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Topic: Ramification group

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	Ramification - Wikipedia, the free encyclopedia
		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.
		The ramification is tame when the e(i) are all less than the residue characteristic p of P. This condition is important in Galois module theory.
		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.
	en.wikipedia.org /wiki/Ramification (530 words)

	PlanetMath: ramification index
		Ramification points or branch points in complex geometry are merely a special case of the high-flown terminology of Definition 6.
		The algebraic-analytic correspondence of ramification points is itself only one manifestation of the wide ranging identification between algebraic geometry and analytic geometry which is explained to great effect in the seminal paper of Serre [6].
		This is version 14 of ramification index, born on 2002-04-23, modified 2005-03-15.
	planetmath.org /encyclopedia/RamificationPoint.html (1092 words)

	Galois module - Wikipedia, the free encyclopedia
		In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group — equivalently for a Galois group G and a group ring R[G] of G with respect to some ring R, it is some R[G]-module M.
		Certainly G can either be a profinite Galois group, or a finite one for a finite extension L/K of fields.
		In the case of G profinite, there is a large supply of G-modules available in the theory of étale cohomology, which is an algebraic theory (and therefore exhibits 'covariance' with respect to Galois symmetry).
	en.wikipedia.org /wiki/Galois_module (474 words)

	Graduate Study in Algebra
		The research strengths of the faculty are in the theory of rings (commutative and noncommutative), the theory of groups, algebraic number theory, the representation theory of groups and algebras, and algebraic geometry.
		The "symmetric" groups of all permutations of a set are investigated, and the Cayley theorem (showing an arbitrary abstract group may be regarded as a subgroup of some symmetric group) is proved.
		Ramification groups, the different, and the discriminant of an extension are studied.
	www.math.uiuc.edu /GraduateProgram/researchmath/gradalgebra.html (1660 words)

	PlanetMath: algebraic number theory
		The structure of the unit group is described by Dirichlet's unit theorem, which asserts the existence of a system of fundamental units.
		The regulator is an important invariant of the unit group (it appears in the class number formula).
		The cyclotomic units are a subgroup of the group of units of a cyclotomic field with very interesting properties.
	planetmath.org /encyclopedia/AlgebraicNumberTheory.html (949 words)

	PlanetMath: Galois representation
		The equivalence of these two definitions is as described in the entry for the group algebra.
		Galois representations play a fundamental role in algebraic number theory, as many objects and properties related to global fields and local fields may be determined by certain Galois representations and their properties.
		cohomology group of a motive), and also to prove that it arises from an automorphic form.
	planetmath.org /encyclopedia/GaloisCohomology.html (759 words)

	HonorsBio
		For the PBL part of the class, groups of 5-6 students worked together to solve complex problems which had a real-world context and were often open ended.
		I was able to assign an honors student to one or two groups as their permanent peer tutor whose responsibilities included assisting the group with construction of learning issues and providing some guidance as the group solved the problem.
		However, in order for PBL to be entirely effective, it is necessary that each group member fully participate and pitch in her or his two cents.
	cte.udel.edu /aboutteach/spring00/schmieg.html (1706 words)

	The Hurwitz action and braid group orderings (Site not responding. Last check: 2007-10-29)
		In connection with the so-called Hurwitz action of homeomorphisms in ramified covers we define a groupoid, which we call a ramification groupoid of the 2-sphere, constructed as a certain path groupoid of the universal ramified cover of the 2-sphere with finitely many marked-points.
		The Artin group of braids of $n$ strands has an order-invariant action in the ramification groupoid of the sphere with $n+1$ marked-points.
		In particular, we show that the underlying set of a free group on countably many generators (minus the identity element) can be linearly ordered in such a way that the classical Artin representation of a braid as an automorphism of the free group is an order-preserving action.
	www.tac.mta.ca /tac/volumes/9/n7/9-07abs.html (171 words)

	Contents of The One-dimensional Case. (Site not responding. Last check: 2007-10-29)
		The first 2 groups in (7) are the Brauer groups of
		for the group of units in a ring.
		But the exact sequence (9) implies that the ramifications ``sum to zero''.
	www.math.fau.edu /ford/preprints/darff/node2_ct.html (702 words)

	Logic and Artificial Intelligence
		Probably at this time both the mathematicians and the philosophers shared a sense that their subject was considered to be somewhat marginal by their colleagues, and may have felt a primary loyalty to logic as a subject rather than to any academic discipline.
		But the solution is presented in the context of an ambitious general project in nonmonotonic logic that not only develops properties of the preferred model approach and shows how to apply it to a number of reasoning problems, but that relates nonmonotonic logic to probabilities, using ideas deriving from Adams 1975.
		Ramification is induced by the presence of static laws which relate the direct consequences of actions to other changes.
	www.seop.leeds.ac.uk /entries/logic-ai (16979 words)

