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Topic: Canonical decomposition

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	Canonical equivalence - Wikipedia, the free encyclopedia
		Canonical equivalence refers to the state in which two things are in every significant respect identical.
		In Unicode, the canonical equivalence of two strings means that they can be reduced to identical strings by recursive application of canonical decomposition routines.
		(Compare compatibility equivalence.) Canonically equivalent strings always have the same meaning and behaviour; they should also look the same and be indistinguishable to the user.
	en.wikipedia.org /wiki/Canonical_equivalence (91 words)

	UAX #15: Unicode Normalization
		Canonical equivalence is a basic equivalency between characters or sequences of characters.
		Canonical decomposition is the process of taking a string, recursively replacing composite characters using the Unicode canonical decomposition mappings (including the algorithmic Hangul canonical decomposition mappings, see Annex 10: Hangul), and putting the result in canonical order.
		Compatibility decomposition is the process of taking a string, replacing composite characters using both the Unicode canonical decomposition mappings and the Unicode compatibility decomposition mappings, and putting the result in canonical order.
	www.unicode.org /reports/tr15/tr15-21.html (6327 words)

	Unicode Architecture: Not Just a Pile of Code Charts
		There are two broad categories of decomposing characters: those with canonical decompositions (these characters are often referred to as "precomposed characters" or "canonical composites") and those with compatibility decompositions (the term "compatibility characters" is frequently used to refer specifically to these characters; a more specific term, "compatibility composite," is better).
		A canonical composite can be replaced with its canonical decomposition with no loss of data: the two representations are strictly equivalent, and the canonical decomposition is the character's preferred representation.
		The Unicode standard enshrines this principle in law: On systems that support both the canonical composites and the combining characters that are included in their decompositions, the two different representations of the same character (composed and decomposed) are required to be treated as identical.
	www.informit.com /articles/article.asp?p=30893&seqNum=4&rl=1 (1193 words)

	ipedia.com: Spectral theorem Article (Site not responding. Last check: 2007-10-08)
		The spectral decomposition of an operator A which has an orthonormal basis of eigenvectors, is obtained by grouping together all vectors corresponding to the same eigenvalue.
		Jordan decomposition, an "algebraic" analogue to spectral decomposition.
		Singular value decomposition, a generalisation of spectral theorem to arbitrary matrices.
	www.ipedia.com /spectral_theorem.html (990 words)

	Magnus
		Similarly, the number of cyclic groups in the canonical decomposition, the primes occurring in the primary decomposition and the orders of the cyclic factors are also invariants.
		These decompositions can also be chosen so that they dovetail with a decompsition of the containing group, which then allows for answering such questions as to whether a subgroup is isolated, is a direct factor and so on.
		The canonical decomposition of a subgroup can be obtained by first using the presentation of its overgroup to obtain a presentation of the subgroup.
	zebra.sci.ccny.cuny.edu /web/caiss/MAGNUS/website/pages/checkinabeliangroup.html (1968 words)

	Unicode case mapping
		Conversely, the presence of a formatting tag also indicates that the decomposition is a compatibility decomposition and not a canonical decomposition.
		For a canonical decomposition, this indicates that the character is a canonical equivalent of another single character.
		For a compatibility decomposition, this indicates that the character is a compatibility equivalent of another single character.
	lists.w3.org /Archives/Public/w3c-sgml-wg/msg02124.html (1288 words)

	Unicode Composition
		A combining character C can be canonically combined with a base character B if there is a character X such that the sequence of characters [B, C] is a canonical equivalent, according to the rules in The Unicode Standard, Chapter 3 and the decomposition mappings in UCD2.1.3.
		In such a case, X is said to be the canonical composition of B and C. A sequence of Unicode characters is canonically composed if it is bit-for-bit identical with its canonical composition, as described in the specification below.
		Generate the canonical decomposition for the source string S, according to the rules in The Unicode Standard, Chapter 3 and the decomposition mappings in the latest supported version of the Unicode Character Database.
	www.unicode.org /reports/tr15/pdtr15.html (1357 words)

	S-PLUS help
		Canonical correlation seeks a linear combination of one set of variables and a linear combination of a second set of variables such that the correlation is maximized.
		The second and higher canonical correlations find linear combinations that maximize the correlation subject to being uncorrelated with previous canonical variables.
		The number of canonical correlations is the minimum of the number of variables in the two sets.
	www.uni-muenster.de /ZIV/Mitarbeiter/BennoSueselbeck/s-html/helpfiles/cancor.html (284 words)

	Collator (Java 2 Platform SE 5.0)
		This is the default setting and provides the fastest collation but will only produce correct results for languages that do not use accents.
		This causes not only accented characters to be collated, but also characters that have special formats to be collated with their norminal form.
		Adjusting decomposition mode allows the user to select between faster and more complete collation behavior.
	java.sun.com /j2se/1.5.0/docs/api/java/text/Collator.html (1180 words)

