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Topic: Spectral decomposition

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	Spectral theorem - Wikipedia, the free encyclopedia
		In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).
		In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find.
		There is also a spectral theorem for normal operators on Hilbert spaces, though, in which the sum in the finite-dimensional spectral theorem is replaced by an integral of the coordinate function over the spectrum against a projection-valued measure.
	en.wikipedia.org /wiki/Spectral_theorem (1124 words)

	Spectral theorem -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that is, represented as a (A square matrix with all off-diagonal elements equal to zero) diagonal matrix in some basis).
		In general, the spectral theorem identifies a class of (An operator that obeys the distributive law: A(f+g) = Af + Ag (where f and g are function)) linear operators that can be modelled by (additional info and facts about multiplication operator) multiplication operators, which are as simple as one can hope to find.
		Examples of operators to which the spectral theorem applies are (additional info and facts about self-adjoint operator) self-adjoint operators or more generally (additional info and facts about normal operator) normal operators on (A metric space that is linear and complete and (usually) infinite-dimensional) Hilbert spaces.
	www.absoluteastronomy.com /encyclopedia/s/sp/spectral_theorem.htm (1379 words)

	Porges and Bohrer (1991) (Site not responding. Last check: 2007-10-09)
		Spectral analysis provides a method to extract information about periodicities, even when they are embedded in a complex pattern and may represent only a small proportion of the total variance of the time series.
		Cross- spectral analysis is the appropriate frequency domain method of assessing the covariation of respiration and heart period at the breathing frequencies.
		For spectral analysis to appropriately function it is necessary that the data are stationary and that the data are not aliased.
	www.wam.umd.edu /~sporges/bohrer/bohrer4.htm (4614 words)

	SpectralDecomposition
		Spectral Decomposition increases the resolution needed to detect a particular stratigraphic unit.
		Spectral decomposition allows the summing and averaging of amplitudes over a selected frequency range.
		Spectral decomposition will only sum the amplitudes within that frequency band and suppress those amplitudes outside this band thus enhancing your ability to see the objective.
	www.homestead.com /3dseis/SpectralDecomposition.html (172 words)

	Geophysical Corner 05:2002 EXPLORER (Site not responding. Last check: 2007-10-09)
		Although spectral decomposition is a relatively new technique, some companies are experiencing great success in many basins around the world.
		The concept behind spectral decomposition is that a reflection from a thin bed has a characteristic expression in the frequency domain that is indicative of temporal bed thickness.
		Spectral decomposition provides this by generating a series of maps or cubes that observe the response of the reservoir to different frequencies.
	www.aapg.org /explorer/geophysical_corner/2002/05gpc.cfm (871 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		We prove the spectral decomposition theorem for eigenfunctions of a linear operator associated with the Ishimori--II equations.
		Nevertheless, spectral decomposition theorems for time-dependent Schr\"{o}dinger and Dirac operators with lump potentials were proved in \cite{PS1,PS2} by methods of complex analysis based on the Riemann--Hilbert and $\bar{\partial}$ problems.
		As a result, a proof of the spectral decomposition theorem is particularly difficult for the linear operators of the IS-II equations.
	www.ma.utexas.edu /mp_arc/papers/03-5 (3981 words)

	Spectral theorem
		In mathematics, the spectral theorem is an important theorem in linear algebra and functional analysis.
		Since the class of normal operators includes a number of special kinds of operators such as self-adjoint operators or unitary operators or operators represented by special kinds of matrices such as symmetric matrices, the spectral theorem can be applied in a wide range of situations.
		In broad terms, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find.
	www.brainyencyclopedia.com /encyclopedia/s/sp/spectral_theorem.html (982 words)

	Schur decomposition - Wikipedia, the free encyclopedia
		In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation (named after Issai Schur) is an important matrix decomposition.
		If A is a normal matrix, then U is even a diagonal matrix and the column vectors of Q are the eigenvectors of A and the Schur decomposition is called the spectral decomposition.
		Furthermore, if A is positive definite, the Schur decomposition of A is the same as the singular value decomposition of the matrix.
	en.wikipedia.org /wiki/Schur_decomposition (338 words)

	Blind Spectral Decomposition of Single-Cell Fluorescence by Parallel Factor Analysis -- Shirakawa and Miyazaki 86 (3): ...
		Linear spectral decomposition, or unmixing, is a straightforward
		Spectral data to be decomposed had a 3-nm bin size, but only every other point in the emission spectra obtained as the solution was plotted in B for clarity.
		Intensities in the spectral data from oocytes in C–E were averaged for 460 ± 5 nm emission at 340 nm excitation (blue, on the left axis) and for 530 ± 5 nm emission at 450 nm excitation (green, on the right axis).
	www.biophysj.org /cgi/content/full/86/3/1739 (5935 words)

