Refinement (sigma Algebra)
Refinement may refer to: Mathematics * Equilibrium refinement, the identification of actualized equilibria in game theory * Refinement of an equivalence relation, in mathematics ** Refinement (topology), the refinement of an open cover in mathematical topology * Refinement (category theory) Other uses * Refinement (computing), computer science approaches for designing correct computer programs and enabling their formal verification * Refining, a process of purification ** Refining (metallurgy) * Refinement (culture), a quality of cultural sophistication * Refinement (horse), a racehorse ridden by jockey Tony McCoy Sir Anthony Peter McCoy (born 4 May 1974), commonly known as AP McCoy or Tony McCoy, is a Northern Irish former National Hunt horse racing jockey. Based in Ireland and the UK, McCoy rode a record 4,358 winners, and was Champion Jockey a reco ...

Equilibrium Refinement
In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium. Many solution concepts, for many games, will result in more than one solution. This puts any one of the solutions in doubt, so a game theorist may apply a refinement to narrow down the solutions. Each successive solution concept presented in the following improves on its predecessor by eliminating implausible equilibria in richer games. Formal definition Let \Gamma be the class of all games and, for each game G \in \Gamma, let S_G be the set of strategy profiles of G. A ''solution concept'' is an element of the direct product \Pi_2^; ''i.e''., a function F: \Gamma \rightarrow \bigcup\nolimits_ 2^ such that F(G) \subseteq S_G for all G \in \Gamma. ...
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Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definition A binary relation \,\sim\, on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a, b, and c in X: * a \sim a ( ...
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Refinement (topology)
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\subset X, then C is a cover of X if \bigcup_U_ = X. Thus the collection \lbrace U_\alpha : \alpha \in A \rbrace is a cover of X if each element of X belongs to at least one of the subsets U_. Cover in topology Covers are commonly used in the context of topology. If the set X is a topological space, then a ''cover'' C of X is a collection of subsets \_ of X whose union is the whole space X. In this case we say that C ''covers'' X, or that the sets U_\alpha ''cover'' X. Also, if Y is a (topological) subspace of X, then a ''cover'' of Y is a collection of subsets C=\_ of X whose union contains Y, i.e., C is a cover of Y if :Y \subseteq \bigcup_U_. That is, we may cover Y with either open sets in Y itself, or cover Y by open sets in the p ...
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Refinement (category Theory)
In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification or saturation of a locally convex space. A dual construction is called envelope. Definition Suppose K is a category, X an object in K, and \Gamma and \Phi two classes of morphisms in K. The definition of a refinement of X in the class \Gamma by means of the class \Phi consists of two steps. * A morphism \sigma:X'\to X in K is called an ''enrichment of the object X in the class of morphisms \Gamma by means of the class of morphisms \Phi'', if \sigma\in\Gamma, and for any morphism \varphi:B\to X from the class \Phi there exists a unique morphism \varphi':B\to X' in K such that \varphi=\sigma\circ\varphi'. * An enrichment \rho:E\to X of the object X in the class of morphisms \Gamma by means of the class of morphisms \Phi is called a ''refinement of X in \Gamma by means of \Phi'', if for any other enrichment \sigma ...
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Refinement (computing)
Refinement is a generic term of computer science that encompasses various approaches for producing correct computer programs and simplifying existing programs to enable their formal verification. Program refinement In formal methods, program refinement is the verifiable transformation of an ''abstract'' (high-level) formal specification into a ''concrete'' (low-level) executable program. ''Stepwise refinement'' allows this process to be done in stages. Logically, refinement normally involves implication, but there can be additional complications. The progressive just-in-time preparation of the product backlog (requirements list) in agile software development approaches, such as Scrum, is also commonly described as refinement. Data refinement Data refinement is used to convert an abstract data model (in terms of sets for example) into implementable data structures (such as arrays). Operation refinement converts a specification of an operation on a system into an implementab ...
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Refining
{{Unreferenced, date=December 2009 Refining (also perhaps called by the mathematical term affining) is the process of purification of a (1) substance or a (2) form. The term is usually used of a natural resource that is almost in a usable form, but which is more useful in its pure form. For instance, most types of natural petroleum will burn straight from the ground, but it will burn poorly and quickly clog an engine with residues and by-products. The term is broad, and may include more drastic transformations, such as the reduction of ore to metal (for which see Refining (metallurgy)). The refining of liquids is often accomplished by distillation or fractionation; this process is useful, for example, for isolating different fractions of petroleum. Gases can be refined in this way as well, by being cooled and/or compressed until they liquefy. Gases and liquids can also be refined by extraction with a selective solvent that dissolves away either the substance of interest, or th ...
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Refining (metallurgy)
In metallurgy, refining consists of purifying an impure metal. It is to be distinguished from other processes such as smelting and calcining in that those two involve a chemical change to the raw material, whereas in refining, the final material is usually identical chemically to the original one, only it is purer. The processes used are of many types, including pyrometallurgical and hydrometallurgical techniques. Lead Cupellation One ancient process for extracting the silver from lead was cupellation. Lead was melted in a bone ash 'test' or 'cupel' and air blown across the surface. This oxidised the lead to litharge, and also oxidised other base metals present, the silver (and gold if present) remaining unoxidised. In the 18th century, the process was carried on using a kind of reverberatory furnace, but differing from the usual kind in that air was blown over the surface of the molten lead from bellows or (in the 19th century) blowing cylinders. Pattinson proces ...
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Refinement (culture)
Virtue ethics (also aretaic ethics, from Greek ἀρετή arete_(moral_virtue).html"_;"title="'arete_(moral_virtue)">aretḗ''_is_an_approach_to_ethics_that_treats_the_concept_of_virtue.html" ;"title="arete_(moral_virtue)">aretḗ''.html" ;"title="arete_(moral_virtue).html" ;"title="'arete (moral virtue)">aretḗ''">arete_(moral_virtue).html" ;"title="'arete (moral virtue)">aretḗ'' is an approach to ethics that treats the concept of virtue">moral virtue as central. Virtue ethics is usually contrasted with two other major approaches in ethics, consequentialism and deontology, which make the goodness of outcomes of an action (consequentialism) and the concept of moral duty (deontology) central. While virtue ethics does not necessarily deny the importance of goodness of states of affairs or moral duties to ethics, it emphasizes moral virtue, and sometimes other concepts, like ''eudaimonia'', to an extent that other ethical dispositions do not. Key concepts Virtue and vice ...
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Refinement (horse)
Refinement may refer to: Mathematics * Equilibrium refinement, the identification of actualized equilibria in game theory * Refinement of an equivalence relation, in mathematics ** Refinement (topology), the refinement of an open cover in mathematical topology * Refinement (category theory) Other uses * Refinement (computing), computer science approaches for designing correct computer programs and enabling their formal verification * Refining, a process of purification ** Refining (metallurgy) * Refinement (culture), a quality of cultural sophistication * Refinement (horse), a racehorse ridden by jockey Tony McCoy Sir Anthony Peter McCoy (born 4 May 1974), commonly known as AP McCoy or Tony McCoy, is a Northern Irish former National Hunt horse racing jockey. Based in Ireland and the UK, McCoy rode a record 4,358 winners, and was Champion Jockey a reco ...