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Topic: Cayley transform

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	PlanetMath: Cayley's parameterization of orthogonal matrices
		In conclusion, it might be worth pointing out that the Cayley transform generalizes to the case of infinite dimensions, if one replaces matrices with operators on a Hilbert space.
		For instance, it is often easier to obtain the spectral decomposition of a Hermitean operator or study symmetric extensions of a symmetric operator by first performing a Cayley transform and dealing with the resulting bounded operator.
		This is version 16 of Cayley's parameterization of orthogonal matrices, born on 2004-11-27, modified 2006-10-05.
	planetmath.org /encyclopedia/CayleysParameterizationOfOrthogonalMatrices.html (572 words)

	Cayley Differential Unitary Space-Time Codes (Site not responding. Last check: 2007-10-17)
		We propose a class of Cayley codes that works with any number of antennas, and has efficient encoding and decoding at any rate.
		The codes are named for their use of the Cayley transform, which maps the highly nonlinear Stiefel manifold of unitary matrices to the linear space of skew-Hermitian matrices.
		This transformation leads to a simple linear constellation structure in the Cayley transform domain and to an information-theoretic design criterion based on emulating a Cauchy random matrix.
	mars.bell-labs.com /cm/ms/what/mars/papers/cayley (245 words)

	Distance Tutorial
		Cayley distance measure disorder of ordinal variables by counting the minimum number of transposition of any pair digits on disorder-vectors that at least one digit does not match to the pattern-vector.
		Cayley distance is rank invariant that the distance will not change after renumbering of the rank.
		Since we count three number of interchange, thus the Cayley distance between A and B is 3.
	people.revoledu.com /kardi/tutorial/Similarity/CayleyDistance.html (412 words)

	Earlier Works
		Family of Cayley transforms of a homogeneous Siegel domain parametrized by admissible linear forms, Diff.
		Spherical Fourier transforms of the Berezin kernels on symmetric Siegel domains,Unpublished manuscript (with H. Ishi).
		Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains --- commutativity, symmetry of the domain and a Cayley transform ---, talk at the "International Conference on Lie Groups, Geometric Structures and Differential Equations --- One Hundred Years after Sophus Lie ---" at RIMS on Dec. 16, 1999, DVI file.
	www.math.kyushu-u.ac.jp /~tnomura/earlwks.html (960 words)

	atlas: atlas::cartanclass::helper::Helper Class Reference (Site not responding. Last check: 2007-10-17)
		Explanation: to each root datum involution, we may associate a transformation from the fundamental cartan to the current cartan, that factors as the composition of a cross action and a composite Cayley transform.
		We find a cross-action followed by a composite Cayley transform, taking the fundamental cartan to the new one.
		Precondition: gr is a grading of the roots in rl; so is a set of strongly orthogonal roots, orthogonal to the roots in rl.
	www-math.mit.edu /~dav/html/classatlas_1_1cartanclass_1_1helper_1_1_helper.html (1129 words)

	[No title]
		Feb24: We elaborate on Remarks 5 in [Sk, Cayley transform].
		The needed convolution estimates for Lemmas 3.2.1 and 3.2.2 are given in the note Convolution.
		We showed that the Fourier transform is one-one on L^1 and consistently defined on L^1\cap L^2, cf.
	home.imf.au.dk /skibsted/specSpring03.html (719 words)

	Bass Resonance
		John Cayley is a London-based poet, translator, publisher and bookdealer.
		Cayley was the winner of the Electronic Literature Organization's Award for Poetry 2001 (www.eliterature.org).
		He is an Honorary Research Associate in the Department of English, Royal Holloway College, University of London, and has taught and direct research at the University of California San Diego and Brown University, amongst other institutions.
	www.electronicbookreview.com /thread/electropoetics/dynamic (2106 words)

	atlas: /Users/dav/math/atlas2/sources/gkmod/kgb.h Source File
		Instead, the 00057 class retains the cross action of (each simple reflection in) W on 00058 orbits, and the Cayley transform.
		Each of these is stored as a vector 00059 (indexed by orbit numbers) of lists of KGBElt's, one for each simple 00060 reflection (often the KGBElt UndefKGB ~0 in the case of the Cayley 00061 transform) 00062 00063 The actual construction of the orbit is carried out by the derived 00064 class Helper.
		What 00067 is retained by the class kgb is only the Cayley transforms (always 00068 going from more compact to less compact involutions) and cross 00069 actions.
	www-math.mit.edu /~dav/html/kgb_8h-source.html (1139 words)

