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Topic: Irreducible component

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	Irreducible (mathematics) - Wikipedia, the free encyclopedia
		In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space.
		The notions of irreducibility in algebra and manifold theory are related.
		An irreducible manifold is thus prime, although the converse does not hold.
	en.wikipedia.org /wiki/Irreducible_(mathematics) (393 words)

	PlanetMath: irreducible
		So this space is reducible, and thus not irreducible.
		This is version 8 of irreducible, born on 2001-12-20, modified 2005-02-06.
		irreducible / reducible by jan on 2005-12-07 10:09:54
	planetmath.org /encyclopedia/IrreducibleClosedSet.html (105 words)

	PlanetMath: irreducible component
		An irreducible component of a topological space is a maximal irreducible subset.
		If a subset is irreducible, its closure is, so irreducible components are closed.
		This is version 3 of irreducible component, born on 2002-06-14, modified 2002-08-23.
	planetmath.org /encyclopedia/IrreducibleComponent.html (69 words)

	[No title]
		Since all the components of $\Cal P^* \cap \Cal Q$ are components of $\Cal P \cap \Cal Q$, we have that the isotopy is fixed on $\Cal P^* \cap \Cal Q$.
		Then let $\Cal A$ be the set of all those components of $\Cal P \cap \Cal Q$ which are \bparallel\ arcs in $\Cal P$ (or equivalently in $\Cal Q$.) Infinite nesting among the components of $\Cal A$ implies by Lemma 1.1 (2) that one of these surfaces has a component which is a trivial halfplane.
		The components of its intersection with $R^_i$ are horizontal surfaces with negative Euler characteristics whose complements in the boundaries of both 3-manifolds have no components* with closure a disk.
	www.math.okstate.edu /preprint/1996/temp.4b (14252 words)

	UBC Mathematics Department - Colloquium (Site not responding. Last check: 2007-11-04)
		In particular, the algorithms lay out the decomposition of the solution set into irreducible components, and give upper bounds for the multiplicities of the irreducible components, with the upper bound of an irreducible component equal to one if and only if the irreducible component is reduced.
		The basic data in a numerical primary decomposition are generic points, that certify the existence of irreducible components of the solution set their dimensions, and their degrees.
		The algorithms, which make essential use of generic projection and interpolation, can also be used to produce, for each irreducible component of the reduced solution set, a finite number of polynomials that vanish precisely on the irreducible component.
	www.math.ubc.ca /Dept/Events/colloquia/sommese.html (231 words)

	Rank 2 Spin Tensors
		An irreducible rank 2 space/spin tensor or a component of that tensor.
		This irreducible rank 2 tensor corresponds to a single spin because it is formulated from the product spin angular momentum vector (rank1 spin tensor) and a spatial vector (rank 1 spatial tensor) as given by the coordinate point.
		Equation has a normalization in which these components result from a product of two of rank 1 tensors given in equation (10-12) as shown in the discussion Section at the end of this Chapter.
	gamma.ethz.ch /html/modules/level1/spint3.htm (3519 words)

	Irreducible component - Wikipedia, the free encyclopedia
		In mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation
		In algebraic geometry, any algebraic set, in affine space or projective space, is the union of a finite number of irreducible components, which are algebraic varieties in the strict sense of being irreducible (in the affine case, this is the same as the condition that their coordinate rings are integral domains).
		As a matter of commutative algebra, the primary decomposition of an ideal gives rise to the decomposition into irreducible components; and is somewhat finer in the information it gives, since it is not limited to radical ideals.
	www.wikipedia.org /wiki/Irreducible_variety (147 words)

	Normal scheme - Wikipedia, the free encyclopedia
		This is done by first separating the scheme into its irreducible components and normalizing each of them individually.
		An irreducible and reduced scheme X has the property that every affine chart is a domain.
		If your initial scheme was not irreducible, it's normalization is just equal to the disjoint union of the normalizations of the irreducible components.
	www.wikipedia.org /wiki/Normal_scheme (177 words)

	PlanetMath: irreducible component
		irreducible analytic variety, irreducible locally analytic set, irreducible analytic variety, reducible locally analytic set, reducible analytic variety
		Cross-references: connected components, closures, complex analytic manifold, connected, regular, maximal, irreducible, analytic variety, locally analytic, open set
		This is version 2 of irreducible component, born on 2005-02-22, modified 2005-02-23.
	planetmath.org /encyclopedia/Ircomp.html (126 words)

	Rank 1 Tensors
		Given the Cartesian components, the function A1 returns a rank 1 spatial tensor or a specified irreducible spherical component of that tensor.
		There are three irreducible spherical components of a rank 1 tensor.
		For a spatial rank 1 tensor these components are explicitly given by equation (9-5).
	gamma.ethz.ch /html/modules/level1/spacet2.htm (824 words)

	1. primary decomposition
		The result is a list of three pairs of ideals (for each pair, the first ideal is the primary component, the second ideal the corresponding prime component).
		The second prime component [2] : [2] is embedded in the first [1] : [2].
		The first primary component [1] : [1] is already prime, the other two are not.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/29/paper_html/node6.html (558 words)

