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Highly Optimized Tolerance
In applied mathematics, highly optimized tolerance (HOT) is a method of generating power law behavior in systems by including a global optimization principle. It was developed by Jean M. Carlson and John Doyle in the early 2000s. For some systems that display a characteristic scale, a global optimization term could potentially be added that would then yield power law behavior. It has been used to generate and describe internet-like graphs, forest fire models and may also apply to biological systems. Example The following is taken from Sornette's book. Consider a random variable, X, that takes on values x_i with probability p_i. Furthermore, let’s assume for another parameter r_i :x_i = r_i^ for some fixed \beta. We then want to minimize : L = \sum_^ p_i x_i subject to the constraint : \sum_^ r_i = \kappa Using Lagrange multipliers, this gives : p_i \propto x_i^ giving us a power law. The global optimization of minimizing the energy along with the power law dependence b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the profession, professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of Mathematical analysis, applied analysis, most notably differential equations; approximation theory (broadly construed, ... [...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] |
Global Optimization
Global optimization is a branch of operations research, applied mathematics, and numerical analysis that attempts to find the global minimum or maximum of a function or a set of functions on a given set. It is usually described as a minimization problem because the maximization of the real-valued function g(x) is equivalent to the minimization of the function f(x):=(-1)\cdot g(x). Given a possibly nonlinear and non-convex continuous function f:\Omega\subset\mathbb^n\to\mathbb with the global minimum f^* and the set of all global minimizers X^* in \Omega, the standard minimization problem can be given as :\min_f(x), that is, finding f^* and a global minimizer in X^*; where \Omega is a (not necessarily convex) compact set defined by inequalities g_i(x)\geqslant0, i=1,\ldots,r. Global optimization is distinguished from local optimization by its focus on finding the minimum or maximum over the given set, as opposed to finding ''local'' minima or maxima. Finding an arbitrary local m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean M
Jean may refer to: People * Jean (female given name) * Jean (male given name) * Jean (surname) Fictional characters * Jean Grey, a Marvel Comics character * Jean Valjean, fictional character in novel ''Les Misérables'' and its adaptations * Jean Pierre Polnareff, a fictional character from ''JoJo's Bizarre Adventure'' * Jean Luc Picard, fictional character from ''Star Trek Next Generation'' Places * Jean, Nevada, United States; a town * Jean, Oregon, United States Entertainment * Jean (dog), a female collie in silent films * Jean (song), "Jean" (song) (1969), by Rod McKuen, also recorded by Oliver * Jean Seberg (musical), ''Jean Seberg'' (musical), a 1983 musical by Marvin Hamlisch Other uses * JEAN (programming language) * USS Jean (ID-1308), USS ''Jean'' (ID-1308), American cargo ship c. 1918 * Sternwheeler Jean, a 1938 paddleboat of the Willamette River See also *Jehan * * Gene (other) * Jeanne (other) * Jehanne (other) * Jeans (disambiguat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Doyle (engineer)
John Comstock Doyle is the Jean-Lou Chameau Professor of Control and Dynamical Systems, Electrical Engineering, and BioEngineering at the California Institute of Technology. He is known for his work in control theory and his current research interests are in theoretical foundations for complex networks in engineering, biology, and multiscale physics. Education He earned Bachelor of Science and Master of Science degrees in electrical engineering from the Massachusetts Institute of Technology in 1977 and a Ph.D. in mathematics from the University of California, Berkeley, in 1984 with his thesis titled ''Matrix interpolation theory and optimal control''. Career Doyle's early work was in the mathematics of robust control, linear-quadratic-Gaussian control robustness, (structured) singular value analysis, and H-infinity methods. He has co-authored books and software toolboxes, and a control analysis tool for high performance commercial and military aerospace systems, as well as othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forest Fire