	Leaf - Wikipedia, the free encyclopedia
		The epidermis is the outer multi-layered group of cells covering the leaf.
		The veins are the vascular tissue of the leaf and are located in the spongy layer of the mesophyll.
		They are typical examples of pattern formation through ramification.
	en.wikipedia.org /wiki/Leaf (2719 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		We then prove, as I have said, cut elimination for this system; as with most such proofs, the proof of cut elimination proceeds by rather elementary stages but is extremely long-winded (28 subcases or so).
		One can also prove that, in many cases, one can apply the rules in any order one likes; one can formalise this in so-called permutability theorems, saying that the applications of certain rules can be permuted around each other.
		And in the "standard case", as it were, of the ramification scenario, we start with a situation A which is already stable under ramification transitions, so no initial applications of (*) are possible; we have no choice but to apply the rules for and -o first.
	www.ida.liu.se /ext/etai/received/actions/012/!ar2r.txt (603 words)

	IEEE P1149.4 Working Group Meeting Minutes for June 2-3, 1997
		He added that he himself is part of a mixed signal circuit core working group and was trying to think of a way to take both hard and soft mixed signal cores into account in our mixed signal test bus scheme.
		Group 1 would contain switches 1 to 4, 9, and 10 which would all be functionally tied together.
		Group 2 would contain switches 5 to 8 all of which would also be functionally tied together.
	grouper.ieee.org /groups/1149/4/min0697.html (7543 words)

	Publications and Preprints
		With Q. Liu and M. Raynaud, On the Brauer group of a surface, Invent.
		On the group of components of a Néron model, J.
		On a finite group associated to the Laplacian of a graph, Discr.
	www.math.uga.edu /~lorenz/paper.html (247 words)

	Constancea 83.6: Recent Additions to the Subfamily Ceramioideae (Rhodophyta)
		A cladistic analysis of six primitive ceramiaceous genera indicates that the ceramialean ancestor was probably similar to Warrenia, exhibiting a bilaterally ramified thallus with lateral filaments of unlimited (or latent) growth, transverse apical divisions, gland cells, carpogonial branches borne on intercalary axial cells, procarpic post-fertilization stages, and tetrasporophytes with cruciately-decussately divided tetrasporangia.
		Transverse ramification is present in all Ceramioideae, but also in the remaining ceramiaceous tribes, which indicates that this character has either developed independently twice or as a synapomorphy for all Ceramiales except the Delesseriopseae and Warrenieae.
		Production of two groups of ('sterile') cells from the supporting cell itself is known to occur in a variety of families, including members of the Dasyaceae, Rhodomelaceae, and Delesseriaceae, but it is unlike what occurs in the Ceramiaceae.
	ucjeps.berkeley.edu /constancea/83/athanasiadis/Athanasiadis.html (8390 words)

	Mathematics MATH Courses - Graduate Catalog Spring 2003 - University of Maryland
		Groups, rings, integral domains and fields, detailed study of several groups; properties of integers and polynomials.
		Students will work in groups on specific projects involving real-life problems that are accessible to their existing mathematical backgrounds.
		Differential forms, the Euler characteristic, Gauss-Bonnet theorem, the fundamental group; an outline of the topological classification of compact surfaces, vector fields, geodesics and Jacobi fields; classical calculus of variations, global differential geometry of surfaces and elementary Riemann surface theory.
	www.math.umd.edu /graduate/courses/MATHcoursedescps.shtml (2747 words)

	Nigel Byott (Site not responding. Last check: 2007-10-29)
		The classes obtained from all tame extensions of a given field with a given Galois group are the realisable classes.
		In certain cases, the ramification numbers for an extension of local fields are enough to determine its Galois module structure.
		In other cases, the ramification breaks do not give sufficient information to do so, but one can define a family of "refined" ramification breaks which contain more information.
	www.maths.ex.ac.uk /~NPByott/research/res-interests.html (498 words)

	Talk 4085 data/Spring_2002/0121
		Minkowski proved that G_Q is trivial, and that in general G_K^ab (which is equal to the ideal class group of K by class field theory) is finite.
		It is a finitely generated prop-p group which can be infinite (Golod- Shafarevich).
		I will describe two recent parallel developements which have shed light on G_K(p): an emerging conjectural classification of finitely generated pro-p groups, and a conjecture of Fontaine and Mazur characterizing Galois representations arising from algebraic geometry.
	www.math.duke.edu /mcal?abstract-4085 (160 words)

	IHES PREPRINT M/01/58 (Site not responding. Last check: 2007-10-29)
		We consider the class of complete discretely valued fields such that the residue field is of prime characteristic p and the cardinality of a p-base is 1.
		A new definition of ramification filtration for such fields is given.
		Therefore, a theory of upper ramification groups, as well as the ramification theory of infinite extensions, can be developed.
	www.ihes.fr /PREPRINTS/M01/Resu/resu-M01-58.html (164 words)

