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	Friedrich Heinrich Jacobi (Stanford Encyclopedia of Philosophy)
		Jacobi's veiled message was that the adepts of this new cultural phenomenon had failed to escape the rationalism of the philosophers, since the rebellious new humanism they advocated made sense only on the presupposition that the philosophers' conception of reality was the right one.
		Jacobi rejects it off-hand on the ground that, as a matter of fact, a subject could not be aware of himself — aware also, therefore, of the alleged subjectivity of some of his representations — without defining his ‘self’ in opposition to some admittedly external object, i.e.
		Jacobi responded with a pamphlet of his own (Jacobi, 1782) in which he defended Müller's position — not because he had any sympathy for Catholicism, or because he was opposed to secularism, but because he thought that the Popes' spiritual despotism was much to be preferred to the secular, allegedly enlightened, despotism of the princes.
	plato.stanford.edu /entries/friedrich-jacobi (15415 words)

	Carl Gustav Jacob Jacobi - Wikipedia, the free encyclopedia
		Jacobi suffered a breakdown from overwork in 1843.
		Jacobi is buried at a cemetery in the Kreuzberg section of Berlin, the Friedhof II der Jerusalems- und Neuen Kirchengemeinde (61 Baruther Street).
		Jacobi was also the first mathematician to apply elliptic functions to number theory, for example, proving the 2 square and four-square theorems of Pierre de Fermat.
	en.wikipedia.org /wiki/Carl_Gustav_Jakob_Jacobi (823 words)

	Jacobi identity - Wikipedia, the free encyclopedia
		The Jacobi identity is satisfied by the multiplication (bracket) operation on Lie algebras and Lie rings and these provide the majority of examples of operations satisfying the Jacobi identity in common use.
		Thus, the Jacobi identity for Lie algebras simply becomes the assertion that the action of any element on the algebra is a derivation.
		In analytical mechanics, Jacobi identity is satisfied by Poisson brackets, while in quantum mechanics it is satisfied by operator commutators.
	en.wikipedia.org /wiki/Jacobi_identity (249 words)

	PlanetMath: Jacobi identity interpretations
		Yet another way to formulate the identity is
		Cross-references: derivation, identity, representation, adjoint representation, Lie algebra, Jacobi identity
		This is version 2 of Jacobi identity interpretations, born on 2002-09-20, modified 2002-09-20.
	planetmath.org /encyclopedia/JacobiIdentityInterpretations.html (74 words)

	PlanetMath: Jacobi's identity for $\vartheta$ functions
		Jacobi's identities describe how theta functions transform under replacing the period with the negative of its reciprocal.
		Cross-references: group, modular, transformations, relations, negative, period, Transform, functions, Jacobi identities
		This is version 2 of Jacobi's identity for
	planetmath.org /encyclopedia/JacobisIdentityForVarthetaFunctions.html (69 words)

	Biography of Carl Gustav Jakob Jacobi (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-04)
		Jacobi was also the first mathematician to apply elliptic functions to number theory, for example, proving the polygonal number theorem of Pierre de Fermat.
		His investigations in elliptic functions, the theory of which he established upon quite a new basis, and more particularly his development of the theta function, as given in his great treatise Fundamenta nova theoriae functionum ellipticarum (Königsberg, 1829), and in later papers in Crelle's Journal, constitute his grandest analytical discoveries.
		He was one of the early founders of the theory of determinants; in particular, he invented the functional determinant formed of the n2 differential coefficients of n given functions of n independent variables, which now bears his name (Jacobian), and which has played an important part in many analytical investigations.
	biography-1.qardinalinfo.com.cob-web.org:8888 /j/Jacobi_Carl_Gustav_Jakob.html (620 words)

