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Topic: Archimedean field

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Archimedean solid

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	PlanetMath: real number
		The real numbers can also be defined as the unique (up to isomorphism) ordered field satisfying the least upper bound property, after one has proved that such a field exists and is unique up to isomorphism.
		However, an ordered group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the Completeness section above is a special case.
		But the original use of the phrase “complete Archimedean field” was by David Hilbert, who meant still something else by it.
	www.planetmath.org /encyclopedia/MathbbR.html (919 words)

	Ordered field (Site not responding. Last check: 2007-10-29)
		The smallest subfield is isomorphic to the rationals (as for any field of characteristic 0), and the order on this rational subfield is the same as the order of the rationals themselves.
		For example, the real numbers form an Archimedean field, but every hyperreal field is non-Archimedean.
		Finite fields cannot be turned into ordered fields, because they do not have characteristic 0.
	www.ebroadcast.com.au /lookup/encyclopedia/or/Ordered_field.html (323 words)

	Real number Summary
		The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication.
		The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
		Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean.
	www.bookrags.com /Real_number (4419 words)

	Archimedean property - Wikipedia, the free encyclopedia
		In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers.
		In particular, a linearly ordered group that is Archimedean is an Archimedean group, and an ordered field that is Archimedean is an Archimedean field.
		In a field, it is enough to check only one of these conditions (since if x is infinitesimal, then 1/x is infinite, and vice versa).
	en.wikipedia.org /wiki/Archimedean_property (781 words)

	D. Prasad
		Distinguished representations for quadratic extension of a finite field.
		Representations of division algebras and of Galois groups of local fields: Generalising local class field theory, Langlands has conjectured a correspondence between irreducible representations of $GL(n)$ or of a division algebra of index $n$ to $n$ dimensional representations of the Galois group of the local field.
		The group $SL(n)$ for $n \cong 2 \bmod 4$ does not have such an element for a finite field ${\Bbb F}_q$ for $q \cong 3 \bmod 4$, and for such group there are generic self-dual representations on which the central element acts trivially, although the representation is symplectic, belying a belief at that point.
	www.math.tifr.res.in /~dprasad (2364 words)

	PlanetMath: complete ultrametric field
		is called a non-archimedean field or also an ultrametric field, since the valuation induces the ultrametric
		Thus the partial sums of (1) form a Cauchy sequence, which converges in the complete field.
		This is version 10 of complete ultrametric field, born on 2005-01-03, modified 2006-11-22.
	planetmath.org /encyclopedia/NonArchimedeanField.html (170 words)

	The Turing Closure of an Archimedean Field - Boldi, Vigna (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		The Turing Closure of an Archimedean Field - Boldi, Vigna (ResearchIndex)
		The Turing Closure of an Archimedean Field (1997)
		41.3%: The Turing Closure of an Archimedean Field - Boldi, Vigna (1997)
	citeseer.ist.psu.edu /boldi97turing.html (588 words)

	PlanetMath: real number
		This definition is well-defined and does not depend on the choice of Cauchy sequences used to represent the equivalence classes.
		The real numbers can also be defined as the unique (up to isomorphism) ordered field satisfying the least upper bound property, after one has proved that such a field exists and is unique up to isomorphism.
		He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of
	planetmath.org /encyclopedia/MathbbR.html (861 words)

	Springer Online Reference Works
		Extension of a field) based on the divisibility of integers by a given prime number
		The extension is obtained by completing the field of rational numbers with respect to a non-Archimedean valuation (cf.
		A necessary condition for the solvability of this equation in integers or in rational numbers is its solvability in the rings or, correspondingly, in the fields of
	eom.springer.de /P/p071020.htm (465 words)

	MA 109 College Algebra Chapter 2 Exercises
		Verify that all the field axions except for the existence of inverses holds for the the set of congruence classes modulo n.
		Show that if F is an Archimedean ordered field in which every infinite decimal in which every digit is even has a limit in F, then F is a field of real numbers.
		Let F be an Archimedean ordered field in which every increasing bounded sequence of numbers converges to an element of F. Show that F is a field of real numbers.
	www.msc.uky.edu /ken/ma109/exercises/real.htm (3497 words)

	Real number : Real (Site not responding. Last check: 2007-10-29)
		Construction by Dedekind cuts -- A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A has no maximum.
		The real numbers form the largest subfield that is Archimedean[?] (meaning that no real number is infinitely large).
		Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which isn't a field anymore but retains many of the properties of the real numbers.
	www.termsdefined.net /re/real.html (2470 words)

