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	PlanetMath: symmetric multilinear function
		For example, the permanent is a symmetric multilinear function of its rows (columns).
		The determinant is a skew-symmetric multilinear function of its rows (columns).
		This is version 8 of symmetric multilinear function, born on 2006-08-20, modified 2006-08-20.
	planetmath.org /encyclopedia/SymmetricMultilinearFunction.html (72 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		Enter a search or verse; then click on "V" next to the verse you wish to view in other versions (the results will be placed in the midst of the other verses rather than in parallel (i.e., interlinearly), but only for one verse at a time--the verse whose “V” you clicked on).
		This is a multilinear Bible, allowing you to view multiple translations, notes, further reference information, and commentary verse by verse.
		One of these columns is dedicated to references made by the Central Figures of the Bahá’í Faith, the Báb, Bahá’u’lláh, or ‘Abdu’l-Bahá, as well as to interpretations of Shoghi Effendi and statements of the Universal House of Justice or on their behalf (note: this grouping is different from the multilinear Qur’án).
	bahai-library.com /?file=bible_zamir_multilinear (2044 words)

	Chi-Kwong Li: List of Publications and Preprints (Site not responding. Last check: 2007-10-16)
		{7.} The $c$-spectral, $c$-radial and $c$-convex matrices, Linear and Multilinear Algebra 20 (1986), 5-15.
		{15.} The $C$-convex matrices, Linear and Multilinear Algebra 21 (1987), 303-312.
		Poon), Linear and Multilinear Algebra 50 (2002), 321-326.
	www.math.wm.edu /~ckli/pub.html (4219 words)

	Multilinear map - Wikipedia, the free encyclopedia
		In linear algebra, a multilinear map is a mathematical function of several vector variables that is linear in each variable.
		The determinant of a matrix is a skew-symmetric multilinear function of the columns (or rows) of a square matrix.
		The trace of a matrix is a multilinear function of the columns (or rows) of a square matrix.
	en.wikipedia.org /wiki/Multilinear_map (329 words)

	3473 Multilinear Regression
		The Multilinear Regression process used actual red and green component rasters of the San Francisco area created from the San Francisco SPOT XS raster object to produce the predicted blue component raster shown.
		Note that an inverted mask appears as fl in the output, the same as a mask that is not inverted; the color of the inverted mask shown is edited to a medium gray shade for illustration purposes.
		The red component predicted by multilinear regression replaced the actual red component in one composite (left); the predicted blue component replaced the actual blue component in another composite (middle); the predicted green component replaced the actual green component in a third composite (right).
	www.microimages.com /refman/xhtml/PROCE044.HTM (2068 words)

	Tufts University \| Computer Science Department -- Colloquia >> Colloquia
		In the context of computer vision and graphics, we deal with natural images which are the consequence of multiple factors related to scene structure, illumination, and imaging.
		I will present two multilinear models that learn (nonlinear) manifold representations of image ensembles in which the multiple constituent factors (or modes) are disentangled and analyzed explicitly.
		In graphics, our image-based rendering technique, called TensorTextures, is a multilinear generative model that, from a sparse set of example images of a surface, learns the interaction between viewpoint, illumination and geometry, which determines surface appearance, including complex details such as self-occlusion and self shadowing.
	www.cs.tufts.edu /colloquia/colloquia.php?event=278 (509 words)

	CSAIL Research Abstract
		The key obstacle in constructing a multilinear model is the vast amount of data (in full correspondence) needed to account for every possible combination of attribute settings.
		For the control problem, we show how to connect this model to cheaper sources of data such as video, so that the model can be applied to performances by actors who are not in the original database and not available for detailed measurement.
		An intriguing prospect is that, by scanning faces of different ages, we can build a multilinear model that gives us control over an actor's aparent age (identity x expression x age).
	publications.csail.mit.edu /abstracts/abstracts05/drdaniel/drdaniel.html (611 words)