	Symbolic Test of the Jacobi Identity for Given Generalized 'Poisson' Bracket -- from Mathematica Information Center
		The problem is to evaluate single and nested arbitrary generalized Poisson brackets and the cyclic sum of these in order to test the Jacobi identity on a given state space for systems described in terms of discrete or of continuous variables.
		The Jacobi identity has to be fulfilled for Poisson brackets consistently describing the reversible dynamics of physical systems as desired, e.g., within the framework of nonequilibrium thermodynamics [1-3].
		Jacobi identity, Poisson brackets, GENERIC, nonequilibrium thermodynamics, reversible motion, symbolic programming
	library.wolfram.com /infocenter/Articles/2721 (125 words)

	SPLCenter.org: Justice vs. Justus
		Christian Identity was the cornerstone of the Freemen edifice, even for farmers who may have first embraced the check-writing scheme out of economic distress.
		To Chris Temple, an Identity ideologue writing in June 1996, the Freemen were "at the center of the most significant clash between the...
		Temple, an Identity strategist now working with Liberty Lobby, was friendly to the Freemen's goals and methods.
	www.splcenter.org /intel/intelreport/article.jsp?pid=726 (528 words)

	[No title]
		The Jacobian can be interpreted as a kind of generalized Poisson bracket: it is skew-symmetric with respect to $f$, $g$ and $h$; it is a derivation of the algebra of smooth functions on ${\Bbb R}^3$, i.e., the Leibniz rule is verified in each argument.
		In fact, in the usual Poisson formulation, the Jacobi identity is the infinitesimal form of Poisson theorem which states that the bracket of two integrals of motion is also an integral of motion.
		\end{theorem} For the case $n=2$, the FI is Jacobi identity and one recovers the usual definition of Poisson manifold.
	www.ma.utexas.edu /mp_arc/papers/96-39 (4095 words)

	Paul-Olivier Dehaye - Papers (Site not responding. Last check: 2007-11-04)
		On an identity of ((Bump and Diaconis) and (Tracy and Widom)).
		They obtained this identity by computing asymptotics for the determinants of finite rank reductions of minors of Toeplitz matrices in two different ways, spanning different branches of mathematics: "Toeplitz minors", J.Combinatorial Theory Ser.
		We show that this identity is essentially a differentiated version of the Jacobi-Trudi identity, under the action of a differential operator on symmetric functions reminiscent of vertex operators.
	www.maths.ox.ac.uk /~pdehaye/papers.html (780 words)

	Abstracts (Site not responding. Last check: 2007-11-04)
		From this finite version we are able to obtain new polynomial versions of Jacobi's identity and other fundamental identities in the theory of partitions and q-series, like Lebesgue's identity.
		The asymptotic methods and the shape of 38 of the 40 identities suggest the influence of the 5-dissection of the generating function for the crank of partitions.
		We give a generalization of an identity of Rogers which was used by Rogers and later by Bressoud to prove some of Ramanujan's 40 identities for the Rogers-Ramanujan functions.
	www.theoryofnumbers.com /CANT/2005/abstracts.htm (4450 words)

	SuGra 3-Connection Reloaded \| The n-Category Café
		Jacobiator = 1-chain which replaces the Jacobi identity when we go from Lie algebras to Lie 2-algebras.
		The Jacobi identity says the bracket is a derivation of itself, which is an infinitesimal way of saying that the flow generated by a vector field, acting as an operation on vector fields, preserves the Lie bracket!
		Similarly, as soon as we know the Jacobi identity, we know the Lie bracket operation on vector fields is preserved by small diffeomorphisms, by the argument outlined in the body of this Week.
	golem.ph.utexas.edu /category/2006/08/sugra_3connection_reloaded.html (6911 words)

	Jacobi identity (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-04)
		The Jacobi identity is the name for the following equation:
		[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0 for all X,Y,Z. Lie algebras are the primary example of an algebra which satisfies the Jacobi identity.
		But note that an algebra can satisfy the Jacobi identity but yet not be anticommutative.
	publicliterature.org.cob-web.org:8888 /en/wikipedia/j/ja/jacobi_identity.html (96 words)

