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Infinite Compositions Of Analytic Functions
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a ''single function'' see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system. Although the title of this article specifies analytic functions, there are results for more general functions of a complex variable as well. Notation There are several notations describing infinite compositions, includi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and are composed to yield a function that maps in domain to in codomain . Intuitively, if is a function of , and is a function of , then is a function of . The resulting ''composite'' function is denoted , defined by for all in . The notation is read as " of ", " after ", " circle ", " round ", " about ", " composed with ", " following ", " then ", or " on ", or "the composition of and ". Intuitively, composing functions is a chaining process in which the output of function feeds the input of function . The composition of functions is a special case of the composition of relations, sometimes also denoted by \circ. As a result, all properties of composition of relations are true of composition of functions, such as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metropolis At 30K
A metropolis () is a large city or conurbation which is a significant economic, political, and cultural center for a country or region, and an important hub for regional or international connections, commerce, and communications. A big city belonging to a larger urban agglomeration, but which is not the core of that agglomeration, is not generally considered a metropolis but a part of it. The plural of the word is ''metropolises'', although the Latin plural is ''metropoles'', from the Greek ''metropoleis'' (). For urban centers outside metropolitan areas that generate a similar attraction on a smaller scale for their region, the concept of the regiopolis ("regio" for short) was introduced by urban and regional planning researchers in Germany in 2006. Etymology Metropolis (μητρόπολις) is a Greek word, coming from μήτηρ, ''mḗtēr'' meaning "mother" and πόλις, ''pólis'' meaning "city" or "town", which is how the Greek colonies of antiquity referred t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dream Of Gold
A dream is a succession of images, ideas, emotions, and sensations that usually occur involuntarily in the mind during certain stages of sleep. Humans spend about two hours dreaming per night, and each dream lasts around 5 to 20 minutes, although the dreamer may perceive the dream as being much longer than this. The content and function of dreams have been topics of scientific, philosophical and religious interest throughout recorded history. Dream interpretation, practiced by the Babylonians in the third millennium BCE and even earlier by the ancient Sumerians, figures prominently in religious texts in several traditions, and has played a lead role in psychotherapy. The scientific study of dreams is called oneirology. Most modern dream study focuses on the neurophysiology of dreams and on proposing and testing hypotheses regarding dream function. It is not known where in the brain dreams originate, if there is a single origin for dreams or if multiple regions of the brain a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Continued Fraction Formula
In the analytic theory of continued fractions, Euler's continued fraction formula is an identity connecting a certain very general infinite series with an infinite continued fraction. First published in 1748, it was at first regarded as a simple identity connecting a finite sum with a finite continued fraction in such a way that the extension to the infinite case was immediately apparent. Today it is more fully appreciated as a useful tool in analytic attacks on the general convergence problem for infinite continued fractions with complex elements. The original formula Euler derived the formula as connecting a finite sum of products with a finite continued fraction. : a_0 + a_0a_1 + a_0a_1a_2 + \cdots + a_0a_1a_2\cdots a_n = \cfrac\, The identity is easily established by induction on ''n'', and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diminishing Returns
In economics, diminishing returns are the decrease in marginal (incremental) output of a production process as the amount of a single factor of production is incrementally increased, holding all other factors of production equal ( ceteris paribus). The law of diminishing returns (also known as the law of diminishing marginal productivity) states that in productive processes, increasing a factor of production by one unit, while holding all other production factors constant, will at some point return a lower unit of output per incremental unit of input. The law of diminishing returns does not cause a decrease in overall production capabilities, rather it defines a point on a production curve whereby producing an additional unit of output will result in a loss and is known as negative returns. Under diminishing returns, output remains positive, however productivity and efficiency decrease. The modern understanding of the law adds the dimension of holding other outputs equal, since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-generating Series3
Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when sufficient energy is available, not needing control by any external agent. It is often triggered by seemingly random fluctuations, amplified by positive feedback. The resulting organization is wholly decentralized, distributed over all the components of the system. As such, the organization is typically robust and able to survive or self-repair substantial perturbation. Chaos theory discusses self-organization in terms of islands of predictability in a sea of chaotic unpredictability. Self-organization occurs in many physical, chemical, biological, robotic, and cognitive systems. Examples of self-organization include crystallization, thermal convection of fluids, chemical oscillation, animal swarming, neural circuits, and black marke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Method
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who treated it in his book ''Institutionum calculi integralis'' (published 1768–1870). The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size. The Euler method often serves as the basis to construct more complex methods, e.g., predictor–corrector method. Informal geometrical description Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Active Contour Model
Active contour model, also called snakes, is a framework in computer vision introduced by Michael Kass, Andrew Witkin, and Demetri Terzopoulos for delineating an object outline from a possibly noisy 2D image. The snakes model is popular in computer vision, and snakes are widely used in applications like object tracking, shape recognition, segmentation, edge detection and stereo matching. A snake is an energy minimizing, deformable spline influenced by constraint and image forces that pull it towards object contours and internal forces that resist deformation. Snakes may be understood as a special case of the general technique of matching a deformable model to an image by means of energy minimization. In two dimensions, the active shape model represents a discrete version of this approach, taking advantage of the point distribution model to restrict the shape range to an explicit domain learnt from a training set. Snakes do not solve the entire problem of finding contours in ima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contours In The Vector Field F(z) = -Cos(z)
Contour may refer to: * Contour (linguistics), a phonetic sound * Pitch contour * Contour (camera system), a 3D digital camera system * Contour, the KDE Plasma 4 interface for tablet devices * Contour line, a curve along which the function has a constant value * Contour drawing, an artistic technique * A closed path in the mathematical method of contour integration * Boundary (topology) of a set Contours may refer to: * ''Contours'' (album), by Sam Rivers * The Contours, a soul music group Contouring may refer to: * The makeup technique contouring Other uses * Ford Contour, a motor car * CONTOUR Contour may refer to: * Contour (linguistics), a phonetic sound * Pitch contour * Contour (camera system), a 3D digital camera system * Contour, the KDE Plasma 4 interface for tablet devices * Contour line, a curve along which the function ha ..., a failed NASA space probe * Contour, a North Wing Apache ultralight aircraft variant See also * Contour fort, a type of Briti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virtual Tunnels
Virtual may refer to: * Virtual (horse), a thoroughbred racehorse * Virtual channel, a channel designation which differs from that of the actual radio channel (or range of frequencies) on which the signal travels * Virtual function, a programming function or method whose behaviour can be overridden within an inheriting class by a function with the same signature * Virtual machine, the virtualization of a computer system * Virtual meeting, or web conferencing * Virtual memory, a memory management technique that abstracts the memory address space in a computer * Virtual particle, a type of short-lived particle of indeterminate mass * Virtual reality (virtuality), computer programs with an interface that gives the user the impression that they are physically inside a simulated space * Virtual world, a computer-based simulated environment populated by many users who can create a personal avatar, and simultaneously and independently explore the world, participate in its activities and co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeno's Paradoxes
Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. It is usually assumed, based on Plato's ''Parmenides'' (128a–d), that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides' view. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one." Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point. Some of Zeno's nine surviving paradoxes (preserved in Aristotle's ''Physics'' [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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