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Monte Carlo Method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on Resampling (statistics), repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic system, deterministic in principle. The name comes from the Monte Carlo Casino in Monaco, where the primary developer of the method, physicist Stanisław Ulam, Stanislaw Ulam, was inspired by his uncle's gambling habits. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. They can also be used to model phenomena with significant uncertainty in inputs, such as calculating the risk of a nuclear power plant failure. Monte Carlo methods are often implemented using computer simulations, and they can provide approximate solutions to problems that are otherwise intractable or too complex to analyze mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation D'une Distribution Normale
An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ''ad-'' (''ad-'' before ''p'' becomes ap- by assimilation) meaning ''to''. Words like ''approximate'', ''approximately'' and ''approximation'' are used especially in technical or scientific contexts. In everyday English, words such as ''roughly'' or ''around'' are used with a similar meaning. It is often found abbreviated as ''approx.'' The term can be applied to various properties (e.g., value, quantity, image, description) that are nearly, but not exactly correct; similar, but not exactly the same (e.g., the approximate time was 10 o'clock). Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws. In science, approximation can refer t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., an integral assigns numbers to functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Derivative, differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Risk
In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environment), often focusing on negative, undesirable consequences. Many different definitions have been proposed. The international standard definition of risk for common understanding in different applications is “effect of uncertainty on objectives”. The understanding of risk, the methods of assessment and management, the descriptions of risk and even the definitions of risk differ in different practice areas (business, economics, environment, finance, information technology, health, insurance, safety, security etc). This article provides links to more detailed articles on these areas. The international standard for risk management, ISO 31000, provides principles and generic guidelines on managing risks faced by organizations. Definitions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncertainty
Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable or stochastic environments, as well as due to ignorance, indolence, or both. It arises in any number of fields, including insurance, philosophy, physics, statistics, economics, finance, medicine, psychology, sociology, engineering, metrology, meteorology, ecology and information science. Concepts Although the terms are used in various ways among the general public, many specialists in decision theory, statistics and other quantitative fields have defined uncertainty, risk, and their measurement as: Uncertainty The lack of certainty, a state of limited knowledge where it is impossible to exactly describe the existing state, a future outcome, or more than one possible outcome. ;Measurement of uncertainty: A set of possible states or outc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Theory Of Gases
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and entertainment * Kinetic art, a form of art involving mechanical and/or random movement, including optical illusions. * ''Kinetic'', the 13th episode of the first season of the TV series ''Smallville'' * ''Kinetic'' (comics), a comic by Allan Heinberg and Kelley Pucklett * "Kinetic" (song), a song by Radiohead Companies * Kinetic Engineering Limited, Indian automotive manufacturer * Kinetic Group, Australian-based public transport company Technology * "Kinetic", Seiko's trademark for its automatic quartz technology * The ''Kinetic camera system'' by Birt Acres (1854–1918), photographer and film pioneer * Kinetic projectile Military terminology * Kinetic military action See also * * * Kinetics (other) * Dynamics (disambiguatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
McKean–Vlasov Process
In probability theory, a McKean–Vlasov process is a stochastic process described by a stochastic differential equation where the coefficients of the diffusion depend on the distribution of the solution itself. The equations are a model for Vlasov equation and were first studied by Henry McKean in 1966. Definition McKean–Vlasov processes take the formdX_t = a(X_t, \mu_t) dB_t + b(X_t, \mu_t) dtwhere \mu_t = \mathcal(X_t) describes the law of X and dB denotes the Wiener process. That is the coefficients of the SDE depend on the marginal distribution of the process X. In general, the process can describe non-linear diffusion. Applications * Mean-field theory * Mean-field game theory * Random matrices: including Dyson's model on eigenvalue dynamics for random symmetric matrices and the Wigner semicircle distribution The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on minus;''R'', ''R''whose probability density functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interacting Particle Systems
In probability theory, an interacting particle system (IPS) is a stochastic process (X(t))_ on some configuration space \Omega= S^G given by a site space, a countable-infinite graph G and a local state space, a compact metric space S . More precisely IPS are continuous-time Markov jump processes describing the collective behavior of stochastically interacting components. IPS are the continuous-time analogue of stochastic cellular automata. Among the main examples are the voter model, the contact process, the asymmetric simple exclusion process (ASEP), the Glauber dynamics and in particular the stochastic Ising model. IPS are usually defined via their Markov generator giving rise to a unique Markov process using Markov semigroups and the Hille-Yosida theorem. The generator again is given via so-called transition rates c_\Lambda(\eta,\xi)>0 where \Lambda\subset G is a finite set of sites and \eta,\xi\in\Omega with \eta_i=\xi_i for all i\notin\Lambda. The rates describe exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cellular Potts Model
In computational biology, a Cellular Potts model (CPM, also known as the Glazier-Graner-Hogeweg model) is a computational model of cells and tissues. It is used to simulate individual and collective cell behavior, tissue morphogenesis and cancer development. CPM describes cells as deformable objects with a certain volume, that can adhere to each other and to the medium in which they live. The formalism can be extended to include cell behaviours such as cell migration, growth and division, and cell signalling. The first CPM was proposed for the simulation of cell sorting by François Graner and James Glazier as a modification of a large-Q Potts model. CPM was then popularized by Paulien Hogeweg for studying morphogenesis. Although the model was developed to describe biological cells, it can also be used to model individual parts of a biological cell, or even regions of fluid. Model description The CPM consists of a rectangular Euclidean lattice, where each cell is a subset of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degrees Of Freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical processes are possible. This relates to the philosophical concept to the extent that people may be considered to have as much freedom as they are physically able to exercise. Applications Statistics In statistics, degrees of freedom refers to the number of variables in a statistic calculation that can vary. It can be calculated by subtracting the number of estimated parameters from the total number of values in the sample. For example, a sample variance calculation based on n samples will have n-1 degrees of freedom, because sample variance is calculated using the sample mean as an estimate of the actual mean. Mathematics In mathematics, this notion is formalized as the dimension of a manifold or an algebraic variety. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coupling (physics)
In physics, two objects are said to be coupled when they are interacting with each other. In classical mechanics, coupling is a connection between two oscillating systems, such as pendulums connected by a spring. The connection affects the oscillatory pattern of both objects. In particle physics, two particles are coupled if they are connected by one of the four fundamental forces. Wave mechanics Coupled harmonic oscillator If two waves are able to transmit energy to each other, then these waves are said to be "coupled." This normally occurs when the waves share a common component. An example of this is two pendulums connected by a spring. If the pendulums are identical, then their equations of motion are given by m\ddot = -mg\frac - k(x-y) m\ddot = -mg \frac + k(x-y) These equations represent the simple harmonic motion of the pendulum with an added coupling factor of the spring. This behavior is also seen in certain molecules (such as CO2 and H2O), wherein two of the atoms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