	Graphical Discovery of a New Identity for Jacobi Polynomials -- from Mathematica Information Center
		During the summer of 1995 the authors engaged in an undergraduate research program that investigated various conjectures about orthogonal polynomials.
		While exploring the grahic capabilities of Mathematica, we generated Figure 1, which shows, on a single set of axes, the fifth-degree Jacobi polynomials, for beta = 0.9 and alpha taking the values 0, 1, 2,..., 6.
		We observed that the curves in the figure appear to intersect at extrema, but our advisors were skeptical about whether this actually happened.
	library.wolfram.com /infocenter/Articles/1780 (118 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		In the talk I'll describe my current state of knowledge, and confusion, concerning the algebraic significance of an identity governing a certain family of sparse matrices.
		I ] The Desnanot-Jacobi identity relates six sub-determinants of this matrix: one (n-2)-by-(n-2) determinant, four (n-1)-by-(n-1) determinants, and one n-by-n determinant.
		PROBLEM: Interpret/prove Kuo's formula via linear algebra (a combinatorial proof is already known) so as to clarify its true relation to the Desnanot- Jacobi identity.
	www.math.wisc.edu /~propp/somos/kuo3 (336 words)

	PlanetMath: (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-04)
		Jacobi elliptic function (in elliptic integrals and Jacobi elliptic functions) owned by mathcam
		Jacobi identity (in algebra (module)) owned by mathcam
		Jacobi identity (in Lie algebra) owned by djao
	planetmath.org.cob-web.org:8888 /encyclopedia/J (441 words)

	Lie algebra (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-04)
		The vector space of left-invariant vector fields on a Lie group is closed under this operation and is therefore a finite dimensional Lie algebra.
		One may alternatively think of the underlying vector space of the Lie algebra belonging to a Lie group as the tangent space at the group's identity element.
		The tangent space at the identity matrix may be identified with the space of all real n-by-''n'' matrices with trace 0, and the Lie algebra structure coming from the Lie group coincides with the one arising from commutators of matrix multiplication.
	lie-algebra.kiwiki.homeip.net.cob-web.org:8888 (1253 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		Bruce Berndt - University of Illinois at Urbana-Champaign "Ramanujan's Forty Identities for the Rogers-Ramanujan Functions" Abstract: The Rogers-Ramanujan identities are among the most famous identities in combinatorial number theory.
		For the remaining five identities, we have found nonrigorous arguments using asymptotic analysis to establish the identities.
		Hamza Yesilyurt - University of Florida "A Generalization of a Modular Identity of L.G. Rogers and Applications to Modular Equations" Abstract: We give a generalization of an identity of Rogers which was used by Rogers and later by Bressoud to prove some of Ramanujan's 40 identities for the Rogers-Ramanujan functions.
	www.theoryofnumbers.com /CANT/2005/cant2005_abstract.doc (2488 words)

	index
		We formulated fundamental associativity, commutativity and generalized rationality properties of intertwining operator algebras in terms of both complex and formal variables, and we formulated and proved basic implications among these axioms.
		In the theory of vertex operator algebras, the Jacobi identity is an analogue of the Lie algebra Jacobi identity and plays a particularly important role.
		For intertwining operator algebras, it is natural to ask whether there is also a Jacobi identity, and we found such a Jacobi identity for intertwining operator algebras.
	www.rci.rutgers.edu /~yzhuang/page25.html (369 words)

	Extensions of the Jacobi Identity for Vertex Operators, and Standard a [1 ](1) -modules; Author: Husu, Cristiano; ...
		Extensions of the Jacobi Identity for Vertex Operators, and Standard a [1 ](1) -modules
		This work extends the Jacobi identity, the main axiom for a vertex operator algebra, to multi-operator identities.
		Based on constructions of Dong and Lepowsky, relative Z [2 -twisted vertex operators are then introduced, and a Jacobi identity for these operators is established.
	www.netstoreusa.com /mabooks/082/0821825712.shtml (216 words)

