WordNet

an arrangement of branching parts
resembling the branches of a tree
the act of branching out or dividing into branches (同)ramification, fork, forking
a part of a forked or branching shape; "he broke off one of the branches" (同)leg, ramification
divide into two or more branches so as to form a fork; "The road forks" (同)ramify, fork, furcate, separate
a division of a stem, or secondary stem arising from the main stem of a plant
a division of some larger or more complex organization; "a branch of Congress"; "botany is a branch of biology"; "the Germanic branch of Indo-European languages" (同)subdivision, arm
a stream or river connected to a larger one
a framework that supports climbing plants; "the arbor provided a shady resting place in the park" (同)arbour, bower, pergola
tree (as opposed to shrub)
having branches (同)branching, ramose, ramous, ramate

PrepTutorEJDIC

枝に分かれること,分岐 / 結果,成り行き
『枝』 / 『支流』,支脈,支線;分家 / 『支店』,『支部』,支局 / (学問の)『部門』,分科 / 枝を出す,枝を広げる / (川・道・鉄道などが)分岐する
(機械の)軸

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/06/12 00:57:15」(JST)

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This article is about mathematics. For ramifications in plants, see ramification (botany). For the ramification problem in AI and philosophy, see Ramification problem. For Bertrand Russell's Ramified Theory of Types, see Type Theory.

Schematic depiction of ramification: the fibers of almost all points in Y below consist of three points, except for two points in Y marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map f is said to be ramified in these points of Y.

In mathematics, ramification is 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. It 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 together of the fibers of the mapping.

In complex analysis [edit]

In complex analysis, the basic model can be taken as the z zⁿ 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 [edit]

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 zⁿ mapping shows this as a local pattern: if we exclude 0, looking at 0 < |z| < 1 say, we have (from the homotopy point of view) the circle mapped to itself by the n-th power map (Euler-Poincaré characteristic 0), but with the whole disk the Euler-Poincaré characteristic is 1, n – 1 being the 'lost' points as the n sheets come together at z = 0.

In geometric terms, ramification is something that happens in codimension two (like knot theory, and monodromy); since real codimension two is complex codimension one, the local complex example sets the pattern for higher-dimensional complex manifolds. In complex analysis, sheets can't simply fold over along a line (one variable), or codimension one subspace in the general case. The ramification set (branch locus on the base, double point set above) will be two real dimensions lower than the ambient manifold, and so will not separate it into two 'sides', locally―there will be paths that trace round the branch locus, just as in the example. In algebraic geometry over any field, by analogy, it also happens in algebraic codimension one.

In algebraic number theory [edit]

In algebraic extensions of [edit]

See also splitting of prime ideals in Galois extensions

Ramification in algebraic number theory means prime numbers factoring into some repeated prime ideal factors. Let R be the ring of integers of an algebraic number field K and P a prime ideal of R. For each extension field L of K we can consider the integral closure S of R in L and the ideal PS of S. This may or may not be prime, but assuming [L:K] is finite it is a product of prime ideals

P₁^e(1) ... P_k^e(k)

where the P_i are distinct prime ideals of S. Then P is said to ramify in L if e(i) > 1 for some i. In other words, P ramifies in L if the ramification index e(i) is greater than one for some P_i. An equivalent condition is that S/PS has a non-zero nilpotent element: it is not a product of finite fields. The analogy with the Riemann surface case was already pointed out by Richard Dedekind and Heinrich M. Weber in the nineteenth century.

The ramification is encoded in K by the relative discriminant and in L by the relative different. The former is an ideal of the ring of integers of K and is divisible by P if and only if some ideal P_i of S dividing P is ramified. The latter is an ideal if the ring of integers of L and is divisible by the prime ideal P_i of S precisely when P_iis ramified.

The ramification is tame when the ramification indices e(i) are all relatively prime to the residue characteristic p of P, otherwise wild. This condition is important in Galois module theory. A finite generically étale extension of Dedekind domains is tame iff the trace is surjective.

In local fields [edit]

Main article: Ramification of local fields

The more detailed analysis of ramification in number fields can be carried out using extensions of the p-adic numbers, because it is a local question. In that case a quantitative measure of ramification is defined for Galois extensions, basically by asking how far the Galois group moves field elements with respect to the metric. A sequence of ramification groups is defined, reifying (amongst other things) wild (non-tame) ramification. This goes beyond the geometric analogue.