	Normalizer (icu4j)
		Multiple strings may be canonically equivalent (their NFDs are identical) and may all conform to FCD without being identical themselves.
		The form is defined such that the "raw decomposition", the recursive canonical decomposition of each character, results in a string that is canonically ordered.
		Canonical equivalence between two strings is defined as their normalized forms (NFD or NFC) being identical.
	icu.sourceforge.net /apiref/icu4j/com/ibm/icu/text/Normalizer.html (4472 words)

	PlanetMath: matrix factorization (Site not responding. Last check: 2007-10-08)
		Note that the process of producing a factorization/decomposition is also called ``factorization'' or ``decomposition''.
		Object id is 5699, canonical name is MatrixFactorization.
		also referred to as right and left decompositions of T, respectively, where U and V are positive symmetric matrices, and R is an orthogonal matrix.
	planetmath.org /encyclopedia/MatrixFactorization.html (206 words)

	Canonical Decomposition of Outerplanar (ResearchIndex)
		Abstract: In this article we de ne a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges.
		This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n nodes where all the nodes of the last branch are colored white.
		As a consequence, for rooted outerplanar maps of n nodes, we derive: an enumeration formula, and an asymptotic of 2 3n (log n) ; ...
	citeseer.ist.psu.edu /644015.html (495 words)

	ICU Userguide
		The decomposition of a character which results from recursively applying the canonical mappings until no characters can be further decomposed and then re-ordering non-spacing marks according to the canonical behavior rules.
		Canonical mappings do not remove formatting information, which is the opposite of what happens during a compatibility decomposition.
		The decomposition of a character which results from recursively applying both compatibility mappings and canonical mappings until no characters can be further decomposed then re-ordering non-spacing marks according to the canonical behavior rules.
	icu.sourceforge.net /userguide/glossary.html (2970 words)

	Spectral theorem
		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.
		As an immediate consequence of the spectral theorem for symmetric operators we get the spectral decomposition theorem: V is the orthogonal direct sum of the spaces V
		If A is a normal operator on a finite dimensional inner product space, A also has a spectral decomposition and the decomposition theorem holds for A.
	www.brainyencyclopedia.com /encyclopedia/s/sp/spectral_theorem.html (982 words)

	Canonical decomposition -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Canonical decomposition -- Facts, Info, and Encyclopedia article
		In (additional info and facts about Unicode) Unicode, canonical decomposition is a process of converting composite characters into canonically-ordered strings of simpler characters using Unicode canonical mappings.
		Any canonical decomposition of a character is a string that is (additional info and facts about canonically equivalent) canonically equivalent to the original character.
	www.absoluteastronomy.com /encyclopedia/c/ca/canonical_decomposition.htm (111 words)

	Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition
		Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition: SIAM Journal on Matrix Analysis and Applications Vol.
		The canonical decomposition of higher-order tensors is a key tool in multilinear algebra.
		Necessary and sufficient conditions for the uniqueness of these simultaneous matrix decompositions are derived.
	epubs.siam.org /sam-bin/dbq/article/39786 (223 words)

	Collator (Java 2 Platform SE v1.4.2)
		Additionally, differences between pre-composed accents such as "\u00C0" (A-grave) and combining accents such as "A\u0300" (A, combining-grave) will be considered significant at the tertiary level if decomposition is set to NO_DECOMPOSITION.
		With CANONICAL_DECOMPOSITION set, characters that are canonical variants according to Unicode 2.0 will be decomposed for collation.
		Decomposition mode determines how Unicode composed characters are handled.
	java.sun.com /j2se/1.4.2/docs/api/java/text/Collator.html (1167 words)

	Matrix Decompositions and Entanglement Dynamics (Site not responding. Last check: 2007-10-08)
		A matrix decomposition is an algorithm for factoring matrices, and in this context such an algorithm splits a quantum computation into a sequence of smaller, hopefully simpler subcomputations.
		This poster describes the concurrence canonical decomposition (C.C.D.), a unitary matrix decomposition developed by the authors.
		Thus, we may use the decomposition to study how entanglement changes over the course of a given quantum computation.
	www.nist.gov /sigmaxi/Posters04/bullock.html (200 words)

	NORM_FORM
		Transforms each decomposed grouping, consisting of a base character plus combining characters, to the canonical precomposed equivalent.
		Transforms each base plus combining characters to the canonical precomposed equivalent and all compatibility characters to their equivalents.
		Transforms each precomposed character to its canonical decomposed equivalent and all compatibility characters to their equivalents.
	msdn.microsoft.com /library/en-us/intl/nls_norm_form.asp (197 words)

	CPT Dictionary
		The composition forms are "shrinking" the letter representation and the decomposition forms are "expanding" it.
		The normalization of this letter is done as follows: If the input is angstrom-sign, A-ring, or A + ring: for canonical decomposition, the normalized form will be A + ring; for canonical composition, the normalized form will be A-ring.
		Jamo/Hangul are ignored, the combining characters, which are not defined in the specification of canonical decomposition, are not removed.
	www.geocities.com /cptshareware/cpt_dc13/cpt_dc.html (3880 words)