	Rank 2 Tensors (Site not responding. Last check: 2007-11-04)
		In contrast to this internal storage format, this function assumes that the tensor is being input in terms of its Cartesian components and will automatically convert the given values into the analogous irreducible spherical values.
		The spherical irreducible components are then determined from the equations labeled (9-7).
		The spherical irreducible components in this situation are then determined from the equations labeled (9-7) on page 169.
	gamma.magnet.fsu.edu /html/modules/level1/spacet3.htm (1047 words)

	Symmetries of Boundary Values Problems in Mathematical Physics
		An irreducible component is a linear combination of the function under consideration taken at symmetric positions.
		The irreducible components of the spatial moments are given in Table 6.
		The elements belonging to the components of a multi (i.e., two or three) dimensional representation are equal.
	www.kfki.hu /~/makai/makai.html (5778 words)

	GAMMA Online Class Documentation (Site not responding. Last check: 2007-11-04)
		component (int l, int m) - When invoked with a rank index and a component index as arguments the function component returns a complex number which is the irreducible spherical l, m component of the input tensor.
		This returns either one of the irreducible tensors of specified rank which makes up the input tensor or one of the irreducible tensor components.
		A complex number or an irreducible spatial tensor is returned.
	spin.magnet.fsu.edu /local/gamma/classes/space_t4.html (813 words)

	Nonwandering sofic systems
		Let K be the set of words x in F(S) such that there are vertices v,w in G that don't belong to the same irreducible component and vx=w.
		Let G' be a representation of S obtained by erasing every irreducible component of G that recognizes an FTR which is not maximal.
		We show that G'' is synchronizing: Let v be a vertex in G'' and C the irreducible component containing v.
	www.math.usf.edu /~jonoska/symbolic/node22.html (486 words)

	Abstract of positive multiplicity paper (Site not responding. Last check: 2007-11-04)
		For example, in the numerical irreducible decomposition of a solution set for a polynomial system, one first obtains a ``witness point set'' containing generic points on all the irreducible components and then these points are grouped via numerical exploration of the components by path tracking from these points.
		A numerical difficulty arises when a component has multiplicity greater than one, because then all points on the component are singular.
		In the case of the numerical irreducible decomposition, this embedding can be the same embedding that one uses to generate the witness point set.
	www.math.uic.edu /~jan/Articles/posmultabs.html (274 words)

	[No title]
		An {\bf orbital variety} is an irreducible component of the intersection of a nilpotent conjugacy class of $gl_n$ and the upper triangular matrices $B_n$.\\ {\bf Example} For the partition (2,1), there are two orbital varieties corresponding to the following linear subspaces of $B_n$.\\ $ \begin{matrix}1.
		But since the Borel group is irreducible as an algebraic variety, the conjugation action restricts to an action on each orbital variety.\end{proof} \section{A classification of orbital varieties} Descriptions of orbital varieties were given by Spaltenstein \cite{ns2} for $sl_n$ and later by Steinberg \cite{rs1} for any semi-simple Lie algebra.
		Then the irreducible components of $N \cap G$ are parameterized by the following way of filling the partition with $a_1$ ones, $a_2$ 2's, $\dots$ inductively.\\ Suppose that all of the numbers $< i$ have been entered in $P^$ to fill a subtableau $P_{i-1}^$.
	math.berkeley.edu /%7Erothbach/orbequ.tex (2482 words)

	Solution, Other Works and Applications (Site not responding. Last check: 2007-11-04)
		For the remaining graphical and hierarchical models the problem of finding the estimates is reduced by decomposition of the models into the irreducible components, an approach also suggested in Malvestuto:89.
		The application of the iterative proportional scaling procedure, the IPS procedure, is limited to these irreducible components, and by representing the distributions of the irreducible components in an economical way as described in Jirouvsek:91, the iterative procedure is optimized.
		In chapter 2 it is argued that by these methods it is possible to compute the deviance in hierarchical models for several hundreds of variables in each irreducible component (under some constraints).
	www.math.aau.dk /~jhb/Thesis/PartI/node20.html (235 words)

	Design Arguments for the Existence of God [Internet Encyclopedia of Philosophy]
		Accordingly, the argument from irreducible biochemical complexity is more plausibly construed as showing that the design explanation for such complexity is more probable than the evolutionary explanation.
		The problem, however, is that the claim that a complex system has some property that would be valued by an intelligent agent with the right abilities, by itself, simply does not justify inferring that the probability that such an agent exists and brought about the existence of that system is not vanishingly small.
		While the argument from irreducible biochemical complexity focuses on the probability of evolving irreducibly complex living systems or organisms from simpler living systems or organisms, the argument from biological information focuses on the problem of generating living organisms in the first place.
	www.iep.utm.edu /d/design.htm (8076 words)