In algebra [edit]

Main article: Ramification theory of valuations

In valuation theory, the ramification theory of valuations studies the set of extensions of a valuation of a field K to an extension field of K. This generalizes the notions in algebraic number theory, local fields, and Dedekind domains.

In algebraic geometry [edit]

There is also corresponding notion of unramified morphism in algebraic geometry. It serves to define étale morphisms.

Let be a morphism of schemes. The support of the quasicoherent sheaf is called the ramification locus of and the image of the ramification locus, , is called the branch locus of . If we say that is formally unramified and if is also of locally finite presentation we say that is unramified [see Vakil's notes].

References [edit]

Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR1697859
Vakil, Ravi, "Foundations of Algebraic Geometry", Lecture Notes, http://math.stanford.edu/~vakil/216blog/

External links [edit]

Ramification in number fields, PlanetMath.org.

English Journal

Semaphorin 5A and plexin-B3 regulate human glioma cell motility and morphology through Rac1 and the actin cytoskeleton.

Li X, Law JW, Lee AY.SourceDepartment of Physiology, Yong Loo Lin School of Medicine; Neurobiology/Ageing Programme, Life Sciences Institute, National University of Singapore, Singapore.
Oncogene.Oncogene.2011 Jun 27. doi: 10.1038/onc.2011.256. [Epub ahead of print]
Semaphorins are implicated in glioma progression, although little is known about the underlying mechanisms. We have reported plexin-B3 expression in human gliomas, which upon stimulation by Sema5A causes significant inhibition of cell migration and invasion. The concomitant inactivation of Rac1 is o
PMID 21706053

Selective vulnerability of rat brain regions to unconjugated bilirubin.

Vaz AR, Silva SL, Barateiro A, Falcao AS, Fernandes A, Brito MA, Brites D.AbstractHippocampus is one of the brain regions most vulnerable to unconjugated bilirubin (UCB) encephalopathy, although cerebellum also shows selective yellow staining in kernicterus. We previously demonstrated that UCB induces oxidative stress in cortical neurons, disruption of neuronal network dynamics, either in developing cortical or hippocampal neurons, and that immature cortical neurons are more prone to UCB-induced injury. Here, we studied whereas immature rat neurons isolated from cortex, cerebellum and hippocampus present distinct features of oxidative stress and cell dysfunction upon UCB exposure. We also explored whether oxidative damage and its regulation contribute to neuronal dysfunction induced by hyperbilirubinemia, considering neurite extension and ramification, as well as cell death. Our results show that UCB induces nitric oxide synthase expression, as well as production of nitrites and cyclic guanosine monophosphate in immature neurons, mainly in those from hippocampus. After exposure to UCB, hippocampal neurons presented the highest content of reactive oxygen species, disruption of glutathione redox status and cell death, when compared to neurons from cortex or cerebellum. In particular, the results indicate that cells exposed to UCB undertake an adaptive response that involves DJ-1, a multifunctional neuroprotective protein implicated in the maintenance of cellular oxidation status. However, longer neuronal exposure to UCB caused down-regulation of DJ-1 expression, especially in hippocampal neurons. In addition, a greater impairment in neurite outgrowth and branching following UCB treatment was also noticed in immature neurons from hippocampus. Interestingly, pre-incubation with N-acetylcysteine, a precursor of glutathione synthesis, protected neurons from UCB-induced oxidative stress and necrotic cell death, preventing DJ-1 down-regulation and neuritic impairment. Taken together, these data point to oxidative injury and disruption of neuritic network as hallmarks in hippocampal susceptibility to UCB. Most importantly, they also suggest that local differences in glutathione content may account to the different susceptibility between brain regions exposed to UCB.
Molecular and cellular neurosciences.Mol Cell Neurosci.2011 Jun 24. [Epub ahead of print]
Hippocampus is one of the brain regions most vulnerable to unconjugated bilirubin (UCB) encephalopathy, although cerebellum also shows selective yellow staining in kernicterus. We previously demonstrated that UCB induces oxidative stress in cortical neurons, disruption of neuronal network dynamics,
PMID 21708263

Japanese Journal

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

Kawakami Yu,川上裕,カワカミユウ
Geometriae Dedicata, 2013-08-00
… We provide an effective ramification theorem for the ratio of canonical forms of a weakly complete flat front in the hyperbolic three-space. …
NAID 120005315152