A wildfire, forest fire, or a bushfire is an unplanned and uncontrolled fire in an area of combustible vegetation. Depending on the type of vegetation present, a wildfire may be more specifically identified as a bushfire ( in Australia), desert fire, grass fire, hill fire, peat fire, prairie fire, vegetation fire, or veld fire. Some natural forest ecosystems depend on wildfire. Modern forest management often engages in prescribed burns to mitigate fire risk and promote natural forest cycles. However, controlled burns can turn into wildfires by mistake. Wildfires can be classified by cause of ignition, physical properties, combustible material present, and the effect of weather on the fire. Wildfire severity results from a combination of factors such as available fuels, physical setting, and weather. Climatic cycles with wet periods that create substantial fuels, followed by drought and heat, often precede severe wildfires. These cycles have been intensified by climate cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function (mathematics), function in which * the Domain of a function, domain is the set of possible Outcome (probability), outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the Range of a function, range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the Real number, real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Multipliers
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). It is named after the mathematician Joseph-Louis Lagrange. Summary and rationale The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function or Lagrangian. In the general case, the Lagrangian is defined as \mathcal(x, \lambda) \equiv f(x) + \langle \lambda, g(x)\rangle for functions f, g; the notation \langle \cdot, \cdot \rangle denotes an inner product. The value \lambda is called the Lagrange multiplier. In simple ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-organized Criticality
Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune control parameters to a precise value, because the system, effectively, tunes itself as it evolves towards criticality. The concept was put forward by Per Bak, Chao Tang and Kurt Wiesenfeld ("BTW") in a paper , following an earlier paper by Jonathan Katz published in 1987 in ''Physical Review Letters'', and is considered to be one of the mechanisms by which complexity arises in nature. Its concepts have been applied across fields as diverse as geophysics, physical cosmology, evolutionary biology and ecology, bio-inspired computing and optimization (mathematics), economics, quantum gravity, sociology, solar physics, plasma physics, neurobiology and others. SOC is typically observed in slowl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Review E
''Physical Review E'' is a peer-reviewed, scientific journal, published monthly by the American Physical Society. The main field of interest is collective phenomena of many-body systems. It is edited by Dario Corradini as of December 2024. While original research content requires subscription, editorials, news, and other non-research content is openly accessible. Scope Although the focus of this journal is many-body phenomena, the broad scope of the journal includes quantum chaos, soft matter physics, classical chaos, biological physics and granular materials. Also emphasized are statistical physics, equilibrium and transport properties of fluids, liquid crystals, complex fluids, polymers, chaos, fluid dynamics, plasma physics, classical physics, and computational physics. About Physical Review E APS. July 2010 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Review Letters
''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. The journal is considered one of the most prestigious in the field of physics. Over a quarter of Physics Nobel Prize-winning papers between 1995 and 2017 were published in it. ''PRL'' is published both online and as a print journal. Its focus is on short articles ("letters") intended for quick publication. The Lead Editor is Hugues Chaté. The Managing Editor is Robert Garisto. History The journal was created in 1958. Samuel Goudsmit, who was then the editor of '' Physical Review'', the American Physical Society's flagship journal, organized and published ''Letters to the Editor of Physical Review'' into a new standalone journal'','' which became ''Physical Review Letters''. It was the first journal intended for the rapid publication of short articles, a format that eventually became popular in many other fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Science News
''Science News'' (''SN'') is an American monthly magazine devoted to articles about new scientific and technical developments, typically gleaned from recent scientific and technical journals. The periodical has been described as having a scope across "all sciences" and as having "up to date" coverage. History ''Science News'' has been published since 1922 by the Society for Science & the Public, a non-profit organization founded by E. W. Scripps in 1920. American chemist Edwin Slosson served as the publication's first editor. From 1922 to 1966, it was called ''Science News Letter''. The title was changed to ''Science News'' with the March 12, 1966, issue (vol. 89, no. 11). Tom Siegfried was the editor from 2007 to 2012. In 2012, Siegfried stepped down, and Eva Emerson became the Editor in Chief of the magazine. In 2017, Eva Emerson stepped down to become the editor of a new digital magazine, ''Annual Reviews''. On February 1, 2018, Nancy Shute became the Editor in Chief of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