	Between Linearity and Nonlinearity
		One approach to the hypertext essay that attempts to steer a course between the linear and nonlinear is a multilinear approach to writing hypertext.
		An alternative multilinear approach might afford the user more control over initial navigation through a less suggestive organizational structure, while imposing greater control over the user's options once a node is selected.
		The argument is both textual and visual, making it something of a new media multilinear research essay and a good example of Davis and Shadle's "multi-writing" (434).
	enculturation.gmu.edu /4_2/cripps/files/multilinear.html (322 words)

	Havlicek/Multilinear Algebra
		This is a series of lectures and exercises for graduate students of mathematics.
		Multilinear mappings of vector spaces, Existence and universal property of the tensor product, Commutativity and associativity of the tensor product, The tensor product in terms of coordinates, Tensor products and spaces of linear mappings, Kronecker product of linear mappings (matrices), Contraction, Lowering and raising of indices, Duality.
		Covariant, contravariant and mixed tensors, Classical definition and notation of a tensor in terms of coordinates, Structure tensor of an algebra, Mixed tensor algebra, Universal property of the tensor algebra.
	www.geometrie.tuwien.ac.at /havlicek/multilineare.html (179 words)

	MERL – Multilinear Face Models
		A multilinear face model is a simple function that can generate a wide variety of realistic face images.
		Therefore we set out to develop methods to estimate multilinear models from facial scan data, to recover multilinear facial parameters from ordinary video, and to resynthesize such videos with interesting changes.
		Technical Discussion: We estimated a multilinear model from a set of high-resolution 3D face scans taken of several people in various expressions and speech articulations.
	www.merl.com /projects/multiface (320 words)

	Publications of Russell Merris
		The irreducibility of K(A), Linear & Multilinear Algebra 3 (1975), 299-303.
		On vanishing decomposable symmetrized tensors, Linear & Multilinear Algebra 5 (1977), 79-86.
		A Hadamard inequality for the third and fourth immanants (with R. Grone), Linear & Multilinear Algebra 21 (1987), 201-209.
	www.mcs.csuhayward.edu /~merris/publ.html (1172 words)

	Multilinear Subspace Analysis of Image Ensembles (Site not responding. Last check: 2007-10-16)
		Multilinear algebra, the algebra of higher-order tensors, offers a potent mathematical framework for analyzing ensembles of images resulting from the interaction of any number of underlying factors.
		This N-mode orthogonal iteration algorithm is based on a tensor decomposition known as the N-mode SVD, the natural extension to tensors of the conventional matrix singular value decomposition (SVD).
		We demonstrate the power of multilinear subspace analysis in the context of facial image ensembles, where the relevant factors include different faces, expressions, viewpoints, and illuminations.
	csdl.computer.org /comp/proceedings/cvpr/2003/1900/02/190020093abs.htm (258 words)

	Cyber-Rhetoric
		Literature dealing with the hypertextuality, (multi)linearity, and the roles of authors and readers of web texts is reviewed.
		The individual reader is empowered with multiple options of how to proceed through the material; he or she may easily choose an idiosyncratic manner of reading it and so experience an utterly different "text" than does another reader.
		Multilinearity: Theorists have pointed out that "nonlinear" does not adequately describe reading of any kind because letters and words are parsed sequentially regardless of the length of the textual chunk.
	www.uwm.edu /Course/com813/foster9.htm (4765 words)

	On multilinear oscillatory integrals, nonsingular and singular
		Basic questions concerning nonsingular multilinear operators with oscillatory factors are posed and partially answered.
		L-p norm inequalities are established for multilinear integral operators of Calderon-Zygmund type which incorporate oscillatory factors e(iP), where P is a real-valued polynomial.
		M Christ, X C. Li, T Tao, and C Thiele, "On multilinear oscillatory integrals, nonsingular and singular" (2005).
	repositories.cdlib.org /postprints/1691 (98 words)

	The Math Forum - Math Library - Multilinear Algebra (Site not responding. Last check: 2007-10-16)
		A short article designed to provide an introduction to linear and multilinear algebra and matrix theory.
		Mgfun is a group of Maple packages intended for calculations with multivariate generating functions, in particular for their symbolic summation and integration, and for the proof of special function and combinatorial identities.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/topics/multilinear (305 words)