	Biophysical Journal: Blind Spectral Decomposition of Single-Cell Fluorescence by Parallel Factor Analysis (Site not responding. Last check: 2007-10-09)
		It is based on the recording of two-dimensional single-cell fluorescence spectra and on the blind spectral decomposition of fluorescence data by method of parallel factor analysis.
		Linear spectral decomposition, or unmixing, is a straight-forward approach to determine the contributions of each fluorophore in the overall fluorescence data, considering the mixed spectrum as the weighted sum of each component, and is practicable when every component spectrum can be measured separately (Dickinson et al., 2001; Zimmermann et al., 2002).
		PARAFAC is one of blind decomposition methods for multiway data, which is originating from psychometrics (Carroll and Chang, 1970) and is recently gaining more interest in the field of chemometrics (Geladi, 1989; Leurgans and Ross, 1992; Wu et al., 2001) for several reasons.
	www.24hourscholar.com /p/articles/mi_qa3938/is_200403/ai_n9376616 (1356 words)

	Geophysical Corner 01:2003 EXPLORER (Site not responding. Last check: 2007-10-09)
		Restrict the usefulness of spectral decomposition to the inspection of single-frequency maps to try to relate amplitude maxima to geologic events.
		A new type of spectral decomposition has been shown to be useful as a simple tool to isolate the reflectivity of hydrocarbons in a Frio sand reservoir using migrated data.
		The ISA method of spectral decomposition does not mix the reflections in time, thus allowing the investigation of reflectivity from individual seismic events.
	www.aapg.org /explorer/geophysical_corner/2003/01gpc.cfm (1327 words)

	The SDC algorithm with Newton iteration
		is the spectral projector onto the invariant subspace corresponding to the eigenvalues of
		is the spectral projector corresponding to the eigenvalues of
		Dealing with ill-conditioned matrices and instability in the Newton iteration for computing the matrix sign function and the subsequent spectral decomposition is discussed in [4][6][11] and the references therein.
	www.netlib.org /utk/papers/sign/node3.html (389 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations (Site not responding. Last check: 2007-10-09)
		The spectral decompostion of the Frobenius-Perron operator of maps composed of many tents is determined from symmetry considerations.
		The construction uses the knowledge of the spectral decomposition of the r-adic map, which involves Bernoulli polynomials and their duals.
		It will be seen that the spectral decomposition of the tent maps involves both Bernoulli polynomials and Euler polynomials along with the appropriate dual states.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=468996 (230 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		The authors consider the problem of computing the eigenvalues and eigenvectors of a matrix $\hat H=H+D$ which is obtained from an indefinite Hermitian low rank modification D of a Hermitian matrix $\hat H$ with known spectral decomposition.
		The results are applied to the singular value decomposition of arbitrary modified matrices and to the spectral decomposition of modified unitary and of Hermitian Toeplitz matrices.
		For both the singular value decomposition and the unitary eigenvalue problem, divide and conquer algorithms based on rank one modifications are also presented.
	www.math.niu.edu /~rusin/known-math/01_incoming/SVD_update (629 words)

	Spectral Decomposition for Seismic Stratigraphic Patterns, by Kenny Laughlin, Paul Garossino, and Greg Partyka, #40096 ...
		Spectral Decomposition for Seismic Stratigraphic Patterns, by Kenny Laughlin, Paul Garossino, and Greg Partyka, #40096 (2003)
		Figure 1– Spectral decomposition images combined to highlight channel edges and thins as well as overbank heterogeneity.
		In thin reservoirs with varying thickness (left) seismic data with higher dominant frequency would highlight the thinner parts of the reservoir on amplitude maps (middle), while seismic with a lower dominant frequency would highlight the thicker parts on an amplitude map (right).
	www.searchanddiscovery.com /documents/geophysical/2003/laughlin/index.htm (957 words)

	College Mathematiccs Journal
		The spectral basis for the commutative algebra generated by a linear operator t on a finite dimensional space consists of orthogonal idempotents and related nilpotent elements that can be determined quickly from the minimal polynomial of the operator.
		The generalized spectral decomposition expresses t in terms of this spectral basis--it is a more detailed form of the so-called Jordan decomposition of t into a sum of commuting semisimple and nilpotent operators.
		Numerical examples illustrating the theory are provided, and it is shown how the generalized spectral decomposition makes accessible several relatively advanced topics such as the Jordan form, the polar factorization, and the classical spectral decomposition for self-adjoint operators.
	www.maa.org /pubs/cmj_toc.html (914 words)

