
	Where results make sense

Topic: Algebraically independent

Related Topics

abstract algebra

algebraic geometry

linear algebra

algebraic topology

Boolean algebra

Lie algebra

algebraic varieties

basis linear algebra

ring algebra

associative algebra

algebraic notation

homological algebra

In the News (Sun 27 May 12)

EXECUTIVE POWER

Iran

ANALYSIS

Pakistan

Family compact

	3 Fermion Generations, E6, E7, E8, Rosenfeld Planes (Site not responding. Last check: 2007-11-04)
		Since the complex numbers C have only one algebraically independent imaginary i, (CxO)P2 corresponds to a plane of one independent octonion for each of the two dimensions of the plane.
		Since the quaternions Q have two algebraically independent imaginaries i and j, (QxO)P2 corresponds to a plane of two independent octonions for each of the two dimensions of the plane.
		Since the octonions have three algebraically independent imaginaries i, j, and E, (OxO)P2 corresponds to a plane of three independent octonions for each of the two dimensions of the plane.
	www.valdostamuseum.com /hamsmith/E678.html (588 words)

	Transcendence degree - Wikipedia, the free encyclopedia
		In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the "size" of the extension.
		Specifically, it is defined as the largest cardinality of an algebraically independent subset of L over K.
		The dictionary matches algebraically independent sets with linearly independent sets; sets S such that L is algebraic over K(S) with spanning sets; transcendence bases with bases; and transcendence degree with dimension.
	en.wikipedia.org /wiki/Transcendence_degree (707 words)

	Algebraic independence - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-04)
		In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K.
		In general, all the elements of an algebraically independent set over K are by necessity transcendental over K, but that is far from being a sufficient condition.
		Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension.
	en.wikipedia.org /wiki/Algebraically_independent (254 words)

	Transcendence degree: Encyclopedia topic (Site not responding. Last check: 2007-11-04)
		In abstract algebra (abstract algebra: more facts about this subject), the transcendence degree of a field extension (field extension: more facts about this subject) L/K is a certain rather coarse measure of the "size" of the extension.
		The dictionary matches algebraically independent sets with linearly independent sets (linearly independent sets: more facts about this subject) ; sets S such that L is algebraic over K(S) with spanning sets (spanning sets: in the mathematical subfield of linear algebra, the linear span of a set of vectors is...
		The field L is the algebraic closure (algebraic closure: in mathematics, an algebraic closure of a field k is an algebraic extension of...
	www.absoluteastronomy.com /reference/transcendence_degree (1013 words)

	[No title]
		The algebraic closure of Q(S), cl(Q(S)), must contain R, so it is C. What's the size of S? Since C has cardinality 2^(aleph_0) and the cardinality of an infinite field is the same as the cardinality of its algebraic closure, Q(S) must have cardinality 2^(aleph_0).
		If a field f(R) is obtained by the extension of a bijection between Q(S) and Q(T) where S is a maximal algebraically independent subset of R and T is an algebraically independent subset of R, then the field Q(T) is in f(R) intersect R, so that f(R) intersect R is of cardinality 2^aleph_0.
		Suppose that r is in the intersection and r is not algebraic.
	www.lightlink.com /lark/cfield.isor (872 words)

	The Transcendent Space of an Extension
		In an earlier section we saw that an element v is algebraic over the base field K if it satisfies a polynomial (in one variable) with coefficients drawn from K. Failing this, v is transcendental.
		Algebraic and integral are synonymous when K is a field, since one can always divide p through by its lead coefficient to make it monic.
		A basis is a maximal algebraically independent set under set theoretic inclusion, which must exist by zorn's lemma.
	www.mathreference.com /fld,algind.html (1563 words)

	Transcendental number - (Site not responding. Last check: 2007-11-04)
		In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients.
		However, an algebraic function of several variables may yield an algebraic value when applied to transcendental numbers if these numbers are not algebraically independent: π and 1−π are both transcendental, but π+(1−π)=1 is obviously not.
		Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, which happen to be a and b, must be algebraic.
	psychcentral.com /psypsych/Transcendental_number (1132 words)

	PlanetMath: Lindemann-Weierstrass theorem
		are algebraically independent is still an open question today.
		Schanuel's conjecture is a generalization of the Lindemann-Weierstrass theorem.
		Cross-references: algebraic independence, Schanuel's conjecture, open question, iff, easy to see, transcendental, equivalent, algebraically independent, algebraic numbers, linearly independent
	planetmath.org /encyclopedia/LindemannWeierstrassTheorem.html (131 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		In any Lorentzian manifold, the fourth rank Riemann curvature tensor R^a_(bcd) (20 algebraically independent components at each event) decomposes into the second rank Ricci curvature tensor R_(ab) (10 algebraically independent components at each event) and the fourth rank Weyl conformal curvature tensor C^a_(bcd) (10 algebraically independent components at each event).
		Petrov classified the algebraic symmetries which the Weyl tensor can have like this: I II D III N O The type I situation is the most generic: the Weyl tensor has no "extra algebraic symmetries".
		There are known examples of vacuum solutions to the EFE which exhibit each of these different types of algebraic symmetries in the spacetime curvature.
	math.ucr.edu /home/baez/PUB/PetrovTypes (664 words)

