Ramification Theory Of Valuations
Ramification may refer to: * Ramification (mathematics), a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. * Ramification (botany), the divergence of the stem and limbs of a plant into smaller ones * Ramification group, filtration of the Galois group of a local field extension * Ramification theory of valuations, studies the set of extensions of a valuation v of a field K to an extension L of K * Ramification problem, in philosophy and artificial intelligence, concerned with the indirect consequences of an action. *Type theory, Ramified Theory of Types by mathematician Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...

Ramification (mathematics)
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping. In complex analysis In complex analysis, the basic model can be taken as the ''z'' → ''z''''n'' mapping in the complex plane, near ''z'' = 0. This is the standard local picture in Riemann surface theory, of ramification of order ''n''. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus. See also branch point. In algebraic topology In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The ''z'' → ''z''''n'' mapping shows this as a local ...
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Ramification (botany)
In botany, ramification is the divergence of the stem and limbs of a plant into smaller ones, i.e., trunk into branches, branches into increasingly smaller branches, and so on. Gardeners stimulate the process of ramification through pruning, thereby making trees, shrubs, and other plants bushier and denser. Short internodes (the section of stem between nodes, i.e., areas where leaves are produced) help increase ramification in those plants that form branches at these nodes. Long internodes (which may be the result of over-watering, the over-use of fertilizer, or a seasonal "growth spurt") decrease a gardener's ability to induce ramification in a plant. A high degree of ramification is essential for the creation of topiary as it enables the topiary artist to carve a bush or hedge into a shape with an even surface. Ramification is also essential to practitioners of the art of bonsai as it helps re-create the form and habit of a full-size tree in a small tree grown in a contai ...
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Ramification Group
In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension. Ramification theory of valuations In mathematics, the ramification theory of valuations studies the set of extensions of a valuation ''v'' of a field ''K'' to an extension ''L'' of ''K''. It is a generalization of the ramification theory of Dedekind domains. The structure of the set of extensions is known better when ''L''/''K'' is Galois. Decomposition group and inertia group Let (''K'', ''v'') be a valued field and let ''L'' be a finite Galois extension of ''K''. Let ''Sv'' be the set of equivalence classes of extensions of ''v'' to ''L'' and let ''G'' be the Galois group of ''L'' over ''K''. Then ''G'' acts on ''Sv'' by σ 'w''nbsp;=  'w'' ∘ σ(i.e. ''w'' is a representative of the equivalence class 'w''nbsp;∈&nb ...
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Ramification Theory Of Valuations
Ramification may refer to: * Ramification (mathematics), a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. * Ramification (botany), the divergence of the stem and limbs of a plant into smaller ones * Ramification group, filtration of the Galois group of a local field extension * Ramification theory of valuations, studies the set of extensions of a valuation v of a field K to an extension L of K * Ramification problem, in philosophy and artificial intelligence, concerned with the indirect consequences of an action. *Type theory, Ramified Theory of Types by mathematician Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...

Ramification Problem
In philosophy and artificial intelligence (especially, knowledge based systems), the ramification problem is concerned with the indirect consequences of an action. It might also be posed as ''how to represent what happens implicitly due to an action'' or how to control the secondary and tertiary effects of an action. It is strongly connected to, and is opposite the qualification side of, the frame problem. Limit theory helps in operational usage. For instance, in KBE derivation of a populated design (geometrical objects, etc., similar concerns apply in shape theory), equivalence assumptions allow convergence where potentially large, and perhaps even computationally indeterminate, solution sets are handled deftly. Yet, in a chain of computation, downstream events may very well find some types of results from earlier resolutions of ramification as problematic for their own algorithms. See also *Non-monotonic logic A non-monotonic logic is a formal logic whose conclusion relatio ...
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Type Theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that were proposed as foundations are Alonzo Church's typed λ-calculus and Per Martin-Löf's intuitionistic type theory. Most computerized proof-writing systems use a type theory for their foundation. A common one is Thierry Coquand's Calculus of Inductive Constructions. History Type theory was created to avoid a paradox in a mathematical foundation based on naive set theory and formal logic. Russell's paradox, which was discovered by Bertrand Russell, existed because a set could be defined using "all possible sets", which included itself. Between 1902 and 1908, Bertrand Russell proposed various "theories of type" to fix the problem. By 1908 Russell arrived at a "ramified" theory ...
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