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Topic: Dual basis

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	Covariant transformation - Wikipedia, the free encyclopedia
		The transformation that describes the new basis vectors in terms of the old basis, is defined as a covariant transformation.
		(the basis vectors are tangent vectors to the coordinate grid).
		Since vectors (and dual vectors) are defined coordinate independently, this definition of a tensor is also free of coordinates and does not depend on the choice of a coordinate system.
	en.wikipedia.org /wiki/Covariant_transformation (1439 words)

	Dual space - Wikipedia, the free encyclopedia
		In the language of category theory, taking the dual of vector spaces and the pullback of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself.
		The idea of a dual vector as an infinite sum should not be taken too literally; in general infinite sums are defined in terms of a limit, which only makes sense in a topological space, and even then not all sums will be convergent.
		The continuous dual V′ of a normed vector space V (e.g., a Banach space or a Hilbert space) forms a normed vector space.
	en.wikipedia.org /wiki/Dual_space (1692 words)

	Dual space
		In mathematics the existence of a 'dual' vector space reflects in an abstract way the relationship between row vectors (1xn) and column vectors (nx1).
		Given any vector space V over some field F, we define the dual space V* to be the set of all linear functions from V to F. These linear functions to the base field are also called linear functionals.
		In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself.
	www.ebroadcast.com.au /lookup/encyclopedia/du/Dual_space.html (940 words)

	PlanetMath: dual space
		The notions of duality extend, in part, from vector spaces to modules, especially free modules over commutative rings.
		A related notion is the duality in projective spaces.
		This is version 10 of dual space, born on 2002-02-03, modified 2004-09-17.
	planetmath.org /encyclopedia/DualBasis.html (241 words)

	AMT: Dual Basis Assets
		Your basis in an asset, such as stock or real property, is used to determine how much gain or loss you report when you sell that asset.
		(Basis may be used for other purposes as well.) In some situations an asset may have one basis for regular income tax purposes and a different basis (usually higher) for alternative minimum tax purposes.
		When you sell a dual basis asset, you report the difference between the regular tax gain or loss and the AMT gain or loss as an adjustment.
	www.fairmark.com /amt/dual.htm (791 words)

	SPxBasis
		The dimension of the vectors is refferred to as the basis' dimension, whereas the number of vectors belonging to the LP is called the basis' codimension.
		The first type specifies that a variable is set on a primal bound, while the later type indicates a dual variable to be set on a bound.
		When a new LP is loaded, the basis matrix and vectors become invalid and possibly also of the wrong dimension.
	web.bilkent.edu.tr /Online/Soplex/SPxBasis.html (1026 words)

	The Seysen Basis Reduction Algorithm
		Seysen's basis reduction algorithm (or just Seysen's algorithm) differs from the LLL algorithm and its variants in that it considers all vectors in the lattice simultaneously, and performs operations on those vectors which will reduce the lattice according to some measure.
		Seysen was motivated to create a new method for basis reduction by a desire to find a better way to simultaneously reduce a lattice and its reciprocal (or dual) lattice.
		It is easy to see that if lattice L is reduced with the LLL algorithm, the resulting reduced lattice may have a dual in which some basis vector is quite large, as no attempt is ever made to consider the size of the dual basis when row moves are being performed.
	www.farcaster.com /papers/sm-thesis/node8.html (626 words)

	McCune/Padmanabhan Monograph Problems (Site not responding. Last check: 2007-10-23)
		Lemma DUAL-BA-7 Dual BA 3-basis, existence of 0 and 1.
		Lemma IL-3 Inverse loop schema and basis imply beta=alpha.
		Theorem TC-9 TC basis for double inversion in Abelian groups (1).
	www-unix.mcs.anl.gov /~mccune/papers/monograph/problems.html (1178 words)

	Simplex-specific Methods
		is 1 then the method does a values pass so variables not in basis are given their current values and one pass of variables is done to clean up the basis with an equal or better objective value.
		The return code is 0 if the basis was not slacks in first case, it is negative if dual is preferred or positive if primal.
		The CLP dual algorithm declares all problems to be dual feasible by putting non-basic variables to correct bounds for the reduced cost.
	www.coin-or.org /Clp/userguide/ch02s07.html (376 words)

	USC School of Social Work: Dual Degree Programs (Site not responding. Last check: 2007-10-23)
		Dual degree programs are based on the premise that certain topics covered in the School of Social Work are also addressed in the curriculum of other schools.
		This dual degree program is a four-year program that requires the completion of 117 units (43 units in social work and 74 units in law).
		This dual degree program is highly valuable to students interested in pursuing a career in services for the aging.
	www.usc.edu /dept/socialwork/academic/dual.php (759 words)

	Dual Space theory
		Dual Space theory aims to cure many of the problems of physics.
		#7 is Dual Space' explanation of starlight bending as it passes the Sun.
		May 6, 2005 Paper #10 elaborates on properties in primal cell which is basis for Espace.
	www.dualspace.net (400 words)