	A/Prof N J Wildberger Personal Pages
		The moment map of symplectic geometry can be applied to the projective space of a finite dimensional unitary representation of a Lie group to yield a map into the dual of the Lie algebra which intertwines with the coadjoint action.
		Construction of a geometric Fourier transform for a compact Lie group, involving a particular function on the integral part of the cotangent bundle of the group which is a kernel for the Fourier transform taking functions on the group to functions on the integral coadjoint orbits.
		In the case of SU(2), this kernel is closely related to the Cayley transform, as well as interesting formulas from classical spherical trigonometry.
	web.maths.unsw.edu.au /~norman/research.htm (2260 words)

	Algebraic Differential Cayley Space-Time Codes (Site not responding. Last check: 2007-10-17)
		Cayley Space-Time Codes have been introduced by Hassibi and Hochwald for coding over the noncoherent MIMO channel using differential modulation.
		Differential modulation is a technique for coding when the channel is unknown, and it requires the design of unitary matrices that are fully diverse.
		Cayley codes are based on the Cayley transform, which maps a hermitian matrix on a unitary matrix.
	www.systems.caltech.edu /~frederique/Cayley.html (95 words)

	[No title] (Site not responding. Last check: 2007-10-17)
		The authors choose the so-called generalized Cayley transform: C(A;\alpha_1,\alpha_2) := (A - \alpha_1 I)^{-1} (A - \alpha_2 I), \alpha_1 < \alpha_2, \alpha_1 not an eigenvalue of A. This is nothing other than the well-known Mobius transform for matrices.
		Obviously the eigenvalues of A are transformed by the corresponding Mobius transformation.
		The authors describe extensively the mapping properties of the Mobius transformation and the corresponding relation to the convergence behavior of the iterative solvers.
	www.math.niu.edu /~rusin/known-math/00_incoming/mobius (381 words)

	[No title] (Site not responding. Last check: 2007-10-17)
		Coding: Each project should include a brief (but complete) writeup (say 2-5 pages) The write up should state the purpose of the project, describe the experiment, derive any equations that are needed, and discuss the results.
		Show the mapping induced by the Cayley transform visually in the complex plane with respect to the eigenvalues.
		(Cayley is (A - mu I)^{-1} (A + mu I) or you can do Mobius (A - sigma I)^{-1} (A + mu I)) This project should include several experiments illustrating the use of the Cayley transformation and what effect good and bad choices of the parameter has on convergence.
	www.caam.rice.edu /~caam551/projects_04.txt (609 words)

	AMCA: The Cayley transform and uniformly bounded representations by Michael Cowling (Site not responding. Last check: 2007-10-17)
		AMCA: The Cayley transform and uniformly bounded representations by Michael Cowling
		The aim of this talk is to review this, and to extend the results to Cayley transformations between the Iwasawa N and K/M for other simple Lie groups of real rank one.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/m/o/65.htm (134 words)

	Geometric Integration and Computation on Manifolds - Abstracts
		Methods similar to the RKMK method are developed based on the different coordinatizations of the Lie group manifold, given by the Cayley transform, diagonal Pad\'e approximants of the exponential map, canonical coordinates of the second kind, etc.
		Time-reversible integrators are typically constructed with the aid of a splitting of the vector field into integrable parts, but for multiple time-scale integration, some modifications of the standard splitting framework are needed.
		Examples of approaches are: Replacing the exponential by the Cayley transform for certain types of Lie groups, approximation by nilpotent Lie groups, and splitting methods for computing the exponential.
	www.ii.uib.no /FoCM99GIW/abstracts.html (2324 words)