	Scientists Succeed At (Cryogenically Enhanced Magneto-Archimedes) Levitation: Science Fiction in the News
		The materials are separated due to differences in their density, and float at vertical positions where the total Archimedean and magnetic buoyancy is equal to their weight.
		Diamagnetism occurs as a result of a magnetic field's interference with the motion of electrons orbiting the atoms or molecules of a material.
		When matter is placed in a magnetic field, the magnetic force acts upon the moving electrons in the matter, causing the electrons to be deflected.
	www.technovelgy.com /ct/Science-Fiction-News.asp?NewsNum=388 (725 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		We give some applications to the field of formal Laurent series in $n$ variables, where the non--analytic inversion formula gives explicit formal solutions of general semilinear differential and $q$--difference equations.
		When $k$ is the field of formal Laurent series ${\mathbb C}((z))$, we consider the vector space: $\mathbb{C}^n\left(\left(z_1,\ldots,z_n\right)\right)$; the non-analytic inversion problem can be applied to obtain the solution of semilinear differential or $q$--difference equations in an explicit (i.e.
		More precisely they proved that the analytic vector field \begin{equation} \label{eq:anavf1} \begin{cases} \dot z_1 &= -z_1 (1+\dots) \\ \dot z_2 &= \omega z_2(1+\dots) \,, \end{cases} \end{equation} where $\omega >0$ and the suspension points mean terms of order bigger than $1$, has the same analytical classification that the germs of $(\C,0)$: $f(z)=e^{2\pi i \omega}z+\bigo{z^2}$.
	www.ma.utexas.edu /mp_arc/papers/01-370 (5564 words)

	PlanetMath: ultrametric triangle inequality
		in the field is equivalent with the condition
		The metrics defined by non-archimedean valuation of the field
		Cross-references: inequality, limit, root, bounded, unity, multiples, archimedean, isosceles, vertices, triangle, implies, ultrametrics, valuation, metrics, non-archimedean, Krull valuation, postulates, satisfies, function, ordered group equipped with zero, field
	planetmath.org /encyclopedia/NonArchimedeanTriangleInequality.html (185 words)

	MA 109 College Algebra Chapter 2
		We begin by defining a field; it is basically a set like the rational numbers or the real numbers where one can operate in the usual way with the four basic arithmetic operations.
		The problem with working in the field of rational numbers is that it is relatively sparse; so, when you go to solve equations of degree greater than one, we often find that what would have been a solution is not rational.
		Since it is a field which contains both the field of real numbers and the element i, it must also contain expressions of the form z = a + bi where a and b are real numbers.
	www.msc.uky.edu /ken/ma109/lectures/real.htm (9248 words)

	Archimedean field - Art History Online Reference and Guide
		Then, an Archimedean field F is one such that for any x in F there exists n in the natural numbers N for which x
		Moreover, it can be proved that any archimedean field is isomorphic (as an ordered field) to a subfield of the real numbers.
		Archimedean fields are important in the axiomatic construction of real numbers.
	www.arthistoryclub.com /art_history/Archimedean_field (145 words)

	Global Fields
		The following lemma essentially says that the denominator of an element of a global field is only ``nontrivial'' at a finite number of valuations.
		Any valuation on a global field is either archimedean, or discrete non-archimedean with finite residue class field, since this is true of
		We will use the following lemma later (see Lemma 20.3.3) to prove that formation of the adeles of a global field is compatible with base change.
	www.math.harvard.edu /archive/129_spring_04/ant/html/node81.html (509 words)

	ORDINAL REAL NUMBERS 2
		In this topology, as it is known, the field F is a topological field.
		As in the case of fields that are classes, we may permit topological spaces that are classes and the open sets is a class of subclasses closed to union and finite intersection.
		In other words the fields of ordinal real numbers are Archemidean complete fields (although they may be non-Archemidean).But this is a characteristic property of the fields of transfinite real numbersb of Glayzal.
	softlab.ntua.gr /~kyritsis/PapersInMaths/InfinityandStochastics/OR2.htm (2766 words)

	Publications (Site not responding. Last check: 2007-10-29)
		Power series with rational exponents on the real number field and the Levi-Civita field are studied.
		We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series.
		Then we study a class of functions that are given locally by power series with rational exponents, which are shown to form a commutative algebra over the Levi-Civita field; and we study the differentiability properties of such functions within their domain of convergence.
	bt.pa.msu.edu /cgi-bin/displaytest.pl?name=RSGPS05 (173 words)

	5.2 Completeness (Site not responding. Last check: 2007-10-29)
		If real fields do exist, there is a question of uniqueness; i.e., is it the case that any two real fields are ``essentially the same"?
		All I can say about consistency is that no contradictions have been found to follow from the real field axioms.
		There exist proofs that any two real fields are essentially the same, cf.
	www.reed.edu /~mayer/html1/node32.html (314 words)

	Real Numbers
		The set R is a field, i.e., addition, subtraction, multiplication and division are defined and have the usual properties.
		He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field.
		Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which is not a field but retains many of the properties of the real numbers.
	www.metu.edu.tr /~e128327/real.htm (1866 words)

