
	Where results make sense

Topic: Homogeneous ideal

	PlanetMath: homogeneous polynomial
		A homogeneous polynomial of degree 1 is called a linear form; a homogeneous polynomial of degree 2 is called a quadratic form; and a homogeneous polynomial of degree 3 is called a cubic form.
		In fact, a homogeneous function that is also a polynomial is a homogeneous polynomial.
		This is version 14 of homogeneous polynomial, born on 2004-12-14, modified 2006-02-24.
	planetmath.org /encyclopedia/HomogeneousPolynomial.html (255 words)

	PlanetMath: homogeneous ideal
		is said to be homogeneous if it can be generated by a set of homogeneous elements, or equivalently if it is the ideal generated by the set of elements
		Cross-references: curve, projective varieties, radical, interest, radical ideal, degree, field, sum, ideal generated by, generated by, ideal, graded ring
		This is version 6 of homogeneous ideal, born on 2001-10-15, modified 2004-02-16.
	planetmath.org /encyclopedia/HomogeneousIdeal.html (131 words)

	METU MATHEMATICS DEPARTMENT
		In general, the elementwise homogenisation of a basis of an ideal I is not a basis for I^h.
		A basis B of I for which the set B^h obtained by elementwise homogenisation of B is a basis for I^h is called a homogeneous basis.
		As an application we present an example of a homogeneous ideal of dimension 0 in four variables which cannot be lifted to any radical ideal.
	www.math.metu.edu.tr /seminars/yilmaz_abs.shtml (163 words)

	Homogeneous Ideal
		An ideal h is homogeneous iff every y in h is a sum of homogeneous elements that also lie in h.
		Homogeneous elements in r map to homogeneous elements in q, and each homogeneous element in q has at least one homogeneous pullback in r.
		Therefore a homogeneous ideal is prime iff xy in h implies x or y is in h, for homogeneous* x and y.
	www.mathreference.com /ring-grad,hid.html (584 words)

	Homogeneous coordinates
		There is a separate ideal point associated with each direction in the plane; for example, the points (1,0,0) and (0,1,0) are associated with the horizontal and vertical directions, respectively.
		All the ideal points lie on a line, called the ideal line, or the line at infinity, which, once again, is treated just the same as any other line.
		The concepts of homogeneous coordinates are summarized in Figure 2.
	vision.stanford.edu /~birch/projective/node4.html (515 words)

	Proj r
		The ideal spanned by an arbitrary union of homogeneous ideals is also homogeneous, hence the intersection of closed sets is closed.
		The nil radical is a radical ideal, and is homogeneous.
		Remember that a homogeneous ideal fails to be prime if ab lies in p, while a and b do not, for a and b homogeneous.
	www.mathreference.com /ring-grad,projr.html (1198 words)

	[No title]
		Remark A homogeneous form cannot be thought of as a function on the points of projective space.
		Not every element of a homogeneous ideal is homogeneous.
		J(Y) be the ideal generated by all homogeneous polynomials that vanish at all points of
	odin.mdacc.tmc.edu /~krc/agathos/proj.html (999 words)

	Graded Polynomial Rings
		Given a graded polynomial ring P (or an ideal thereof), return the variable weights of P as a sequence of n integers where n is the rank of P. If P was constructed without specific weights, the sequence containing n copies of the integer 1 is returned.
		Given a polynomial f of the graded polynomial ring P, this function returns whether f is homogeneous with respect to the weights on the variables of P (i.e., whether the weighted degrees of the monomials of f are all equal).
		Given an ideal I of the graded polynomial ring P, this function returns whether I is homogeneous with respect to the weights on the variables of P (i.e., whether I possesses a basis consisting of homogeneous polynomials alone).
	www.umich.edu /~gpcc/scs/magma/text848.htm (952 words)

