 # 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. Random walks have been used in many fields: ecology, economics, psychology, computer science, physics, chemistry, and biology. 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 .

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 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 ). ).

Markov process Markov process example Introduction 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 The general case 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 In the case where is a discrete set with the discrete sigma algebra and , this can be reformulated as follows: Examples Gambling 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 .

Markov chains don’t converge I often hear people often say they’re using a burn-in period in MCMC to run a Markov chain until it converges. But Markov chains don’t converge, at least not the Markov chains that are useful in MCMC. These Markov chains wander around forever exploring the domain they’re sampling from. Not only that, Markov chains can’t remember how they got where they are. When someone says a Markov chain has converged, they may mean that the chain has entered a high-probability region. Burn-in may be ineffective. Why use burn-in? So why does it matter whether you start your Markov chain in a high-probability region? Samples from Markov chains don’t converge, but averages of functions applied to these samples may converge. It’s not just a matter of imprecise language when people say a Markov chain has converged.

Markov renewal process In probability and statistics a Markov renewal process is a random process that generalizes the notion of Markov jump processes. Other random processes like Markov chain, Poisson process, and renewal process can be derived as a special case of an MRP (Markov renewal process). Definition Consider a state space Consider a set of random variables , where are the jump times and are the associated states in the Markov chain (see Figure). . Relation to other stochastic processes If we define a new stochastic process for , then the process is called a semi-Markov process. See also References and Further Reading

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. 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, 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 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.

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.

Markov chain A simple two-state Markov chain A Markov chain (discrete-time Markov chain or DTMC), named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another on a state space. It is a random process usually characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes. Introduction A Markov chain is a stochastic process with the Markov property. In literature, different Markov processes are designated as "Markov chains". The changes of state of the system are called transitions. A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. Many other examples of Markov chains exist. Formal definition . . of states as input, where is defined, while and

Markov model Introduction The most common Markov models and their relationships are summarized in the following table: Markov chain The simplest Markov model is the Markov chain. It models the state of a system with a random variable that changes through time. Hidden Markov model A hidden Markov model is a Markov chain for which the state is only partially observable. Markov decision process A Markov decision process is a Markov chain in which state transitions depend on the current state and an action vector that is applied to the system. Partially observable Markov decision process A partially observable Markov decision process (POMDP) is a Markov decision process in which the state of the system is only partially observed. Markov random field A Markov random field (also called a Markov network) may be considered to be a generalization of a Markov chain in multiple dimensions. See also References

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 The algorithm consists of two steps: creating a histogram and then rendering the histogram. Creating the histogram 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.

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