	UNT Department of Mathematics: AGANT Seminar
		If f(z) is a rational function (or more generally a meromorphic function over the complex numbers or a quotient of convergent power series over a non-Archimedean valued field), then the logarithmic derivative f'/f(z) gets small (or in the case of meromorphic functions does not "grow" very quickly) as z gets large.
		Abstract: The zero locus of a holomorphic vector field has been a subject of study for a long time on a compact complex manifold.
		For example, the residue theorem of Bott relates the zero locus of a vector field with the Chern numbers of the manifold (which, in high-brow terms, explains why a vector field on a sphere has to vanish at some point).
	www.math.unt.edu /seminars/agant.shtml (1240 words)

	On the height constant for curves of genus two, by Michael Stoll (Site not responding. Last check: 2007-10-29)
		Together with some bounds for the archimedean places, this gives an upper bound for the difference h - h^ between a certain naive height and the canonical height on the Jacobian of a curve of genus two defined over a number field.
		In particular, we get a bound on the naive height of a torsion point, which can be used to find the rational torsion subgroup of the Jacobian.
		We give a fairly detailed description of an algorithm that computes the rational torsion subgroup when Q is the base field.
	www.math.uiuc.edu /Algebraic-Number-Theory/0145 (212 words)

	“The WIND Spacecraft and Its Early Scientific Results,” K. W. Ogilvie and M.D. Desch, NASA/Goddard Space Flight ...
		Russell and McPherron (1973) showed that geoeffective interplanetary fields (having a southward component in the magnetospheric coordinate system) occur when the interplanetary field points toward the sun in March and away from the sun in September.
		Subsonic electrons moving into the wake provide an electric field which accelerates ions along the magnetic field direction from both sides of the wake, leading to two ion beams, one moving faster and one slower than the solar wind.
		This is a cartoon showing the way in which electrostatic acceleration along the magnetic field direction produces two ion beams of different velocities as were observed by the SWE instrument on the WIND spacecraft.
	www-ssc.igpp.ucla.edu /IASTP/04/index.htm (2893 words)

	Publications
		Power series with rational exponents on the real number field and the Levi-Civita field are studied.
		We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series.
		Then we study a class of functions that are given locally by power series with rational exponents, which are shown to form a commutative algebra over the Levi-Civita field; and we study the differentiability properties of such functions within their domain of convergence.
	www.bt.pa.msu.edu /cgi-bin/display.pl?name=RSGPS05 (181 words)

	Archimedean ordered fields
		I won't expect you to learn or directly use the axioms of an ordered field in the sequences and series section of the module, and I won't set exam questions on this material, but you might be asked questions about this (or similar) things in reference to the first half of the module.
		There will also be some proofs later on in the course where the Archimedean Property of the reals will be used explicitly to good effect.
		You have seen the axioms for Archimedean ordered fields, the two key examples being the reals and the rationals, and some of the basic consequences.
	web.mat.bham.ac.uk /R.W.Kaye/seqser/archfields (1154 words)

	Papers by Joshua Lansky
		Abstract: Let k be a non-archimedean locally compact field and let G be the set of k-points of a connected reductive group defined over k.
		Let W be the relative Weyl group of G, and let H (G,B) be the Hecke algebra of G with respect to an Iwahori subgroup B of G.
		Abstract: In this paper we develop a theory of newforms for SL) where F is a non-Archimedean local field whose residue characteristic is odd.
	www.american.edu /faculty/lansky/papers/index.html (893 words)

	Math Forum Discussions - Re: Non-Archimedean Field Valuation Property
		Absolute values on fields and real valuations are different things: An absolute value that does not satisfy the ultrametric triangle imequality does not yield a valuation.
		This fact leads to the (sometimes unsatisfactory) situation that one has to develop two theories (one for absolute values, one for valuations) that coincide in the rather large area of real valuations.
		To mention one: an absolute value on a field K in general can not be extended to a field extension L of K. A valuation on the other hand can always be extended.
	www.mathforum.com /kb/thread.jspa?forumID=13&threadID=1321416&messageID=4185595 (189 words)

	Modeling the solar energetic particle events in closed structures of interplanetary magnetic field
		In the framework of focused transport, we perform Monte Carlo simulations of the SEP propagation and adiabatic deceleration caused by an overall expansion of the CME.
		The loop-like structure of the interplanetary magnetic field strongly modifies the intensity-time profiles of high-energy protons.
		The modeling results are similar to the patterns observed with the Energetic and Relativistic Nuclei and Electron (ERNE) particle telescope on board SOHO on 2–3 May 1998.
	www.agu.org /pubs/crossref/2005/2005JA011082.shtml (383 words)

	Franz-Viktor Kuhlmann's recent talks
		I show their connection with the defect of valued field extensions and the meaning of the defect for the model theory of valued fields and for local uniformization in positive (residue) characteristic.
		We prove that every place of an algebraic function field FK of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field over K is equal to the transcendence degree of FK (we call such places Abhyankar places).
		Tame fields are henselian valued fields K for which the ramification field of the extension K^sepK is algebraically closed; here, K^sep denotes the separable-algebraic closure of K. The model theory of tame fields has nice applications to the theory of places of algebraic function fields in arbitrary characteristic.
	math.usask.ca /~fvk/Fvktalks.html (3883 words)

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