	[No title]
		The ideal involved is of a very special type: it is the intersection of squares of ideals generated by families of linear forms in the exterior algebra.
		These ideals arise naturally in representation theory of quivers, and their cohomological invariants are polynomials that give universal topological data on degeneracy loci for sequences of vector bundle morphisms.
		The core of an ideal is the intersection of all reductions of the ideal.
	www.ms.uky.edu /~corso/Sevilla/abstracts.html (2083 words)

	Aristotle -- General Introduction [Internet Encyclopedia of Philosophy]
		The prominent virtue of this list is high-mindedness, which, as being a kind of ideal self-respect, is regarded as the crown of all the other virtues, depending on them for its existence, and itself in turn tending to intensify their force.
		The list seems to be more a deduction from the formula than a statement of the facts on which the formula itself depends, and Aristotle accordingly finds language frequently inadequate to express the states of excess or defect which his theory involves (for example in dealing with the virtue of ambition).
		Ideal preferences aside, then, the constitutional republic is regarded as the best attainable form of government, especially as it secures that predominance of a large middle class, which is the chief basis of permanence in any state.
	www.utm.edu /research/iep/a/aristotl.htm (7053 words)

	Projective Schemes (Site not responding. Last check: 2007-10-20)
		We note in passing that the radical of a homogeneous ideal is also homogeneous and so the associated and minimal primes for a homogeneous ideal are also homogeneous.
		To summarise the “big” irrelevant ideal of R is irrelevant since under all homomorphisms under consideration its image generates the full ring, while the “small” irrelevant ideal of R is irrelevant since its image under all homomorphisms under considertation is 0.
		From this description it is clear that the homogenisation of the primary decomposition of an ideal is a primary decomposition of the homogenisation.
	www.imsc.ernet.in /~kapil/geometry/caag/proj.html (2073 words)

	commalg.org - the center for commutative algebra
		Let I be a homogeneous ideal in R. In this paper we investigate asymptotic behaviour of the quotient between the length of local cohomology group H^0_m(R/I^n) and n^d.
		We determine the ideals $I$ in a finitely generated graded $k$-algebra $A$, whose associated graded rings are isomorphic to $A$.
		We show, by using quadratic transform, that the depth of the associated graded ring to a contracted ideal $I$ is determined by depth of the associated graded rings to a certain family of monomial ideals (indeed lex-segments) which are naturally attached to the $I$.
	www.commalg.org /preprints/2004_03.shtml (2318 words)

	Homogeneous Transformation Matrices
		Example: In homogeneous coordinates [3, 2, 1] represent the same ordinary point P as coordinates [1, 2/3, 1/3], the normalized form of P. P corresponds to the point in the plane with Cartesian coordinates (2/3, 1/3).
		Using homogeneous coordinates the plane is represented by the column matrix h = [D; A; B; C] and points P = [k, x, y, z] on the plane satisfy Ph = 0.
		A "normal" is an ideal point or flat, whereas "normalization" is the unit scaling of homogenous coordinate representations of points, hyperplanes and matrices.
	www.silcom.com /~barnowl/HTransf.htm (3992 words)

	Medical Laboratory Observer: Quality control in the new environment: QC materials; an ideal control closely simulates ...
		Ideally, they will be homogeneous and stable, and contain matrix and analytes in a concentration similar to that of patient specimens.
		Homogeneity in control materials is necessary to attribute variation in analytic data correctly to the analytic system.
		A lack of homogeneity may result from incomplete mixing of pools or from differential stability of unstable analytes within portions of a pool.
	www.findarticles.com /p/articles/mi_m3230/is_v19/ai_4671169 (1444 words)

	Cryptology ePrint Archive (Site not responding. Last check: 2007-10-20)
		Namely, in all these families, the ideal access structures coincide with the vector space ones and, besides, the optimal information rate of a non-ideal access structure is at most $2/3$.
		Specifically, we completely characterize the ideal access structures in this family, we prove that they coincide with the ${\bf Z}_2$-vector space ones and, besides, we demonstrate that there is no structure in this family having optimal information rate between $2/3$ and $1$.
		That is, we establish that the properties that were previously proved for several families also hold for the family of the sparse $3$-homogeneous access structures.
	eprint.iacr.org /2003/151 (280 words)

