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Topic: Cyclic extension

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Cyclotomic field

	Dicyclic group
		In group theory, a dicyclic group is a member of a class of groups which are formed by an extension of a group (generally a cyclic group) by a cyclic group of order 2 (the latter giving the name di-cyclic).
		Since for a cyclic group of even order, there is always a unique element of order 2, we can see that dicyclic groups are just a specific type of generalized dicyclic group.
		Generalized dicyclic groups, in turn, are examples of cyclic extensions[?].
	www.ebroadcast.com.au /lookup/encyclopedia/di/Dicyclic_group.html (328 words)

	The Ekpyrotic Universe and its Cyclic Extension Transparency 2
		The Ekpyrotic Universe and its Cyclic Extension Transparency 2
		The Ekpyrotic Universe and its Cyclic Extension Transparency 3
		The Ekpyrotic Universe and its Cyclic Extension Transparency 1
	www.damtp.cam.ac.uk /talks/ngt1000/annecy/node02.html (80 words)

	Abelian extension - Wikipedia, the free encyclopedia
		In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian.
		In general a cyclotomic extension formed by adjoining roots of unity is abelian.
		The Kummer theory gives a complete description of the abelian extension case, and the Kronecker-Weber theorem tells us that if K is the field of rational numbers, an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity.
	en.wikipedia.org /wiki/Abelian_extension (278 words)

	PlanetMath: group extension
		The extension problem in group theory is the classification of all extension groups of a given group
		is a cyclic group, then the extensions in question are called cyclic extensions.
		This is version 8 of group extension, born on 2005-07-20, modified 2005-08-16.
	planetmath.org /encyclopedia/ExtensionProblem.html (324 words)

	Extension Springs
		Since extension springs have an initial tension in their resting state, they also have a shear stress in their coils while at rest.
		With an extension spring, there is no such safety geometry since the spring is in tension.
		For this and other reasons, extension spring maximum working stresses are typically limited to three-fourths (3/4) of those for compression springs of similar geometry and material.
	www.efunda.com /designstandards/springs/spring_extension.cfm (274 words)

	cylic quadralerals
		A cyclic quadrangle or cyclic quadrialteral is a quadrilateral for which a single circle passes through all four vertices.
		The length of the two diagonals of a cyclic quadrilateral are related to the four sides in Ptolemy's Theorem which states (using m and n for the diagonals lengths) mn=ac+bd.
		The center of this "orthic cyclic quadrilateral" is the reflection of the circumcenter of the original quadrilateral in the anti-center.
	www.pballew.net /cycquad.html (1589 words)

	Extension to cyclic communication:
		This should also hold for cyclic communication patterns assuming that all processes are equal with respect to network topology.
		The overhead for cyclic communication was significant for both kinds of communicators as can be seen in Tab.
		The result was that once cyclic communication is involved the performance of the network decreases.
	www.hlrs.de /people/mueller/papers/pe-mapping/node7.html (517 words)

	Conversion to number fields (Site not responding. Last check: 2007-09-07)
		Although in theory an abelian extension "uniquely" defines a number field and therefore all its properties, not all of them are directly accessible (in Magma at least).
		Given an abelian extension A of a number field, using the algorithm of Fieker ([Fie00], [Coh00]) defining equations for A are computed.
		Depending on the size of the cyclic factors encountered, this may be a very lengthy process.
	www.umich.edu /~gpcc/scs/magma/text663.htm (281 words)

	COASTAL MOORING DESIGN (Site not responding. Last check: 2007-09-07)
		A 15-m storm wave requires a taut mooring tether with an elastic stretch of <1 percent in 2,000 m of water, 8 percent in 200 m of water, 46 percent in 40 m or water, and 120 percent in 20 m of water.
		This extension is required to prevent the buoy being pulled under by the passing wave peak.
		From cyclic tests of synthetic fiber ropes, which are textile structures, not elastomeric bodies, it is known that the long term Young’s modulus in cyclic extension tests can be two to three times the value of the initial response in rapid cyclic loading.
	kelvin.whoi.edu /mooring.html (4560 words)

	Trace and Norm in a Cyclic Extension
		After all, every solvable extension is a tower of cyclic extensions.
		Let a cyclic extension f/k have galois group g, and let the automorphism c generate g.
		This is possible, if f/k is purely inseparable, but we said the extension was cyclic, and that means f/k is galois by definition.
	www.mathreference.com /fld-slv,norm.html (601 words)

	The Ekpyrotic Universe and its Cyclic Extension Transparency 6
		The Ekpyrotic Universe and its Cyclic Extension Transparency 6
		The Ekpyrotic Universe and its Cyclic Extension Transparency 7
		The Ekpyrotic Universe and its Cyclic Extension Transparency 5
	www.damtp.cam.ac.uk /talks/ngt1000/annecy/node06.html (80 words)