	February 2002 Hot Topics: Exercising ISOS and the AMT - The Dual Basis Concept
		Under the regular tax system, you have a basis in your stock which is usually the amount paid for it (the option price).
		Thus you will have a dual basis, one under the AMT and the other under the regular tax system.
		The rules involving dual basis in stock, AMT capital loss carryforwards and use of your AMT credit are extremely complex and require accurate and extensive number-crunching with sophisticated tax software.
	www.taxprophet.com /hot_topic/0202.shtml (1537 words)

	The Foundations of Relativity - Page 3
		There is also a "dual space" V* with the same number of dimensions that the duals of the vectors live in.
		A tensor is just the functional you described which maps a certain number of dual vectors and a certain other number of vectors to a scalar - but all the vectors v_i live in the same vector space V, and all the dual vectors v_j live in the same dual space V*.
		A dual vector is a linear mapping between the vector spaces
	www.physicsforums.com /showthread.php?t=104683&page=3 (3102 words)

	Computation - Operations Research Models and Methods
		The dual variables are shown to the right of the node external flow data.
		Show Basis: For a network flow problem, a basis is defined by a collection of arcs with one arc entering each node of the network.
		The reduced costs are shown on the arc display in column M. The dual variables basis tree information is shown on the node display.
	www.me.utexas.edu /~jensen/ORMM/computation/unit/mp_add/subunits/solver/nfsolver.html (1049 words)

	Duality in vector Padé-Hermite approximantion problems (Site not responding. Last check: 2007-10-23)
		It describes dual basis matrices for the solution spaces of such problems and proves their properties.
		We also show that the classical duality between Pad'e-Hermite problems of type I and type II is a special case of vector Pad'e-Hermite duality.
		The article proposes an algorithm that solves two dual vector Pad'e-Hermite problems at the same time, returning dual basis matrices of the respective solution spaces.
	www.cs.kuleuven.ac.be /cwis/research/nalag/papers/ade/dual (158 words)

	Differential operators on a smooth manifold.
		Let (dim:b:V) be some basis of V, with (dim:p:dual-V) the implied dual basis of V's dual: this is defined by p(i).b(j) being 1 when i=j but 0 otherwise.
		}), dual to bb in the sense that, for each m in M and i,j in dim, pp(m,i).bb(m,j) is 0 unless i=j, in which case it is 1: call this when(i=j).
		So, let (dim: b :gradient fields) be a pointwise basis of the gradient fields in some neighbourhood and look, therein, at the zero derivative of the metric.
	www.chaos.org.uk /~eddy/math/smooth/differ.html (2029 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		This paper presents a new approach for dual basis multiplication.
		In contrast to the conventional approach, the proposed technique assumes that both operands are given in the polynomial basis.
		The space and time complexities of the proposed multiplication algorithm are the same as the conventional dual basis multipliers for irreducible trinomials.
	islab.oregonstate.edu /papers/s01dbmul.html (134 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		%DUAL Dual simplex code for solving an LP problem given an initial % dual feasible basis.
		On entry, % basis is a row vector containing the indices of the basis, which % is assumed to be dual feasible.
		On exit, basis contains the indices % of the final basis and iter denotes the number of iterations.
	www.math.kth.se /~andersf/5B5850/dual.m (157 words)

	Double information keys (Site not responding. Last check: 2007-10-23)
		Dual objective value reported by the interior-point or Qcone optimizer.
		Dual feasibility measure reported by the interior-point and Qcone optimizer.
		Whereas it should converge to infinity when the problem is (strictly) primal or dual infeasible.
	www.mosek.com /products/2/tools/doc/html/tools/node246.html (420 words)

	The Dual Basis in Tensor Math -- Please help me understand
		Take a vector basis function: v = ax + by + cz Its a composition of single vectors where, I think, the single vectors are 1.
		Guess one about the dual basis: we can either choose an abstract unit for A and express B in these units or choose _one_ and some reference (label) to that particular unit type -- say _meter_ so that we can say that B is three meters long.
		I.e., geometric lengths posses a duality of length and density and so are best described via a dual basis.
	www.lns.cornell.edu /spr/2002-03/msg0040641.html (493 words)

	Row Moves Involving Three or Four Vectors
		exhibit a symmetry between the prime and dual lattice which is lost when we consider n-moves with n > 2.
		for a given set of basis vectors is also more complicated as the number of vectors involved in the move increases.
		In all cases one vector was designated the ``target'' vector and integer multiples of the other two or three vectors were added to the target.
	www.farcaster.com /papers/sm-thesis/node20.html (653 words)

	Re: The Dual Basis in Tensor Math -- Please help me understand
		A solid understanding of "abstract" linear algebra (in particular, linear operators on a vector space V and their components [matrix representations] wrt a basis for V) is a prerequisite for multilinear algebra (in particular, tensors on a vector space V and their components wrt a basis for V).
		Coxeter > presents the math of dual basis ^^^^^^^^^^ In several of his books, Coxeter is concerned with dual bases for -lattices-.
		So are shears, incidentally (consider the obvious basis for [b,b], where b is the Lie algebra of the upper triangular group).
	www.lns.cornell.edu /spr/2002-03/msg0040680.html (608 words)