	[No title]
		Recall that the height of an ideal I is defined as ht (I) = min{ht (p)I H p, p F H * prime }.
		We denote by Proj (H ) the spectrum of homogeneous* prime ideals in H , and by Proj P (H ) the spectrum of P invariant homogeneous prime ideals, where an ideal is called P * invariant if it is stable under the action of the Steenrod algebra P *.
		Since this is possible for every prime ideal q of height k, this means that no prime ideal of height k is contained in a prime ideal associated to (d q s n,0,.
	hopf.math.purdue.edu /Neusel/uncoma.txt (4896 words)

	Homogeneous Coordinates
		Converting a degree n homogeneous polynomial in x, y, z and w back to the conventional form is exactly identical to the two-variable case.
		Given a homogeneous coordinate (x,y,w) of a point in the xy-plane, let us consider (x,y,w) to be a point in space whose coordinate values are x, y and w for the x-, y- and w- axes, respectively.
		Therefore, as a homogeneous point moves on a curve defined by homogeneous polynomial f(x,y,w)=0, its corresponding point moves in three-dimensional space, which, in turn, is projected to the plane w=1.
	www.cs.mtu.edu /~shene/COURSES/cs3621/NOTES/geometry/homo-coor.html (1133 words)

	Hilbert Series and Hilbert Polynomial (Site not responding. Last check: 2007-10-20)
		Let I be a homogeneous ideal of the graded polynomial ring P = K[x_1,..., x_n], where K is a field.
		where V_d is the K-vector space consisting of all homogeneous polynomials in P/I (i.e., reduced residues of polynomials of P with respect to I) of weighted degree d.
		Given an homogeneous ideal I of a polynomial ring P over a field, return the Hilbert series H_(P/I)(t) of the quotient ring P/I as an element of the univariate function field Z(t) over the ring of integers.
	www.umich.edu /~gpcc/scs/magma/text849.htm (339 words)

	extracting generators of an ideal (Site not responding. Last check: 2007-10-20)
		Once an ideal has been constructed it is possible to obtain individual elements.
		To obtain a minimal generating set of a homogeneous ideal as a matrix use mingens and as an ideal use trim.
		and the generators are desired in the usual format for input of an ideal, the function toStringis very useful.
	www.math.rutgers.edu /Macaulay2/1169.html (177 words)

	[No title]
		If $I$ is an ideal of a ring $R$, we let $\rad(I)$ denote the radical of $I$, that is $\rad(I) = \{r \in R : r^n \in I \text{ for some positive integer } n \}.$ We say that $I$ is a {\it radical ideal } if $\rad(I) = I$.
		With each ideal $I$ of $R$, let $\widetilde I$ denote the homogeneous ideal of $R[\mathbf X]$ generated by $\{ \widetilde r : r \in I \}$ ($\widetilde r$ is the {\em homogenization} of $r$ and $\widetilde I$ is the homogenization of $I$).
		This behavior where the dimension of the homogeneous spectrum of a graded integral domain $R$ is less than $\dim R$ also occurs in the case where $R$ is an $\mathbb N$-graded integral domain.
	www.math.purdue.edu /~heinzer/preprints/homog42.tex (3790 words)

	Jan Snellman (Site not responding. Last check: 2007-10-20)
		It is shown in Initial ideals of Truncated Homogeneous Ideals that the initial ideals of such truncated ideals converge to the initial ideal of the corresponding ideal in the graded subring.
		This contrast to the lexicographic term order, where for instance the generic ideal generated by a quadratic and a cubic form has an initial ideal wich is minimally generated by an infinite set of monomials, containing two monomials of every (sufficiently high) total degree.
		In Groebner bases for non-homogeneous ideals in we show that the calculations of Groebner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated.
	www.su.se /forskning/disputationer/spikblad/JanSnellman.html (530 words)