	DEPENDENCE OF LOGARITHMS ON COMMUTATIVE ALGEBRAIC GROUPS (Site not responding. Last check: 2007-11-04)
		A well-known conjecture states that linearly independent logarithms of algebraic numbers are algebraically independent over the field of rational numbers.
		So far, it is not yet known that there exist two algebraically independent logarithms of algebraic numbers.
		From this point of view, half of the conjecture is known: the actual rank of such a matrix is at least half the conjectural rank.
	math.la.asu.edu /~rmmc/rmj/vol26-3/wald.htm (129 words)

	[Algebraically independent] \| [All the best Algebraically independent resources at karaoke.velocityincome.com]
		Rank ExtractorsWe establish some basic connections between the min-entropy of such sources and the algebraic rank of the given polynomials (namely the largest number of algebraically independent ones).
		In particular, a one element set {α} is algebraically independent over K if and only if α is transcendental over K. In general, all the elements of an algebraically independent set over K are by necessity transcendental over K, but that is far from being a sufficient condition.
		For example, the subset {√π, 2π 1} of the real numbers R is not algebraically independent over the rationals Q, since the non-zero polynomial
	karaoke.velocityincome.com /Algebraically_independent (545 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		An abstract mathematical system consisting of a vector space together with a multiplication by which two vectors may be combined to yield a third, and some axioms relating this multiplication to vector addition and scalar multiplication.
		A field F is said to be algebraically closed in an extension field K if any root in K of a polynominal with coefficients in F also lies in F.
		] An algebra of subsets of a set S is a family of subsets of S that contains the null set, the complement (relative to S) of each of its members, and the union of any two of its members.
	www.accessscience.com /Dictionary/A/A19/DictA19.html (1509 words)

	[FOM] FOM: Zilber's Program
		A key part of their proof was the observation that in a strongly minimal set algebraic closure satisfies exchange (if x in acl(A,b) and x not in acl(A), then b is in acl(A,x)).
		This means that strongly minimal sets with algebraic closure determine a pregeometry and there is a well defined dimension theory (special cases are the usual dimension in linear algebra or transcendence degree in algebraically closed fields).
		In addition to Shannuel's conjecture (SC) there are axioms asserting algebraic closedness (ACF) the kernel of exponentiation is free on one generator (ker), subsets of the line defined as solutions to systems of equations are countable (CC), and an axiom asserting that certain non-overdetermined systems of equations have solutions (EC).
	www.cs.nyu.edu /pipermail/fom/2003-June/006760.html (1667 words)

	RR-3345 : Algebraic Relations among Matching Constraints of Multiple Images
		The exact algebraic relations among these multi-linear constraints have not been elucidated by previous authors.
		The key result that will be established is that for generic view configurations and generic points, all multi-linear constraints may algebraically be reduced to the algebraically independent bilinear constraints-.
		As a consequence, $2n-3$ algebraically independent bilinearities from pairs of views completely describe the algebraic/geometric structure of $n$ uncalibrated views for generic views and points.
	www.inria.fr /rrrt/rr-3345.html (344 words)

	Pi - Wikipedia, the free encyclopedia
		In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula from 1706 and calculated the first 140 decimal places for π of which the first 126 were correct [1] and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct.
		The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating π to 707 decimal places (accomplished in 1873).
		In 1944, D. Ferguson found (with the aid of a mechanical desk calculator) that Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were fallacious.
	en.wikipedia.org /wiki/Pi (3880 words)

	[No title]
		Furthermore, a* lthough the wn(ff) are again algebraically* independent for fixed ff, various relations * *hold as ff varies.
		For fixed ff the wn(ff) will be algebraically independent, by t* *he argument used for Chern classes, but there will be relations as ff varies.
		BGLF induces a homomorphism of Hopf algebras S(H~RP 1) ~=HBO- !
	hopf.math.purdue.edu /Mitchell/sw.txt (8631 words)