	Bibliography (Site not responding. Last check: 2007-10-17)
		The morphological approach to segmentation: The watershed transformation.
		A transformation for extracting new descriptors of shape.
		A representation for shape based on peaks and ridges in the difference of low-pass transform.
	www.ensc.sfu.ca /people/grad/brassard/personal/THESIS/node198.html (2432 words)

	Journal of Lie Theory, Vol. 11, No. 1, pp. 185-206, 2001 (Site not responding. Last check: 2007-10-17)
		Abstract: In this paper we introduce a Cayley transform $\cal{C}$ of a homogeneous Siegel domain $D$ as a slight modification of Penney's one.
		When $D$ is quasisymmetric, our Cayley transform $\cal{C}$ is shown to be naturally coincident with Dorfmeister's one.
		A phenomenon which does not appear in the case of quasisymmetric domains is presented by an example in the last section.
	www.emis.de /journals/JLT/vol.11_no.1/11.html (111 words)

	Jacobi-Davidson Method with Cayley Transform
		of the correction equation is obtained by the action of the Cayley transform
		Both pole and zero of the Cayley transform lie close to the desired eigenvalue.
		So, the pole of the Cayley transform is the Rayleigh-Ritz quotient of
	www.cs.utk.edu /~dongarra/etemplates/node405.html (124 words)

	Algebra Seminar: The Cayley Transform (Site not responding. Last check: 2007-10-17)
		Abstract: The Cayley transform maps a unitary matrix to an Hermitian matrix.
		Using this transform, I will illustrate how we can learn more about the structure of unitary matrices based on what we know about Hermitian matrices.
		Using the Cayley transform, I will also introduce an algorithm that computes the eigenvalues of a unitary matrix based on an algorithm that computes the eigenavlues of an Hermitian matrix.
	www.sci.wsu.edu /math/Calendar/2006-10-230.xml (90 words)

	DC MetaData for: The Generalized Cayley Map from an Algebraic Group to its Lie Algebra (Site not responding. Last check: 2007-10-17)
		DC MetaData for: The Generalized Cayley Map from an Algebraic Group to its Lie Algebra
		The Generalized Cayley Map from an Algebraic Group to its Lie Algebra
		The paper is published: in "The orbit method in geometry and physics: In honor of A. Kirillov" Eds.
	www.esi.ac.at /Preprint-shadows/esi1066.html (272 words)

	Arnoldi Method with Inexact Cayley Transform
		In this paragraph, we discuss the case of the inexact Cayley transform within a Galerkin projection framework.
		We restrict ourselves to the case where the linear system (11.2) is solved by a preconditioner
		The separation of the eigenvalues of the exact transformation improves when
	www.cs.utk.edu /~dongarra/etemplates/node401.html (141 words)

	Math 132 Applet 3
		A Möbius transformation (also called a fractional linear transformation, projective linear transformation, or a bilinear transformation by some authors) is any map of the form
		The purpose of the rest of the buttons on the applet is to change this equation to a different Möbius transform.
		grid is the "pole" (or "singularity") of the Möbius transformation.
	www.math.ucla.edu /~tao/java/Mobius.html (892 words)

	AMCA: A Numerical Method Based on Cayley Transform for Parapolic Transmission Problem by Nataliya Rossokhata (Site not responding. Last check: 2007-10-17)
		Using the Cayley transform, we convert the original parabolic transmission problem into a recursive sequence of steady-state transmission problems with space operator depending on some positive constant and space operator of the original parabolic transmission problem.
		The numerical examples are shown to demonstrate the accuracy of the method for one and two interface points.
		From the numerical experiments, we can conclude that the method based on Cayley transform is successful for efficient numerical solving of parabolic transmission problem.
	at.yorku.ca /c/a/e/n/22.htm (320 words)

	proweb
		In order to derive these ellipses, a Cayley transform of a skew symmetric matrix
		Further, using a polar decomposition of the Cayley transform as
		This is achieved by using a modified Cayley transform according to
	www.cs.umu.se /~viklands/procrustes (467 words)