	Complex Routines
		If the tensor itself is irreducible only the components with l=k will exist and the function returns zero if asked for components with l
		The irreducible spherical components of A are given by equation (9-15)
		of the function and the L, M component from usage 4, AL,M. When the product of two reducible tensors is performed, each of the input tensors must be expanded in terms of irreducible components.
	www.gamma.ethz.ch /html/modules/level1/spacet4.htm (968 words)

	Stembridge's Way of Producing d-Complete Examples (Site not responding. Last check: 2007-11-04)
		John Stembridge has observed that at least one example of an irreducible component from each of the 15 classes may be produced with the construction described below.
		Of these 19 posets, 14 are maximal irreducible components.
		Another 3 of the 19 posets are slant sums of two slant irreducible posets Q and P. Here Q is the one element poset and P is a maximal irreducible component from Classes 1, 2, and 15 respectively.
	www.math.unc.edu /Faculty/rap/StembObsv.html (415 words)

	[No title]
		The dimension of an irreducible quasiprojective variety X, noted dim X is the transcendence degree of the rational function field EMBED Equation.DSMT4 .
		If X is not irreducible the dimension of will be take as the maximum of the dimension of its irreducible components.
		An irreducible component X of a hypersurface in EMBED Equation.DSMT4 is the zero set of a polynomial P(T) (irreducible).
	igd.univ-lyon1.fr /~altinel/tournier.doc (2527 words)

	Classification of d-Complete Posets (Site not responding. Last check: 2007-11-04)
		Roughly speaking, each slant irreducible component consists of a local region of four element diamonds, together with a few more elements which are required by a certain condition.
		Let's say that a Dynkin diagram is of general type E if it is Y-shaped, has exactly one branch of length 1, and is not of type A. There are 15 classes of possible slant irreducible components.
		In all but Classes 1-3, the top tree of each slant irreducible component is of general type E. Each of the 15 classes is further divided into subclasses, and each subclass possesses a unique maximal member.
	www.math.unc.edu /Faculty/rap/Classif.html (366 words)

	SINUM Volume 38 Issue 6
		In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component.
		The bound is sharp (i.e., equal to one) for reduced components.
		The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroes of a finite number of polynomials.
	epubs.siam.org /sam-bin/dbq/article/37254 (230 words)

	The Dimension of a Space
		A component may be the projection of a product space, an open and closed (disconnected) subspace, or a maximal irreducible subspace.
		This is a contradiction, hence the irreducible components are the same in both representations.
		This is a contradiction, hence the decomposition of c is precisely the components of c.
	www.mathreference.com /top-dim,intro.html (1531 words)

	ipedia.com: Cognitivism (psychology) Article (Site not responding. Last check: 2007-11-04)
		The first is a positivist approach and the belief that psychology can be (in principle) fully explained by the use of experiment, measurement and the scientific method.
		This is also largely a reductionist goal, with the belief that individual components of mental function can be identified and meaningfully understood.
		They feel that cognitivism throws the baby out with the bathwater by using positivist methods on something which is inherently irreducible to component parts.
	www.ipedia.com /cognitivism__psychology_.html (563 words)

	[No title]
		In a joint article [AD], Albarello and De Concini suggest that if $(A,\Theta)$ is an irreducible principally polarized abelian variety, then the singular locus of the theta divisor has codimension at least 3 in $A$.
		\endproclaim The condition that $\lfloor \frac {1}{m}D \rfloor =0$ is equivalent to requiring that the multiplicity of every irreducible component of $D$ is strictly less than $m$.
		Suppose that there exists a component $F$ of $D$ of multiplicity $f>m$, and let $$D=fF+\sum d_iD_i$$ be a decomposition of $D$ into distinct irreducible components.
	www.math.utah.edu /~hacon/pluri3 (2063 words)

	Georg Wilhelm Friedrich Hegel (Site not responding. Last check: 2007-11-04)
		Having posed the question of the ground of the relation of a representation to an object, Kant had answered that where a representation was not made possible by the process of sensory affection, it could be justified as objective only if through it it became possible to cognise something as an object.
		In the family the particularity of each individual tends to be absorbed into the social unit, giving this manifestation of Sittlichkeit a one-sidedness that is the inverse of that found in market relations in which participants grasp themselves in the first instance as separate individuals who then enter into relationships that are external to them.
		These two opposite but interlocking principles of social existence provide the basic structures in terms of which the component parts of the modern state are articulated and understood.
	plato.stanford.edu /entries/hegel (7685 words)

	Rank 1 Spin Tensors (Site not responding. Last check: 2007-11-04)
		The function T1 returns either the standard rank 1 spin tensor or a specific component of that tensor.
		T1(spin_sys andsys, int spin, int l, int m) - When T1 is invoked with a spin system, a spin index, and angular momentum components the function returns a spin operator which is the l,m irreducible spherical component of the standard rank 1 spin tensor.
		All tensor components are returned as spin operators, these are mathematically defined by equation (10-12).
	gamma.magnet.fsu.edu /html/modules/level1/spint2.htm (316 words)

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