	Equalization for DMT-Based Broadband Modems
		This technique works well, provided that the time span of the cyclic extension, inserted for the duration of the guard time, is longer than the channel impulse response.
		The reader may question whether a cyclic extension is still useful when the cyclic prefix single-tap FEQ scheme is not sufficient and additional channel equalization is required (TEQ, per-tone equalizer) for severe dispersive channels.
		The length of the cyclic extension is chosen longer than the sum of the channel impulse response and the transmission delay or echo (whichever is longer).
	www.comsoc.org /ci1/private/2000/may/Pollet.html (4777 words)

	Some congruences on prime factors of class number of algebraic extensions K/Q, by Roland Queme
		This paper is a contribution to the description of the odd prime factors of the class number of the number fields.
		There exists a cyclic extension K/F with beta \in N, 1 \leq beta \leq alpha and [K:F] = g^beta, such an extension K/F_1 not existing for alpha \geq beta_1 > beta.
		Suppose now that p \not= g is an odd prime dividing the class number h(K) of K and not dividing the class number h(F) of F. Let r_p be the rank of the p-class group of K. Then p \equiv 1 \modu g.
	www.math.uiuc.edu /Algebraic-Number-Theory/0372 (265 words)

	p-cyclic over Other Characteristics
		In the last section we described p-cyclic extensions of k when the characteristic of k was p.
		In this section we will describe p-cyclic extensions of k when the characteristic is not equal to p.
		The only group of order p is cyclic, hence f/k is a cyclic extension of order p.
	www.mathreference.com /fld-slv,npcyc.html (696 words)

	Cyclic and non-cyclic attributes (Site not responding. Last check: 2007-09-07)
		For the moment a non-cyclic attribute may be defined as an attribute with a finite number of possible values (i.e.
		The completeness and coherence condition is restricted to cyclic attributes.
		To define this properly it is necessary to define the notion non-cyclic extension.
	odur.let.rug.nl /~vannoord/papers/coling90/node8.html (177 words)

	PlanetMath: cyclic extension
		is said to be a cyclic extension if the Galois group
		This is version 2 of cyclic extension, born on 2002-11-16, modified 2002-11-16.
		(Field theory and polynomials :: Field extensions :: Separable extensions, Galois theory)
	planetmath.org /encyclopedia/CyclicExtension.html (43 words)

	On the class number of cyclic extensions K/Q, by Roland Queme (Site not responding. Last check: 2007-09-07)
		On the class number of cyclic extensions K/Q, by Roland Queme
		In this paper, we give several congruences connecting the prime divisors of the degree g= [K:Q] with the prime divisors of the class number h of K/Q. As an exemple, the theorem :
		Let K/Q be a cyclic extension with [K:Q]=g.
	www.math.uiuc.edu /Algebraic-Number-Theory/0182 (153 words)

	Lambda calculus plus letrec (Site not responding. Last check: 2007-09-07)
		To define the well-formed cyclic lambda-graphs we introduce the notion of a scoped lambda-graph.
		We conclude by presenting a variant of our cyclic lambda calculus, which follows the tradition of the explicit substitution calculi.
		Since most implementations of non-strict functional languages rely on sharing to avoid repeating computations, we develop a variant of our cyclic lambda calculus that enforces the sharing of computations and show that the two calculi are observationally equivalent.
	www.cs.uoregon.edu /~ariola/cycles.html (357 words)

	International Geosynthetics Society
		The load-strain behavior of the geogrid materials is characterized using two distinct features of the cyclic response: 1) non-linear hysteresis unload-reload loops; and 2) a load-strain cap that contains all hysteresis curves.
		The cyclic load-strain response of the HDPE and PET geogrid specimens is demonstrated to be significantly different from the static response of the same materials tested according to the current ASTM D 4595 method of test.
		The load-strain caps for HDPE geogrid specimens are found to be very sensitive to the loading frequency and loading history while those of the PET geogrid appear to be relatively insensitive.
	www.geosyntheticssociety.org /gi/GI_V1N1.htm (1084 words)

	OFDM as a modulation technique for wireless communications, with a CDMA comparison
		The guard period used for the simulations consisted of 256 samples of zero amplitude, and 256 samples of a cyclic extension of the symbol.
		It was verified that the tolerance is due to the cyclic extension not the zeroed period with other simulations.
		This length matches the cyclic extension period of the guard interval, and is due to the guard period maintaining the orthogonality of the signal
	www.skydsp.com /publications/4thyrthesis/chapter2.htm (6771 words)

