|
Multifractal System
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 geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Law
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent: one quantity varies as a power of another. The change is independent of the initial size of those quantities. For instance, the area of a square has a power law relationship with the length of its side, since if the length is doubled, the area is multiplied by 2, while if the length is tripled, the area is multiplied by 3, and so on. Empirical examples The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, cloud sizes, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words ... [...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 dis ... [...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 lacunarit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nota Bene
( ; plural: ) is the Latin language, Latin phrase meaning ''note well''. In manuscripts, ''nota bene'' is abbreviated in upper-case as NB and N.B., and in lower-case as n.b. and nb; the editorial usages of ''nota bene'' and ''notate bene'' first appeared in the English writing style, English style of writing around the year 1711. In Modern English, since the 14th century, the editorial usage of ''NB'' is common to the legal writing, legal style of writing of documents to direct the reader's attention to a thematically relevant aspect of the subject that qualifies the matter being litigated, whereas in academic writing, the editorial abbreviation ''n.b.'' is a casual synonym for ''footnote''. In medieval manuscripts, the editorial marks used to draw the reader's attention to a supporting text also are called marks; however, the catalogue of medieval editorial marks does not include the NB abbreviation. The medieval equivalents to the n.b.-mark are anagrams derived from the f ... [...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 lacunarit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Adelaide
The University of Adelaide is a public university, public research university based in Adelaide, South Australia. Established in 1874, it is the third-oldest university in Australia. Its main campus in the Adelaide city centre includes many Sandstone universities, sandstone buildings of historical and architectural significance, such as Bonython Hall. Its royal charter awarded by Queen Victoria in 1881 allowed it to become the University of London, second university in the English-speaking world to confer degrees to women. It Adelaide University, plans to merge with the neighbouring University of South Australia, is adjacent to the Australian Space Agency headquarters on Lot Fourteen and is part of the Adelaide BioMed City research precinct. The university was founded at the former South Australian Society of Arts, Royal South Australian Society of Arts by the Union College and studies were initially conducted at its State Library of South Australia, Institute Building. The soc ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Cascade
In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process. Definition The plots above are examples of multiplicative cascade multifractals. To create these distributions there are a few steps to take. Firstly, we must create a lattice of cells which will be our underlying probability density field. Secondly, an iterative process is followed to create multiple levels of the lattice: at each iteration the cells are split into four equal parts (cells). Each new cell is then assigned a probability randomly from the set \lbrace p_1,p_2,p_3,p_4 \rbrace without replacement, where p_i \in ,1/math>. This process is continued to the ''N''th level. For example, in constructing such a model down to level 8 we produce a 48 array of cells. Thirdly, the cells are filled as follows: We take the probability of a cell being occupied as the product of the cell's own ''p''''i'' and those of all its paren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Transform
In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent real variables, then the Legendre transform with respect to this variable is applicable to the function. In physical problems, the Legendre transform is used to convert functions of one quantity (such as position, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively). In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism (or vice versa) and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables. For sufficiently smooth functions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