	Commutator - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-04)
		The subgroup of G generated by all commutators is called the derived group or the commutator subgroup of G.
		It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section).
		The fourth identity follows from the first and third.
	en.wikipedia.org.cob-web.org:8888 /wiki/Commutator (421 words)

	Henriques: "A Lie algebra for the String group"
		The bracket\nandgt;\nandgt; [(X,a), (Y,b)] := ([X,Y], 0)\nandgt;\nandgt; satisfies the Jacobi identity on the nose.\nandgt;\nandgt; The Jacobiator in this example degenerates to an automorphism 2-morphism on\nandgt; the on 1-morphism [[(X,a),(Y,b)],(Z,c)].\n\nThe Jacobi identity holds of course for the bilinear bracket, but how\ncan then [(X,a),(Y,b),(Z,c)] != 0 ?
		The Jacobi identity holds of course for the bilinear bracket, but how
		Because the trinary bracket is not equal to the Jacobi operation on the
	www.physicsforums.com /showthread.php?t=62139 (1330 words)

	week238
		Suppose you have a tangent vector at any point of the group G. Then you can translate it to the identity element of G and get a tangent vector at the identity of G. But, this is nothing but an element of Lie(G)!
		To be a bit more formal about this, let's think of L as a graded vector space where everything is of degree zero.
		= 0 and the Jacobi identity say that d and the Lie bracket are preserved by diffeomorphisms - but at least they imply these operations are preserved by small diffeomorphisms.
	math.ucr.edu /home/baez/week238.html (2665 words)

	jan jacobi - ResearchIndex document query
		P i6=k a ki x k j a kk and it is called the Jacobi algorithm.
		Z ffiF ffiu v dx: When J is constant on M, the Jacobi identity is trivially satisfied, and one need
		Analysis of a Multigrid Method for a Transport Equation by..
	citeseer.ist.psu.edu /cis?q=Jan+Jacobi (813 words)

	Mathematics
		C, the category of `special coalgebras.’ Finally, we extract the underlying vector space from our special coalgebra and show that it is the Lie algebra of the Lie group we started with.
		The Jacobi identity for the bracket follows from the self-distributive law for the quandle operation, while the antisymmetry of the bracket arises from the idempotence law satisfied by the quandle operation.
		The content of this paper is taken from the second chapter of my dissertation.
	myweb.lmu.edu /acrans/research.html (1132 words)

	[ref] 30 Domains and their Elements (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-04)
		But a domain with operational structure of multiplication, taking the identity, and taking inverses will be treated as a group as soon as the multiplication is found out to be associative for this domain.
		The identity of an object need not be distinct from its zero, so for example a ring consisting of a single element can be regarded as a ring-with-one (see Rings).
		satisfies the Jacobi identity, that is, x * y * z + z * x * y + y * z * x is zero for all x, y, z in F.
	www.gap-system.org.cob-web.org:8888 /Manuals/doc/htm/ref/CHAP030.htm (5263 words)

	[No title]
		2) Jacobi Identity: [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 for all x,y,z in L. The operation [,] is called a "bracket"; sometimes "lie bracket" or "commutator" [17:06]
		Indeed, this is equivalent to the Jacobi identity: [17:38]
		it's an instructive exercise again to check that these are ideals (use Jacobi identity).
	br.endernet.org /~loner/algebra/kliealg.txt (2777 words)

	AddALL.com - Extensions of the Jacobi Identity for Vertex Operators, and Standard A (via CobWeb/3.1 ... (Site not responding. Last check: 2007-11-04)
		AddALL.com - Extensions of the Jacobi Identity for Vertex Operators, and Standard A (via CobWeb/3.1 planetlab2.cs.unc.edu)
		Extensions of the Jacobi Identity for Vertex Operators, and Standard A
		If you cannot find this book in our new and in print search, be sure to try our used and out of print search too!
	www.addall.com.cob-web.org:8888 /detail/0821825712.html (75 words)

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