	[No title]
		Subject: Projective Groups There are a wealth of cool and interesting groups in the group libraries, but I cannot find hide nor hair of functions that could spit out projective groups for me.
		I would like to know if GAP can be used to determine the ramification groups of a prime in a Galois extension of the rationals defined as the splitting field of a given polynomial.
		It seems to me that computing these groups is at least as complicated as to computing the Galois group as a group of field automorphisms which is already quite hard.
	www-groups.dcs.st-and.ac.uk /~gap/ForumArchive/forum95c.txt (11877 words)

	Introduction to "Taming Wild Extensions" of Lindsay N. Childs
		Thus as a module over the group ring KG, L is a free module of rank one with basis s.
		In this case, a congruence condition on the ramification number, or break number, characterizes when the associated order of the valuation ring of L is a Hopf order.
		Chapter 12 then presents the Kummer theory of formal groups, from {CM94}, {Mo94}, {Mo96}, which shows that if a Hopf algebra H is constructed as in Chapter 11, then the principal homogeneous spaces for H, or equivalently, the Galois extensions for H^*, are easily and explicitly described.
	math.albany.edu:8000 /~lc802/mono.html (1829 words)

	Contents of Introduction (Site not responding. Last check: 2007-10-29)
		In the 1960s and 1970s, turning the cohomological crank on the engine of Algebraic Geometry, Grothendieck, M. Artin and Mumford derived exact sequences for the Brauer group of function fields for varieties of dimension 1 and 2.
		These Brauer group theorems can be viewed as generalizations of the results of class field theory and furthermore can be thought of as providing laws of
		Before proving Theorem 1.1, we review the theory underlying the definition of the ramification divisor of a division algebra.
	www.math.fau.edu /ford/preprints/darff/node1_ct.html (512 words)

	RedOrbit NEWS \| Phylogeny of eukaryotes recovered with molecular data: Highlights and pitfalls
		Subsequently, a rapid ramification occurs which is often referred to the "crown" group, including red algae, chlorobionts (green algae, mosses, and vascular plants), fungi, metazoans, and choanoflagellates.
		Nevertheless, many of the groupings suggested by the ssu rRNA tree were readily confirmed, such as the close relationship of fungi and metazoans, the alveolates, and their sister group relationship with the stramenopiles.
		the mycetozoans and lobose amoebae, branching together as amoebozoans, the grouping of opisthokonts (metazoans and fungi) with amoebpzoans, and the grouping of euglenozoans and heteroloboseans.
	www.redorbit.com /modules/news/tools.php?tool=print&id=14395 (4708 words)

	Some papers
		Extends the classification of irreducible finite dimensional representations of almost simple algebraic groups over an algebraically closed field of characteristic zero to certain non-connected groups G where the component group is cyclic.
		If $G$ is a finite subgroup of the automorphism group of a projective curve $X$ and $D$ is a divisor on $X$ stabilized by $G$, then under the assumption that $D$ is nonspecial, we compute a simplified formula for the trace of the natural representation of $G$ on Riemann-Roch space $L(D)$.
		Are all subfields of a cyclotomic field of the form $Q (r)$, for some irreducible character r of a finite group G? In particular, we explicitly determine the Galois action on all irreducible characters of the generalized symmetric groups.
	web.usna.navy.mil /~wdj/papers.html (2557 words)

	Classics in the History of Psychology -- Tolman (1922)
		Just as the concept of the behavior-cue was found to bear a certain relation to a concept of the older psychology (viz., that of sense quality) so the concept of the behavior-object bears an analogous relation to another concept of the older psychology; viz., that of the perceived or apperceived meaning.
		A behavior-object results from a behavior-cue or a group of behavior-cues which, because of a particular behavior situation, possesses for the organization in question a specific behavior-meaning.
		On another occasion, however, this same group of behavior-cues might be apperceived not as a chair, but as a very different sort of behavior-object.
	psychclassics.yorku.ca /Tolman/formula.htm (2790 words)

	Proc. of the Natl. Academy of Sciences of Belarus, Ser. Phys.-Math. Sci., 2003, No.2
		For special ramification point of such algebras the local Phister conjecture is proved.
		It is shown that every of four different groups, coverings of the full Lorentz group, has only two non-equivalent, 4-component, complex, spinor representations, which can be related to certain two sorts of physical fermions.
		It is established that only one group from four coverings of the full Lorentz group can be reduced by a similarity transformation to a real-valued form and therefore just this covering permits existence of Majorana fermions.
	www.ac.by /publications/vestifm/vfm03_2.html (1466 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		This research is in the general area of algebra and is an interesting combination of algebra, number theory and algebraic geometry.
		Given a surface it is possible to associate with it a coordinate ring and with this ring a group.
		This project will examine properties of this group in an effort to determine the geometry of the surface.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a8822944.txt (89 words)

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