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Random walk

Random walk
Example of eight random walks in one dimension starting at 0. The plot shows the current position on the line (vertical axis) versus the time steps (horizontal axis). A random walk is a mathematical formalization of a path that consists of a succession of random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler can all be modeled as random walks, although they may not be truly random in reality. The term random walk was first introduced by Karl Pearson in 1905.[1] Random walks have been used in many fields: ecology, economics, psychology, computer science, physics, chemistry, and biology.[2][3][4][5][6][7][8][9] Random walks explain the observed behaviors of processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity. Various different types of random walks are of interest. . . Lattice random walk[edit] . Related:  Fractals Mandlebrots & Dreams of Electric SheepCollection

Turbulence Flow visualization of a turbulent jet, made by laser-induced fluorescence. The jet exhibits a wide range of length scales, an important characteristic of turbulent flows. Laminar and turbulent water flow over the hull of a submarine In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic property changes. Flow in which the kinetic energy dies out due to the action of fluid molecular viscosity is called laminar flow. Features[edit] Turbulence is characterized by the following features: Irregularity: Turbulent flows are always highly irregular. Turbulent diffusion is usually described by a turbulent diffusion coefficient. Via this energy cascade, turbulent flow can be realized as a superposition of a spectrum of flow velocity fluctuations and eddies upon a mean flow. Integral length scales: Largest scales in the energy spectrum. According to an apocryphal story, Werner Heisenberg was asked what he would ask God, given the opportunity. and pressure where ). ).

Generalized continued fraction In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values. A generalized continued fraction is an expression of the form where the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction. The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas: and in general[1] If the sequence of convergents {xn} approaches a limit the continued fraction is convergent and has a definite value. History of continued fractions[edit] The story of continued fractions begins with the Euclidean algorithm,[4] a procedure for finding the greatest common divisor of two natural numbers m and n. Notation[edit] Pringsheim wrote a generalized continued fraction this way: and If , then

Markov process Markov process example Introduction[edit] A Markov process is a stochastic model that has the Markov property. Note that there is no definitive agreement in literature on the use of some of the terms that signify special cases of Markov processes. Markov processes arise in probability and statistics in one of two ways. Markov property[edit] The general case[edit] Let , for some (totally ordered) index set ; and let be a measurable space. adapted to the filtration is said to possess the Markov property with respect to the if, for each and each with s < t, A Markov process is a stochastic process which satisfies the Markov property with respect to its natural filtration. For discrete-time Markov chains[edit] In the case where is a discrete set with the discrete sigma algebra and , this can be reformulated as follows: Examples[edit] Gambling[edit] Suppose that you start with $10, and you wager $1 on an unending, fair, coin toss indefinitely, or until you lose all of your money. , then the sequence .

Complexity There is no absolute definition of what complexity means, the only consensus among researchers is that there is no agreement about the specific definition of complexity. However, a characterization of what is complex is possible.[1] Complexity is generally used to characterize something with many parts where those parts interact with each other in multiple ways. The study of these complex linkages is the main goal of complex systems theory. In science,[2] there are at this time a number of approaches to characterizing complexity, many of which are reflected in this article. Neil Johnson admits that "even among scientists, there is no unique definition of complexity - and the scientific notion has traditionally been conveyed using particular examples..." Overview[edit] Definitions of complexity often depend on the concept of a "system"—a set of parts or elements that have relationships among them differentiated from relationships with other elements outside the relational regime.

Cantor's diagonal argument An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above. An infinite set may have the same cardinality as a proper subset of itself, as the depicted bijectionf(x)=2x from the natural to the even numbers demonstrates. Nevertheless, infinite sets of different cardinalities exist, as Cantor's diagonal argument shows. In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.[1] Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began. An uncountable set[edit] The set T is uncountable. Interpretation[edit] Real numbers[edit]

Astronomers discover complex organic matter exists throughout the universe -- ScienceDaily Astronomers report in the journal Nature that organic compounds of unexpected complexity exist throughout the Universe. The results suggest that complex organic compounds are not the sole domain of life but can be made naturally by stars. Prof. The researchers investigated an unsolved phenomenon: a set of infrared emissions detected in stars, interstellar space, and galaxies. Not only are stars producing this complex organic matter, they are also ejecting it into the general interstellar space, the region between stars. Most interestingly, this organic star dust is similar in structure to complex organic compounds found in meteorites. Prof.

Fractal flame Fractal flames differ from ordinary iterated function systems in three ways: Nonlinear functions are iterated instead of affine transforms.Log-density display instead of linear or binary (a form of tone mapping)Color by structure (i.e. by the recursive path taken) instead of monochrome or by density. The tone mapping and coloring are designed to display as much of the detail of the fractal as possible, which generally results in a more aesthetically pleasing image. Algorithm[edit] The algorithm consists of two steps: creating a histogram and then rendering the histogram. Creating the histogram[edit] First one iterates a set of functions, starting from a randomly chosen point P = (P.x,P.y,P.c), where the third coordinate indicated the current color of the point. Set of flame functions: In each iteration, choose one of the functions above where the probability that Fj is chosen is pj. Each individual function has the following form: The functions Vk are a set of predefined functions.

Interesting number paradox The interesting number paradox is a semi-humorous paradox which arises from the attempt to classify natural numbers as "interesting" or "dull". The paradox states that all natural numbers are interesting. The "proof" is by contradiction: if there exists a non-empty set of uninteresting numbers, there would be a smallest uninteresting number – but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number, producing a contradiction. Paradoxical nature[edit] Attempting to classify all numbers this way leads to a paradox or an antinomy of definition. However, as there are many significant results in mathematics that make use of self-reference (such as Gödel's Incompleteness Theorem), the paradox illustrates some of the power of self-reference, and thus touches on serious issues in many fields of study. One proposed resolution of the paradox asserts that only the first uninteresting number is made interesting by that fact. See also[edit] Notes[edit]

Monte Carlo method Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results; typically one runs simulations many times over in order to obtain the distribution of an unknown probabilistic entity. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to obtain a closed-form expression, or infeasible to apply a deterministic algorithm. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration and generation of draws from a probability distribution. The modern version of the Monte Carlo method was invented in the late 1940s by Stanislaw Ulam, while he was working on nuclear weapons projects at the Los Alamos National Laboratory. Introduction[edit] Monte Carlo method applied to approximating the value of π. Monte Carlo methods vary, but tend to follow a particular pattern: History[edit] Definitions[edit]

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