	Multilinear operators
		A counterexample to a multilinear endpoint question of Christ and Kiselev
		Bilinear and multilinear estimates also arise in my
		Multilinear expansions also arise in inverse scattering, but I have classified inverse scattering as a branch of
	www.math.ucla.edu /~tao/preprints/multilinear.html (261 words)

	Linear and Multilinear Algebra
		Taylor & Francis Homepage for Linear and Multilinear Algebra
		Appropriate areas include spaces over fields or rings, tensor algebras or subalgebras, nonnegative matrices, inequalities in linear algebra, combinatorial linear algebra, matrix numerical analysis, plus other areas including representation theory, Lie theory, invariant theory, and functional analysis.
		Linear and Multilinear Algebra is addressed to mathematicians in both industrial and academic worlds.
	www.combinatorics.net /journals/jhomes/LinMulAlg.html (142 words)

	Elegant Mathematics Ltd.
		Support the multilinear decomposition, so the total amount of dimensions could be as much as possible, certainly it should be greater than 2.
		The product of all dimensions can be considerably larger than the total main memory of the computer, only defined entries of initial data with their indexes should fit to the main memory of your computer.
		PARAFAC for fully populated and partially defined multilinear data is implemented for massively parallel computers.
	www.elegant-mathematics.com (924 words)

	Multi-Party Authenticated Key Agreement Protocols From Multilinear Forms - Lee, Lee, Lee (ResearchIndex)
		Recently, Boneh and Silverleg [4] studied multilinear forms, which provides a one round multi-party key agreement protocol.
		In this paper, we propose (n + 1) types of one round authenticated multi-party key agreement protocols from multilinear forms based on the application of MTI and MQV...
		Multi-Party Authenticated Key Agreement Protocols from Multilinear Forms.
	citeseer.ist.psu.edu /lee02multiparty.html (542 words)

	I P M - Bulletin Board
		In this lecture we study the multilinear morphisms between group schemes and associated constructions.
		We define in the same fashion the symmetric and alternating morphisms and the group of all such morphisms.
		Finally, we give some results that show the analogy of this theory and multilinear algebra.
	www.ipm.ac.ir /IPM/news/ViewNewsInfo.jsp?NTID=253 (154 words)

	Kitab-i-Aqdas Multilinear Translation project
		available in an online edition, includes extensive notes and commentary, and should be consulted in conjunction with this multilinear translation project.
		The schema of the multilinear translations, complete with brief notations, is given below.
		I have included all diacritics included in print - the only discrepency between the printed text is in final yá' where I have opted to differ between dotted and dotless.
	www.bahai-library.com /provisionals/aqdas (762 words)

	ECCC Report TR04-042 and related Papers (Site not responding. Last check: 2007-10-16)
		Abstract: An arithmetic circuit or formula is multilinear if the polynomial computed at each of its wires is multilinear.
		We give an explicit example for a polynomial $f(x_1,...,x_n)$, with coefficients in ${0,1}$, such that over any field: 1) $f$ can be computed by a polynomial-size multilinear circuit of depth $O(log^2 n)$.
		This gives a super-polynomial gap between multilinear circuit and formula size, and separates multilinear $NC_1$ circuits from multilinear $NC_2$ circuits.
	eccc.hpi-web.de /eccc-reports/2004/TR04-042/index.html (121 words)

	Taylor & Francis Journals: Welcome
		This journal publishes original research papers that advance the study of linear and multilinear algebra, or that apply the techniques of linear and multilinear algebra in other branches of mathematics and science.
		The Journal also publishes research problems, survey articles and book reviews of interest to researchers in linear and multilinear algebra.
		The audience for Linear and Multilinear Algebra includes both industrial and academic mathematicians.
	www.tandf.co.uk /journals/titles/03081087.asp (256 words)

	Multilinear IF
		IF offers a little extra leeway, however: people (at least in my experience) tend to have less rigorous expectations of the plot and pacing than they would if they were dealing with a written story, and if the ending makes sense based on the outcome of the last several turns, they seem relatively content.
		What I didn't anticipate was how many people would be made nervous by the multilinearity.
		Perhaps it's the puzzle-solving nature of IF-players, but much of my feedback has come in the form of the question, "These are the endings I found: have I got them all?"
	www.mindspring.com /~emshort/multi.htm (913 words)