	How to Use INDSCAL, a Computer Program for Canonical Decomposition of N-way Tables and Individual Differences in ...
		In the individual differences analysis, the canonical decomposition analysis is used to identify the perceptual dimensions underlying the stimulus space.
		It should be noted that a separate canonical decomposition analysis may be performed.
		CANDECOMP is used for the more general analysis called 'Canonical decomposition of N-way tables,' while INDIFF performs the analysis of individual differences in multidimensional scaling described by Carroll and Chang (1968).
	marketing.byu.edu /htmlpages/books/pcmds/INDSCAL.html (2005 words)

	RWUNormalizer
		Canonical equivalence is a fundamental equivalence between characters and sequences of characters.
		Another example of canonical equivalence is between Korean hangul syllables and the jamo characters that compose them.
		Note that two of the normalization forms, NFD and NFKD, replace composite characters with their canonical decompositions.
	www.roguewave.com /support/docs/leif/sourcepro/html/i18nref/rwunormalizer.html (664 words)

	Encyclopedia: Kelvin
		However, the "kelvin sign" is canonically decomposed into U+004B, thereby seen as a (preexisting) encoding mistake, and it is better to use U+004B (K) directly.
		Jump to: navigation, search Unicode is an international standard whose goal is to provide the means by which text of all forms and languages can be encoded for use by computers.
		Jump to: navigation, search Web browser shortcuts on an Apple computer A web browser is a software application that enables a user to display and interact with HTML documents hosted by web servers or held in a file system.
	www.nationmaster.com /encyclopedia/Kelvin (1454 words)

	[No title]
		There are two new features: (i) "Save Triangulation", which produces an ideal triangulation of the (hyperbolic) knot complement, and writes it to a file in SnapPea-readable format; (ii) "Canonical Cell Decomposition", which displays the canonical cell decomposition of the knot complement, as viewed from the cusp.
		The canonical cell decomposition can be saved as a PostScript file.
		The source code for computation of fixed point sets and the canonical decomposition is not yet supplied, as the two programs in question have to be compiled against non-standard versions of the SnapPea kernel.
	www.math.utk.edu /~morwen/download/README (824 words)

	rootr.net : man : Unicode::Normalize (Site not responding. Last check: 2007-10-08)
		If the second parameter (a boolean) is omitted or false, decomposes it using the Canonical Decomposition Mapping.
		"$is_non_starter_decomposition = isNonStDecomp($code- point)" Returns a boolean whether the canonical decomposition of the character of the specified codepoint is a Non- Starter Decomposition.
		This module is free software; you can redistribute it and/or modify it under the same terms as Perl itself.
	rootr.net /man/man/Unicode::Normalize (880 words)

	anova_3_by_5.html
		In the following we derive the projection matrices for the two-way decomposition of the sum of squares.
		The decomposition takes the form I=P1+P2+P3+P4 obtained from (A3+Q3)x(A5+Q5), where "x" indicates the Kronecker product, and I=A3+Q3 is the canonical decomposition in dimension of 3, and I=A5+Q5 is the canonical decomposition in dimension of 5.
		First we derive the canonical projections, A and Q=I-A, in dimensions n=3 and n=5:
	tigger.uic.edu /~viana/anova_3_by_51.html (120 words)

	Normalize.pm
		Returns a boolean whether the character of the specified codepoint is a singleton.
		Returns a boolean whether the canonical decomposition of the character of the specified codepoint is a Non-Starter Decomposition.
		Returns a boolean whether the character of the specified codepoint may be composed with the previous one in a certain composition (including Hangul Compositions, but excluding Composition Exclusions and Non-Starter Decompositions).
	www.devdaily.com /scw/perl/perl-5.8.5/ext/Unicode/Normalize/Normalize.pm.shtml (695 words)

	Class java.text.Collator
		This provides the fastest collation but will only produce correct results for languages that do not use accents.
		This is the default setting and should be used to get correct collation of accented characters.
		If the given value is not a valid decomposition mode.
	www.seismo.unr.edu /ftp/web/htdocs/students/ICHINOSE/apibook/api/java.text.Collator.html (1058 words)

	Unicode::Normalize - perldoc.perl.org
		If the character of the specified codepoint is canonically decomposable (including Hangul Syllables), returns the completely decomposed string canonically equivalent to it.
		In older Unicode versions, a small number of characters (all of which are CJK compatibility ideographs as far as they have been found) may have an erroneous decomposition mapping (see NormalizationCorrections.txt).
		Therefore this module running on an older perl with an older Unicode database may use the erroneous decomposition mapping blindly conforming to the Unicode database.
	www.marcin-nolte.de /docs/perl/Unicode/Normalize.html (987 words)

	Joel on Software - Stripping diacritical marks in Java?
		The best discussion of folding, in all its complexity (folding accents, case, canonical duplicates, Greek letterforms, Hiragana ands Katakana, etc., etc.) is at http://www.unicode.org/reports/tr30/.
		The easiest way is probably to do a canonical decomposition of each character, and then take the (first?) one of those that's in ASCII.
		Of course, U+0131 (dotless i) looks vaguely like i, but it doesn't have a decomposition (there's no "white-out" glyph).
	discuss.joelonsoftware.com /?joel.3.91624.4 (689 words)

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