	UC Berkeley Mathematics
		Matrix Computations and Scientific Computing Seminar, Cayley Transforms among the Unitary, Orthogonal, and Skew Matrices, Prof.
		The Gromov-Witten invariants of a blowup of CP at points have a symmetry which arises from the Cremona Transformation.
		We discuss how the interpretation of the PML as a complex-valued coordinate transformation leads to a particularly simple finite element implementation, and describe how to choose good values for the PML parameters.
	math.berkeley.edu /index.php?module=calendar&calendar[view]=day&month=09&year=2005&day=14 (968 words)

	The Wavelet Digest, Volume 7, Issue 7 (July 21, 1998)
		The point at which the test statistic, using the maximal overlap discrete wavelet transform, achieves its maximum value can be used to estimate the time of the unknown variance change.
		Necessary and sufficient conditions are given for convergence at given rates in terms of behavior of Fourier transforms of the wavelet or scaling function near the origin.
		In the whiplash study, it was demonstrated that cervical muscle activity exists shortly after the initial motion of the sled using two different techniques: continuous wavelet transform and multiresolution analysis.
	cm.bell-labs.com /cm/ms/what/wavelet/digest_07/digest_07.07.html (2783 words)

	Stable matrices, the Cayley transform, and convergent matrices
		Stable matrices, the Cayley transform, and convergent matrices
		In passing, the concept of Cayley transform is generalized, and the generalized version is shown closely related to the equation
		As consequences are derived several characterizations of stability (closely related to Lyapunov's result) which involve Cayley transforms.
	www.hindawi.com /GetArticle.aspx?doi=10.1155/S0161171291000078 (152 words)

	AMS Journals :: Print and Electronic
		A Paley-Wiener theorem for the spherical Laplace transform on causal symmetric spaces of rank 1.
		Weighted Laplace transforms and Bessel functions on Hermitian symmetric spaces.
		Characterization of the range of the Radon transform on homogeneous trees.
	e-math.ams.org /joursearch/servlet/DoSearch?f1=msc&v1=43A85 (220 words)

	Beckman Institute: HCII Recent Publications
		Do, M. Vetterli, M., The contourlet transform: An efficient directional multiresolution image representation.
		Zhou, J. Do, M. Kovacevic, J., Multidimensional orthogonal filter bank characterization and design using the Cayley transform.
		Zhou, J. Do, M. Kovacevic, J., Special paraunitary matrices, Cayley transform, and multidimensional orthogonal filter banks.
	www.beckman.uiuc.edu /faculty/pubshcii.html (2776 words)

	Streaming Video - Fall 2001
		Simon Gindikin: The Radon Transform and its Variations - Part II Simon Gindikin: The Radon Transform and its Variations - Part III
		Joachim Hilgert: The dual horospherical Radon transform for polynomials'
		Takaaki Nomura: Geometric connection of the Poisson kernel with a Cayley transform for homogeneous Siegel domains
	www.msri.org /publications/video/index03.html (1277 words)

	The SDC algorithm with inverse free iteration
		Next: Spectral Transformation Techniques Up: Parallel Spectral Divide Previous: The SDC algorithm
		This is analogous to the sign function iteration where computing
		Further explanation of how the algorithm works can be found in [7].
	www.netlib.org /utk/papers/sign/node4.html (342 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-17)
		Between the self-adjoint and the unitary operators on a Hilbert space there is a one-to-one relation, defined by the Cayley transformation (cf.
		Of importance (especially in the theory of linear differential operators) is the class of symmetric operators (cf.
		M.H. Stone, "Linear transformations in Hilbert space and their applications to analysis", Amer.
	eom.springer.de /H/h047380.htm (2478 words)

	Prof. W. Kahan's web pages
		Is there a small skew Cayley transform with zero diagonal ?
		Is there a Small Skew Cayley Transform with Zero Diagonal?
		How Blabber-Mouth U-boats Got Sunk in World War II(PDF file)
	www.cs.berkeley.edu /~wkahan (665 words)

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