	Extension Uci Extension Table (Site not responding. Last check: 2007-09-07)
		It is the first stage of capitalism and thus in it there is plenty of room for the extension of capitalist relations of production, meaning, of wage labour and therefore of commodity production.
		Abelian extension - In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian.
		UCI World Cup - The UCI World Cup was a season-long competition for European professional racing cyclists, organised by the Union Cycliste Internationale (UCI), the governing body of competitive cycling.
	fpuser.com /extensionuci.html (196 words)

	CJM - Descending Rational Points on Elliptic Curves to Smaller Fields
		We consider an elliptic curve $E$ defined over a number field $K$ whose Mordell-Weil rank over a Galois extension $F$ is $1$, $2$ or $3$.
		Suppose that the $\o$-rank over a Galois extension $F$ is $1$ or $2$.
		If $\k\neq\Q(\sqrt{-1})$ and $\Q(\sqrt{-3})$ and $h_{\k}$ (class number of $\k$) is odd, we show that $E$ acquires positive $\o$-rank over a cyclic extension of $K$ or over a field whose Galois group is one of $\SL_2(\Z/3\Z)$, an extension of $\SL_2(\Z/3\Z)$ by $\Z/2\Z$, or a central extension by the dihedral group.
	journals.cms.math.ca /cgi-bin/vault/view/akbary1279 (224 words)

	Computation of the Fundamental Units and the Regulator of a Cyclic Cubic Function Field, Y. Lee, R. Scheidler, C. ...
		This paper presents algorithms for computing the two fundamental units and the regulator of a cyclic cubic extension of a rational function field over a field of order {$q \equiv 1 \pmod{3}$}.
		The procedure is based on a method originally due to Voronoi that was recently adapted to purely cubic function fields of unit rank one.
		Our numerical examples show that the two fundamental units tend to have large degree, and frequently, the extension has a very small ideal class number.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1067634732 (218 words)

	[GrpConst] 5 The Cyclic Split Extension Method
		Thus it is useful to install and load this share package before using the functions described in this chapter.
		As in Chapter The Frattini Extension Method all groups are described as codes.
		It is possible to construct the cyclic extensions of a single group of order
	www-groups.dcs.st-and.ac.uk /gap/Manuals/pkg/grpconst/htm/CHAP005.htm (267 words)

	Oxidative Stress and the Mechanical Properties of Naturally Occurring Chimeric Collagen-Containing Fibers -- Sun et al. ...
		Cyclic tensile stress-strain tests were performed on a Bionix 200 instrument (MTS Systems, Cary, NC) equipped with a 10-N
		measurement of stress and modulus at 50% extension.
		in modulus concomitant with cyclic extension is referred to as
	www.biophysj.org /cgi/content/full/81/6/3590 (2952 words)

	Math 250: Higher Algebra (2001-2)
		Let E/F be a cyclic extension, with Galois group G generated by e.
		Its kernel is the unit group of the commutant of L. The commutant is L itself by Theorem 9 (Cor.2).
		The quotient group is an extension of G by A, which we may regard again as an element of H
	www.math.harvard.edu /~elkies/M250.01/index.html (9028 words)

	Math 601 Homework (Abstract Algebra II)
		Show that [Q(u):Q] = 8 and that Q(u) is a Galois extension of Q with cyclic Galois group.
		Let K be a field of characteristic not equal to 2, and let D be an element of K that is not a perfect square.
		Show that K is a cyclic extension of Q(w) of degree 3, and use the Lagrange resolvant method to show it's obtained by adjoining a cube root of something.
	www.math.umd.edu /users/jmr/601/homework.html (1377 words)

	XSL FO Document
		Marking an RTF document's tables and lists with a special text sequence to provide cyclic extension of the XML data file.
		If a field with the same name was inserted in many places of RTF file, only one instance of nsdc:field is created in XML, and the content of this field is referred in XSL stylesheet in all places where the field was inserted in the RTF document.
		A point for cyclic expansion is a place in RTF document where an unlimited number of entries will be later inserted.
	www.novosoft.net /RTF_TO_XMLQuickStart.html (1368 words)

	Math 250: Higher Algebra (Fall 2004)
		``Lemma 3'' in Jacobson 4.7 generalizes to normal extensions with a cyclic Galois group of arbitrary order n (not necessarily prime) over a field F containing the n-th roots of unity (that is, F such that the polynomial X
		Besides Jacobson's application to solvability in radicals -- for which the case of prime n is sufficient -- this generalization also leads to Kummer theory of any finite normal extension K/F of such a field F such that the Galois group G is abelian and of exponent dividing n.
		In a Galois extension E/F, the norm and trace of any element x are the product and sum of its conjugates; more precisely, they are the product and sum of g(x), g ranging over Gal(E/F).
	www.math.harvard.edu /~elkies/M250.04/index.html (12646 words)

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