	Multilinear IF
		IF offers a little extra leeway, however: people (at least in my experience) tend to have less rigorous expectations of the plot and pacing than they would if they were dealing with a written story, and if the ending makes sense based on the outcome of the last several turns, they seem relatively content.
		What I didn't anticipate was how many people would be made nervous by the multilinearity.
		Perhaps it's the puzzle-solving nature of IF-players, but much of my feedback has come in the form of the question, "These are the endings I found: have I got them all?"
	home.mindspring.com /~emshort/multi.htm (913 words)

	multilinear data modeling
		Robert T. Ross Department of Biochemistry, The Ohio State University, 484 West 12th Avenue, Columbus OH 43210 USA Internet: rtr+@osu.edu edition of 01 Feb 95 This document describes a family of programs written by the author to fit multilinear models to data in 2-way, 3-way, and 4- way arrays.
		CHOICES OF ARRAY WAYS: In many multilinear models, two or more ways can be interchanged and the resulting permuted model is the same type of multilinear model.
		In many circumstances, including the unmodified use of general multilinear models such as Parafac, T2, and T3, the values of the independent variables, XI, XJ, and XK, are not used in the model, and thus may be assigned dummy values.
	www.biosci.ohio-state.edu /~rtr/multilin/muldoc.html (1165 words)

	Research - M. Alex O. Vasilescu
		In addressing this problem, we take advantage of the assets of multilinear algebra, the algebra of higher-order tensors, to obtain a parsimonious representation that separates the various constituent factors.
		Our new representation of facial images, called TensorFaces, leads to improved recognition algorithms for use in the aforementioned applications.
		``Multilinear Image Analysis for Facial Recognition,'' M. Vasilescu, D. Terzopoulos, Proceedings of International Conference on Pattern Recognition (ICPR 2002), Quebec City, Canada, Aug, 2002.
	www.cs.toronto.edu /~maov/research_index.html (832 words)

	Multilinear -- from Wolfram MathWorld
		A basis, form, function, etc., in two or more variables is said to be multilinear if it is linear in each variable separately.
		SEE ALSO: Bilinear Function, Linear Operator, Multilinear Basis,
		Weisstein, Eric W. "Multilinear." From MathWorld--A Wolfram Web Resource.
	mathworld.wolfram.com /Multilinear.html (68 words)

	Separation of Multilinear Circuit and Formula Size: Theory of Computing: An Open Access Electronic Journal in ... (Site not responding. Last check: 2007-10-16)
		An arithmetic circuit or formula is multilinear if the polynomial computed at each of its wires is multilinear.
		Any multilinear formula for f is of size n
		This gives a super-polynomial gap between multilinear circuit and formula size, and separates multilinear NC
	theoryofcomputing.org /articles/main/v002/a006/index.html (95 words)

	NSDL Metadata Record -- Alternating Multilinear Form -- from MathWorld
		NSDL Metadata Record -- Alternating Multilinear Form -- from MathWorld
		An alternating multilinear form on a real vector space V is a multilinear form F:V \otimes \cdots \otimes V \rightarrow \mathbb{R} such that F(x_1, \dots, x_i, x_{i+1}, \dots, x_n) = -F(x_1, \dots, x_{i+1}, x_i, \dots, x_n) for any index i.
		An alternating multilinear form is defined on a module in a similar...
	nsdl.org /mr/698155 (107 words)

	Miscellaneous Contributions
		Editor for the special issues of Linear & Multilinear Algebra devoted to Multilinear Algebra and Combinatorics, v.
		One of three editors for the special issue of Linear & Multilinear Algebra devoted to the 1993 Edinburgh Workshop on Algebraic Graph Theory, v.
		Fonseca, A survey of the multilinearity partition, Linear Algebra Appl.
	www.mcs.csuhayward.edu /~merris/other.html (593 words)

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