	The Spectral Decomposition of Nonsymmetric Matrices on Distributed Memory Parallel Computers
		The algorithms perform spectral divide and conquer, i.e.
		Both algorithms are simply constructed from a small set of highly parallelizable matrix building blocks, including matrix multiplication, QR decomposition and matrix inversion.
		To take advantage of the geometric nature of the spectral decomposition algorithm, we have also designed a graphical user interface to let the user choose which eigenvalues to compute.
	www.netlib.org /utk/papers/sign/sign.html (291 words)

	Noise reduction by using spectral decomposition and non-linear transformation - Patent 5133013
		A noise reduction system for enhancing noisy speech signals by performing a spectral decomposition on the signal, passing each spectral component through a non-linear stage which progressively attenuates lower intensity spectral components (uncorrelated noise) but passes higher intensity spectral components (correlated speech) relatively unattenuated, and reconstituting the signal.
		The relationship between the magnitudes of all spectral components having a magnitude larger than X dB is therefore undisturbed by the processing stage, since all such components are amplified or attenuated by an equal factor.
		The input is resolved into its spectral components, so that for the triangular signal the signal power is concentrated in spectral components except at odd multiples of the fundamental frequency of the signal.
	www.freepatentsonline.com /5133013.html (4416 words)

	IRAF/STECF 2-D spectrum point source / background decomposition package
		When the background is simple, such as a flat or sloping background, a linear interpolation of the values from both sides of the source is sufficient, but in the case of complex backgrounds, such as galactic nuclei with strong gradients, higher order fitting is required.
		The general problem is one of spectral decomposition - of separating the spectra of the point sources from the background.
		The spectral range is 8275 to 8850A to include the Ca II triplet absorption lines.
	ecf.hq.eso.org /~jwalsh/specres/specres.html (1732 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Spectral decomposition of Operators on Banch Spaces./Ali Akbar Jafarian.
		Spectral decomposition of operators on Banach spaces is investigated.
		A new characterization of decomposable operators is used to show that if direct sun of a finite number of operators is a decomposable operator, then each of summand is decomposable.
	dbase.irandoc.ac.ir /00223/00223731.htm (120 words)

	Thermo Electron Corporation - - Algorithms - Partial Least Squares (Site not responding. Last check: 2007-10-09)
		Instead of first decomposing the spectral matrix into a set of eigenvectors and scores, and regressing them against the concentrations as a separate step, PLS actually uses the concentration information during the decomposition process.
		In other words, the results of the spectral decomposition for both of these techniques give one set of scores and one set of eigenvectors for calibration.
		The variable descriptions are as above, except that Au is the 1 byp vector of the spectral responses of the sample being predicted, and Cu is the 1 by m vector of the predicted constituent concentrations.
	nicolet.com /com/cda/resources/resources_detail/1,2166,13429,00.html (1606 words)

	Emerging Technologies Dinner - Thursday, 6 November, 2003 - 1-Nov-03: Bulletin On-Line Technology article: contact ...
		Various wavelet transform techniques have resulted in spectral decomposition with improved vertical resolution and more accurate frequency spectra.
		Spectral decomposition has been used successfully in both direct hydrocarbon indication and stratigraphic interpretation.
		The next major hurdle to overcome in the interpretation of spectrally decomposed data is separating fluid and stratigraphic effects.
	www.hgs.org /en/articles/printview.asp?89 (333 words)

	Spectral Decomposition Approaches to Analyzing Text Data (Site not responding. Last check: 2007-10-09)
		We investigate several approached to analyzing text data based on the spectral decomposition of matrices created from the documents.
		Using this vector representation, traditional spectral decomposition approaches such as principal components analysis are used to visualize the data.
		In addition, graphs are constructed from the data and the spectral decomposition of the Laplacian of the graph is used to visualize the data.
	www.galaxy.gmu.edu /stats/colloquia/AbstractsFall2004/CollNov12.html (98 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Spectral decomposition technique for time delay systems with application to optimal control/Hamid Shorooghi ; Supervised by: W.M. Wonham.
		Computational aspects of spectral decomposition techniques for linear time-invariant delay systems, particularly for computation of eigenvalnes, eigen functions, and scaling are studied to obtain a finite dimensional approximation.
		A solution for simulation of closed-loop system, using optimal control solution, is obtained from spectral decomposition.
	dbase.irandoc.ac.ir /00229/00229886.htm (120 words)

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