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Fuzzy Measure Theory
In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also ''capacity'', see ), which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different classes of fuzzy measures including plausibility/belief measures; possibility/necessity measures; and probability measures, which are a subset of classical measures. Definitions Let \mathbf be a universe of discourse, \mathcal be a class of subsets of \mathbf, and E,F\in\mathcal. A function g:\mathcal\to\mathbb where # \emptyset \in \mathcal \Rightarrow g(\emptyset)=0 # E \subseteq F \Rightarrow g(E)\leq g(F) is called a ''fuzzy measure''. A fuzzy measure is called ''normalized'' or ''regular'' if g(\mathbf)=1. Properties of fuzzy measures A fuzzy measure is: * additive if for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and mathematical analysis, analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Weighted Averaging Aggregation Operator
In applied mathematics – specifically in fuzzy logic – the ordered weighted averaging (OWA) operators provide a parameterized class of mean type aggregation operators. They were introduced by Ronald R. Yager. Many notable mean operators such as the max, arithmetic average, median and min, are members of this class. They have been widely used in computational intelligence because of their ability to model linguistically expressed aggregation instructions. Definition Formally an OWA operator of dimension \ n is a mapping F: R_n \rightarrow R that has an associated collection of weights \ W = _1, \ldots, w_n lying in the unit interval and summing to one and with : F(a_1, \ldots , a_n) = \sum_^n w_j b_j where b_j is the ''j''th largest of the a_i . By choosing different ''W'' one can implement different aggregation operators. The OWA operator is a non-linear operator as a result of the process of determining the ''b''''j''. Properties The OWA operator is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exotic Probabilities
Exotic may refer to: Mathematics and physics * Exotic R4, a differentiable 4-manifold, homeomorphic but not diffeomorphic to the Euclidean space R4 *Exotic sphere, a differentiable ''n''-manifold, homeomorphic but not diffeomorphic to the ordinary ''n''-sphere *Exotic atom, an atom with one or more electrons replaced by other negatively charged particles *Exotic hadron **Exotic baryon, bound states of 3 quarks and additional particles **Exotic meson, non-quark model mesons *Exotic matter, a hypothetical concept of particle physics Music * "Exotic" (1963 song), a song by The Sentinals from the 1963 album ''Surf Crazy - Original Surfin' Hits'' * "Exotic" (Lil Baby song), 2018 * "Exotic" (Priyanka Chopra song), a 2012 song by Priyanka Chopra featuring Pitbull Flora and fauna *Exotic pet *Exotic Shorthair, a breed of cat *Exotic species (or introduced species), a species not native to an area Other *Exotic dancer, a type of dancer or stripper *Exotic derivative, a type of financia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George J
George may refer to: People * George (given name) * George (surname) * George (singer), American-Canadian singer George Nozuka, known by the mononym George * George Washington, First President of the United States * George W. Bush, 43rd President of the United States * George H. W. Bush, 41st President of the United States * George V, King of Great Britain, Ireland, the British Dominions and Emperor of India from 1910-1936 * George VI, King of Great Britain, Ireland, the British Dominions and Emperor of India from 1936-1952 * Prince George of Wales * George Papagheorghe also known as Jorge / GEØRGE * George, stage name of Giorgio Moroder * George Harrison, an English musician and singer-songwriter Places South Africa * George, Western Cape ** George Airport United States * George, Iowa * George, Missouri * George, Washington * George County, Mississippi * George Air Force Base, a former U.S. Air Force base located in California Characters * George (Peppa Pig), a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shapley Value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a unique distribution (among the players) of a total surplus generated by the coalition of all players. The Shapley value is characterized by a collection of desirable properties. Hart (1989) provides a survey of the subject. The setup is as follows: a coalition of players cooperates, and obtains a certain overall gain from that cooperation. Since some players may contribute more to the coalition than others or may possess different bargaining power (for example threatening to destroy the whole surplus), what final distribution of generated surplus among the players should arise in any particular game? Or phrased differently: how important is each player to the overall cooperation, and what payoff can he or she reasonably expect? The Shapley va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' joins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Sets And Systems
''Fuzzy Sets and Systems'' is a peer-reviewed international scientific journal published by Elsevier on behalf of the International Fuzzy Systems Association (IFSA) and was founded in 1978. The editors-in-chief (as of 2010) are Bernard De Baets of the Department of Data Analysis and Mathematical Modelling (at Ghent University in Belgium), Didier Dubois (of IRIT, Université Paul Sabatier in Toulouse, France) and Eyke Hüllermeier (of the Department of Mathematics, Statistics and Computer Science, Ludwig-Maximilians Universität München, Germany). The journal publishes 24 issues a year. ''Fuzzy Sets and Systems'' is abstracted and indexed by Scopus and the Science Citation Index. According to the Journal Citation Reports released in 2010, its 2-year impact factor calculated for 2020 is 3.343 and its 5-year impact factor for 2020 is 3.213. References {{reflist See also * Fuzzy control system * Fuzzy Control Language * Fuzzy logic Fuzzy logic is a form of many-value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tokyo Institute Of Technology
is a national research university located in Greater Tokyo Area, Japan. Tokyo Tech is the largest institution for higher education in Japan dedicated to science and technology, one of first five Designated National University and selected as a Top Type university of Top Global University Project by the Japanese government. It is generally considered to be one of the most prestigious universities in Japan. Tokyo Tech's main campus is located at Ōokayama on the boundary of Meguro and Ota, with its main entrance facing the Ōokayama Station. Other campuses are located in Suzukakedai and Tamachi. Tokyo Tech is organised into 6 schools, within which there are over 40 departments and research centres. Tokyo Tech enrolled 4,734 undergraduates and 1,464 graduate students for 2015–2016. It employs around 1,100 faculty members. Tokyo Institute of Technology produced a Nobel Prize laureate in Chemistry Hideki Shirakawa Ph.D. History Foundation and early years (1881–1922) Toky ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-criteria Decision Analysis
Multiple-criteria decision-making (MCDM) or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision making (both in daily life and in settings such as business, government and medicine). Conflicting criteria are typical in evaluating options: cost or price is usually one of the main criteria, and some measure of quality is typically another criterion, easily in conflict with the cost. In purchasing a car, cost, comfort, safety, and fuel economy may be some of the main criteria we consider – it is unusual that the cheapest car is the most comfortable and the safest one. In portfolio management, managers are interested in getting high returns while simultaneously reducing risks; however, the stocks that have the potential of bringing high returns typically carry high risk of losing money. In a service industry, customer satisfaction and the cost of providing service are fundamen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . The notation , meaning the set of all functions from S to a given set of two elements (e.g., ), is used because the powerset of can be identified with, equivalent to, or bijective to the set of all the functions from to the given two elements set. Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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