	How does matter couple to space-time so that space-time becomes curved?
		Furthermore, the curvature of space-time at each event is completely described by a multilinear operator (a generalization of a linear operator) called the Riemann curvature tensor, which has 20 algebraically independent components at each event.
		The components of the Riemann tensor identically satisfy a differential equation (the Bianchi identity), which is why the metric tensor (ten algebraically independent components at each event) can and does completely determine the Riemann curvature tensor (20 algebraically independent components at each event).
		Physics enters via the stress-energy tensor T_(ab), which you can think of as a 4x4 symmetric matrix (so it has 10 algebraically independent components at each event); this tensor completely describes the amount of (non-gravitational) mass-energy at each event, and also any momentum (mass-energy flow) and stresses (such as the pressures in a fluid).
	www.physlink.com /Education/AskExperts/ae98.cfm (689 words)

	The linear 4-point algorithm
		This can effectively be done with any computer algebra system and will directly give the unique solution of the problem for general configurations of the points.
		However this algebraic method is almost useless for practical situations as the successive elimination will ultimately give complicated coefficients for the final linear polynomial which compromise the numerical stability of the solution.
		Instead of doing it algebraically, our goal is to develope a numerical linear method which indeed gives the unique solution if it does exist.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/QUAN/node7.html (366 words)

	[No title]
		We consider the algebraic equations that result from specifying the relative distances between points in a plane.
		It is a well-known result of Laman [2] that maximal independence, minimal rigidity and infinitesimal rigidity are all equivalent.
		In case (a) the subgraph is replaced in its entirety by three new edges and we show that the new edges are again algebraically independent.
	www-calfor.lip6.fr /ICPSS/papers/24OP/24OP.html (1398 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		Of course, there must be other symmetries so that the number of algebraically independent components of *R_(abcd) is 20, the same as for R_(abcd).
		Together, ER(X) and BR(X) account for 6+8=14 of the 20 algebraically independent components of the Riemann tensor R. There is a third tensor, the Bel tensor LR(X), which accounts for the other 6 components, and has the same symmetries as ER(X).
		Recall that in contrast, the conformal curvature or Weyl tensor, the completely traceless part of the Riemann tensor, has 10 algebraically independent components at each event, and splits into electro-Weyl tensors EW(X) and magneto-Weyl tensors BW(X), each of which can be thought of as symmetric traceless three by three matrices (five algebraically independent components each).
	math.ucr.edu /home/baez/PUB/electromagneto (5847 words)

	Orðasafn: A
		algebraic distance stefnubundin fjarlægð, = directed distance, = signed distance.
		algebraic equation algebrujafna, algebruleg jafna, margliðujafna, -> polynomial equation.
		algebraic operation aðgerð, reikniaðgerð, = law of composition, = operation 2, -> binary operation, -> internal composition, -> internal operation.
	www.hi.is /~mmh/ord/safn/safnA.html (2405 words)

	Noether Normalization
		Since 1 maps to 1, the map on K is injective, and K maps to K. We only need map the indeterminants into A. Before diving into this theorem, you might want to review a similar theorem from field theory.
		The ring R can be described as a transcendental extension of K, followed by an algebraic extension.
		In the previous section we proved that such a field is a finite extension of K, hence it is algebraic.
	www.mathreference.com /ag,nnl.html (613 words)

	Nishioka: Algebraic independence by Mahler's method and $S$-unit equations
		Algebraic independence by Mahler's method and $S$-unit equations.
		: Algebraic independence of the values of certain functions at a transcendental number.
		: On the algebraic independence of holomorphic solutions of certain functional equations and their values.
	www-mathdoc.ujf-grenoble.fr /numdam-bin/item?id=CM_1994__92_1_87_0 (159 words)

	Complexity Digest - Use Of Algebraically Independent Numbers For Zero Recognition (Site not responding. Last check: 2007-11-04)
		Use Of Algebraically Independent Numbers For Zero Recognition, J.
		We attempt to decide whether or not such a tree represents the zero polynomial by substituting algebraically independent real numbers for the variables and attempting to decide whether or not the resulting constant is zero.
		The result is that if a polynomial term has s(T) nodes, then the bit complexity of deterministic zero recognition is bounded by O(s(T)M(s(T) length(T))), where length(T) measures the length of the term T, and M(n) is the bit complexity of multiplication of two n-digit natural numbers.
	www.comdig.org /article.php?id_article=12842 (147 words)

	Lindemann-Weierstrass theorem: Encyclopedia topic (Site not responding. Last check: 2007-11-04)
		The transcendence of e (e: The 5th letter of the Roman alphabet) and π (π: more facts about this subject) are direct corollaries of this theorem.
		Therefore, if π were algebraic, cos(π) = −1 and sin(π) = 0 would be transcendental, which proves by contradiction π is not algebraic, and hence is transcendental.
		are a set of algebraic numbers linearly independent over the rationals such that for some prime p, then the p-adic exponentials are algebraically independent transcendentals.
	www.absoluteastronomy.com /reference/lindemann-weierstrass_theorem (276 words)

Try your search on: Qwika (all wikis)