	Generation of an exact three-dimensional quadrupole electric field and superposition of a homogeneous electric field in ...
		The possibility of creating such a homogeneous electric field not interfering with the quadrupole electric field is one of the major advantages of the method according to the invention.
		In a variant of the method according to the invention, the above mentioned second, homogeneous electric field inside the boundary of the quadrupole electric field or the electric field of higher multipole moments is used for a mass-to-charge specific excitation of the fundamental frequencies of the ions to be analyzed.
		In addition a homogeneous field can be generated in the same interior region by applying a second potential which varies linearly along the rhombic boundaries in a way different from the first, for example along the line AB, given in FIG.
	www.freepatentsonline.com /5283436.html (4908 words)

	Maybe this can forecast the market trend... best Homogeneous Ideal (Site not responding. Last check: 2007-10-20)
		Some Results On Normal Homogeneous Ideals Some Results On Normal Homogeneous Ideals In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed.
		Homogeneous Ideal -- from MathWorld Homogeneous Ideal -- from MathWorld A homogeneous ideal I in a graded ring R = \oplus A_i is an ideal generated by a set of homogeneous elements, i.e., each one is contained in only one of the A_i.
		On the homogeneous ideal of a projective nonsingular toric variety On the homogeneous ideal of a projective nonsingular toric variety Reason for withdrawal: There is a serious mistake in the calculation of the divisor of the rational...
	www.ascot.pl /th/Fourier4/Homogeneous-Ideal.htm (571 words)

	Ancient Eugenics by Allen G. Roper
		The Romantic ideal was the discovery of the late Greek world under the Roman Empire, but any sentiment that existed at Sparta was as unhampered as romance to-day in the theory of modern Eugenists.
		In this respect, therefore, the Spartan practice was not remote from modern ideals, but infanticide, eliminating the unfit at birth, offered a solution of the problem which we can only hope to solve by the scientific application of the principles of heredity.
		As in the animal body, the homogeneous are for the sake of the heterogeneous.
	www.euvolution.com /articles/ancient.htm (16710 words)

	ideals
		The ideal generated by a list of ring elements can be constructed with the function ideal.
		Once you have an ideal, then you may construct the quotient ring or the quotient module (there is a difference).
		In general, when an ideal is used as an argument to a function that usually would be given a module, we try to make an informed choice about whether the user intends the ideal to be used as a module directly, or whether the quotient module is more suitable.
	www.math.shu.edu /Manuals/Macaulay2/1193.html (661 words)

	Constructing Schemes (Site not responding. Last check: 2007-10-20)
		As the process of saturation may be quite expensive in higher dimensional ambient spaces, the ideal of X is not saturated until the saturation property is required and once saturation has been performed, this is recorded internally.
		The subscheme of X defined, for an affine scheme X by the trivial polynomial 1, or by maximal ideal (x_1,..., x_n) for a projective scheme X. The returned scheme is marked as saturated.
		The ideal of the result will be the colon ideal of the ideal of X and the ideal of Y. If X is saturated then the result is as well and is marked as such.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1152.htm (856 words)

	Boltzmann constant - Wikipedia, the free encyclopedia
		In classical statistical mechanics, homogeneous ideal gases possess kT/2 per degree of freedom per atom.
		Monatomic ideal gases possess 3 degrees of freedom per atom, corresponding to the three spatial directions, which means a thermal energy of 1.5kT per atom.
		As indicated in the article on heat capacity, this corresponds very well with experimental data.
	en.wikipedia.org /wiki/Boltzmann_constant (444 words)

	NCRVE MDS-934
		Although, managing heterogeneity may be more difficult in the beginning than managing homogeneity, most organizations, communities, and countries do not have the option of working in a homogeneous environment (Thomas, 1992).
		They formed homogeneous and heterogeneous groups of undergraduate students of psychology and human relations and conducted a laboratory study on them.
		Ideally, companies should conduct a needs assessment to identify the particular diversity needs within the content of their organizational goals.
	vocserve.berkeley.edu /AllInOne/MDS-934.html (17976 words)

Try your search on: Qwika (all wikis)