	[No title]
		I'll pass on exhibiting a basis for the dual space of F^N (direct product), and focus on finding a basis for the dual space of F^(N) (direct sum).
		Prove that therefore no infinite- dimensional F-space has a countable Hamel basis." Finally, a proof (using the Hahn-Banach theorem) that the dimension of any infinite dimensional Banach space is at least c is given on page 75 of Goffman and Pedrick's text "First Course in Functional Analysis" (1983 2'nd edition published by Chelsea).
		Brown claimed that the DIMENSION of the dual space of F^(N) is uncountable, not just that the dual space of F^(N)--as a set--is uncountable.
	www.math.niu.edu /~rusin/known-math/00_incoming/duals (2065 words)

	Linear Algebra project (Site not responding. Last check: 2007-10-23)
		The main responsibility of a LinearAmbient is to keep a cannon basis and to respond the coordinates of any vector in that basis.
		A (linear) subspace has a basis and you want to express each vector of the subspace as a linear combination of the basis.
		In other words, each time we want to calculate the coordinates of a vector v in some basis B, we must first be able to express the coordinates of the elements of B and v in terms of another basis.
	mate.dm.uba.ar /~caniglia/Linear (1503 words)

	Numerical Notes: Math Archives (Site not responding. Last check: 2007-10-23)
		In a finite-dimensional space V of dimension n, we can choose a basis of n vectors; then anything in the space can be written uniquely as a linear combination of those vectors.
		To matrix analysts, this might be the distinction between a row and a column vector; to physicists more comfortable with Dirac notation, it is the distinction between a bra and a ket.
		That is, we have a basis for the space of linear operators on V; and where we have a basis, we have a concrete representation.
	www.cs.berkeley.edu /~dbindel/blog/archives/cat_5 (1882 words)

	IEEE P1363a: Additional number-theoretic algorithms
		there is needed no additional work load as basis changing (from standard to the dual basis or from the dual basis to standard basis as the conventional dual based arithmetic does),
		This summary of finite field basis conversion techniques is proposed for inclusion in IEEE P1363 Annex A. Included are some conventional basis conversion techniques, as well as some new storage-efficient basis conversion techniques.
		In this paper we discuss the cost of the communication between such cryptosystems and propose the use of a non-conventional basis representation providing the improved communicaton.
	grouper.ieee.org /groups/1363/P1363a/NumThAlgs.html (715 words)

	The Foundations of Relativity - Page 5
		First we note that a mapping of basis vectors between V and V* defines a general linear map from V to V*.
		I suppose I'm also requiring that g_ij be invertible, but the basis vectors are linearly independent by defintition, and that should be good enough to insure that an inverse exists.
		This is an elementary fact about dual spaces: there does not exist a natural map from a vector space to its dual.
	www.physicsforums.com /showthread.php?p=875985 (2677 words)

	Differential Forms and Chains
		to an expression for the alternating k-tensor in terms of the basis of Proposition 1.
		The linear independense is shown analogously to the argument of Proposition 1.
		In both cases we can express the vectors in terms of a standard basis, and so we have a certain number of coodinate functions.
	www.ms.uky.edu /~ken/ma570/lectures/lecture7/html/forms.htm (679 words)

	Daily Radio Broadcast::The Dual Basis for Christian Service (Site not responding. Last check: 2007-10-23)
		The basis upon which you understand the Word of God is, at some point, you've got to read it.
		Romans 12 [:1], the basis of a sacrificed life, "I beseech you therefore, brethren, by the mercies of God, that you present your bodies a living sacrifice."
		The basis on which Paul makes this statement, the basis upon which he makes this request is knowing full well all that God has done for us, God expects us to step up to the plate.
	www.backtothebible.org /broadcasts/radio/print.php/26147 (3752 words)

	New Results with ANL's ATP software
		Padmanabhan's schema and McKenzie's absorptive basis for lattice theory (see the two preceding sections) produce an axiom with 7 variables and length 355.
		The new schema (preceding sections), a slightly different majority polynomial, and a new technique that relies on properties of the polynomial, allowing a weaker absorptive basis, produce an axiom of length 79 (also with 7 variables).
		With that basis, a single axiom was constructed using a majority reduction schema (similar to the lattice theory case above).
	www.mcs.anl.gov /AR/new_results (3321 words)

	Relapse Prevention - Literature
		Our literature and recovery books are an ever available resource that provides us insight and believable hope on our journey of dual recovery.
		Many D.R.A. members find that when read on a daily basis, the literature further reinforces their dual recovery and continues to bring new meaning to living the Twelve Steps.
		When our sponsors are unavailable, our support network can't be reached, and no meetings are currently scheduled, we can always turn to our recovery literature.
	draonline.org /relapse_reading.html (254 words)

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