more. The origin of multifractality in sequential (time series) data has been attributed to mathematical convergence effects related to the central limit theorem that have as foci of convergence the family of statistical distributions known as the Tweedie exponential dispersion models, as well as the geometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Analysis
Fractal analysis is assessing fractal characteristics of data. It consists of several methods to assign a fractal dimension and other fractal characteristics to a dataset which may be a theoretical dataset, or a pattern or signal extracted from phenomena including topography, natural geometric objects, ecology and aquatic sciences, sound, market fluctuations, heart rates, frequency domain in electroencephalography signals, digital images, molecular motion, and data science. Fractal analysis is now widely used in all areas of science. An important limitation of fractal analysis is that arriving at an empirically determined fractal dimension does not necessarily prove that a pattern is fractal; rather, other essential characteristics have to be considered. Fractal analysis is valuable in expanding our knowledge of the structure and function of various systems, and as a potential tool to mathematically assess novel areas of study. Fractal calculus was formulated which is a generaliza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer. The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed ''fractional dimensions''. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used ( see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singularity Exponent
Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiable Geometry * Singular point of a curve, where the curve is not given by a smooth embedding of a parameter * Singular point of an algebraic variety, a point where an algebraic variety is not locally flat * Rational singularity Complex analysis * Essential singularity, a singularity near which a function exhibits extreme behaviour * Isolated singularity, a mathematical singularity that has no other singularities close to it * Movable singularity, a concept in singularity theory * Removable singularity, a point at which a function is not defined but at which it can be so defined that it is continuous at the singularity Natural sciences * Singularity (system theory), in dynamical and social systems, a context in which a small change can cau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distort
In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signal representing sound or a video signal representing images, in an electronic device or communication channel. Distortion is usually unwanted, and so engineers strive to eliminate or minimize it. In some situations, however, distortion may be desirable. For example, in noise reduction systems like the Dolby system, an audio signal is deliberately distorted in ways that emphasize aspects of the signal that are subject to electrical noise, then it is symmetrically "undistorted" after passing through a noisy communication channel, reducing the noise in the received signal. Distortion is also used as a musical effect, particularly with electric guitars. The addition of noise or other outside signals ( hum, interference) is not considered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distort
In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signal representing sound or a video signal representing images, in an electronic device or communication channel. Distortion is usually unwanted, and so engineers strive to eliminate or minimize it. In some situations, however, distortion may be desirable. For example, in noise reduction systems like the Dolby system, an audio signal is deliberately distorted in ways that emphasize aspects of the signal that are subject to electrical noise, then it is symmetrically "undistorted" after passing through a noisy communication channel, reducing the noise in the received signal. Distortion is also used as a musical effect, particularly with electric guitars. The addition of noise or other outside signals ( hum, interference) is not considered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Box Counting
Box counting is a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. The essence of the process has been compared to zooming in or out using optical or computer based methods to examine how observations of detail change with scale. In box counting, however, rather than changing the magnification or resolution of a lens, the investigator changes the size of the element used to inspect the object or pattern (see Figure 1). Computer based box counting algorithms have been applied to patterns in 1-, 2-, and 3-dimensional spaces. The technique is usually implemented in software for use on patterns extracted from digital media, although the fundamental method can be used to investigate some patterns physically. The technique arose out of and is used in fractal analysis. It also has application in related fields such as lacunarity and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nota Bene
(, or ; plural form ) is a Latin phrase meaning "note well". It is often abbreviated as NB, n.b., or with the ligature and first appeared in English writing . In Modern English, it is used, particularly in legal papers, to draw the attention of the reader to a certain (side) aspect or detail of the subject being addressed. While ''NB'' is also often used in academic writing, ''note'' is a common substitute. The markings used to draw readers' attention in medieval manuscripts are also called marks. The common medieval markings do not, however, include the abbreviation ''NB''. The usual medieval equivalents are anagrams from the four letters in the word , the abbreviation DM from ("worth remembering"), or a symbol of a little hand (☞), called a manicule or index, with the index finger pointing towards the beginning of the significant passage.Raymond Clemens and Timothy Graham, Introduction to Manuscript Studies (Ithaca: Cornell University Press, 2007), p. 44. Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Box Counting
Box counting is a method of gathering data for analyzing complex patterns by breaking a dataset, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. The essence of the process has been compared to zooming in or out using optical or computer based methods to examine how observations of detail change with scale. In box counting, however, rather than changing the magnification or resolution of a lens, the investigator changes the size of the element used to inspect the object or pattern (see Figure 1). Computer based box counting algorithms have been applied to patterns in 1-, 2-, and 3-dimensional spaces. The technique is usually implemented in software for use on patterns extracted from digital media, although the fundamental method can be used to investigate some patterns physically. The technique arose out of and is used in fractal analysis. It also has application in related fields such as lacunarity and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Adelaide
The University of Adelaide (informally Adelaide University) is a public research university located in Adelaide, South Australia. Established in 1874, it is the third-oldest university in Australia. The university's main campus is located on North Terrace in the Adelaide city centre, adjacent to the Art Gallery of South Australia, the South Australian Museum, and the State Library of South Australia. The university has four campuses, three in South Australia: North Terrace campus in the city, Roseworthy campus at Roseworthy and Waite campus at Urrbrae, and one in Melbourne, Victoria. The university also operates out of other areas such as Thebarton, the National Wine Centre in the Adelaide Park Lands, and in Singapore through the Ngee Ann-Adelaide Education Centre. The University of Adelaide is composed of three faculties, with each containing constituent schools. These include the Faculty of Sciences, Engineering and Technology (SET), the Faculty of Health and Medical S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymous adjective ''Gaussian'' is pronounced . Mathematics Algebra and linear algebra Geometry and differential geometry Number theory Cyclotomic fields *Gaussian period *Gaussian rational *Gauss sum, an exponential sum over Dirichlet characters ** Elliptic Gauss sum, an analog of a Gauss sum **Quadratic Gauss sum Analysis, numerical analysis, vector calculus and calculus of variations Complex analysis and convex analysis *Gauss–Lucas theorem *Gauss's continued fraction, an analytic continued fraction derived from the hypergeometric functions * Gauss's criterion – described oEncyclopedia of Mathematics* Gauss's hypergeometric theorem, an identity on hypergeometric series *Gauss plane Statistics *Gauss–Kuzmi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
