
	Where results make sense

Topic: Monomial basis

Related Topics

Bernstein polynomial

Linear algebra

In the News (Fri 25 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Monomial - Wikipedia, the free encyclopedia
		The most obvious fact about monomials is that any polynomial is a linear combination of them, so they can serve as basis vectors in a vector space of polynomials - a fact of constant implicit use in mathematics.
		In group representation theory, a monomial representation is a particular kind of induced representation.
		In propositional logic, a monomial is a conjunction of literals.
	en.wikipedia.org /wiki/Monomial (307 words)

	Representation and Monomial Orders
		Multivariate polynomial rings are constructed in Magma such that the monomials of any polynomial are sorted with respect to a specified monomial order, with the greatest monomial first.
		A Gröbner basis of an ideal with respect to the graded reverse lexicographical order is usually the easiest to compute so it is recommended that this order be used when just any Gröbner basis for an ideal is desired.
		Thus all variables but the i-th are "eliminated" so that a Gröbner basis of a zero-dimensional ideal I with this ordering will contain the unique monic generator of the elimination ideal consisting of all the polynomials in I containing the i-th variable alone.
	www.umich.edu /~gpcc/scs/magma/text835.htm (1472 words)

	Monomial basis - Wikipedia, the free encyclopedia
		In mathematics a monomial basis is a way to uniquely describe a polynomial using a linear combination of monomials.
		This description, the monomial form of a polynomial, is often used because of the simple structure of the monomial basis.
		of polynomials with degree n is the polynomial sequence of monomials
	en.wikipedia.org /wiki/Monomial_basis (126 words)

	R: Define a Functional Basis Object
		For example, if it is required that a function be zero at the left boundary, this is achieved by dropping the first basis function, the only one that is nonzero at that point.
		The "create" function names are rather long, and users who mind all that typing might be advised to modify these to versions with shorter names, such as "splbas", "conbas", and etc. However, a principle of good programming practice is to keep the code readable, preferably by somebody other than the programmer.
		Normally only developers of new basis types will actually need to use this function, so no examples are provided.
	rss.acs.unt.edu /Rdoc/library/fda/html/basisfd.html (425 words)

	AMTH247 Lecture 11 Interpolation I
		Below is a Scilab function to compute the basis matrix for the monomial basis at a given set of data values.
		The key point is to note that the columns of the basis matrix are just powers of the vector of data points considered as a column vector.
		This problem of the basis matrix having a large condition number is typical of monomial bases.
	turing.une.edu.au /~amth247/Lectures_2003/Lecture_11/lecture (1123 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		We construct a "standard monomial basis" for the co-ordinate ring of the variety of nilpotent matrices in the space M(n) of n by n matrices (over a field of characteristic 0).
		For this construction, using a result of Lusztig, we identify the variety of nilpotent matrices with a open subset of a Schubert variety in the affine grassmannian \hat{Gr}_n.
		We then use the Plucker co-ordinates on the affine grassmannian \hat{Gr}_n, and construct the required basis as certain monomials in the Plucker co-ordinates (in the same spirit as that of Hodge's work on the classical Grassmannian.).
	www-math.mit.edu /~combin/abstracts/sep02/laks.html (131 words)

	[No title]
		A recursive change of basis formula is produced to c* onvert between the Milnor basis* and each of the bases for which the change of ba* *sis matrix in every grading is upper triangular.
		To convert a element from the Milnor basis to the Arnon C basis we follow the* * now familiar path of showing that the basis of C-admissible monomials is triangular* * with respect to the Milnor basis by defining the appropriate fl and ordering needed* * for the recursive formula of the form (1.1).
		To convert a element from the Milnor basis to the basis of admissible monomia* ls we show that the Arnon A basis* is triangular with respect to the Milnor basis a* nd define the fl and ordering _needed for the recursive formula of the form (1.1) *.
	www.math.purdue.edu /research/atopology/Monks/bases.txt (5697 words)

	Gröbner Bases
		A basis is called reduced if each polynomial in it is monic and, for every monomial of each polynomial in the basis, that monomial is not divisible by the leading monomial of any other polynomial in the basis (equivalently, each leading monomial does not divide any monomial in any of the other polynomials) [CLO96, Chap.
		Over Euclidean rings, the definition of a minimal basis is practically the same as for fields (there must be no polynomial in the ideal generated by the others and each polynomial must be normalized), but the definition of a reduced basis is more subtle.
		Given a set or sequence S of polynomials describing a basis of an ideal, return whether the basis is itself a (not necessarily minimal or reduced) Gröbner basis of the ideal.
	www.umich.edu /~gpcc/scs/magma/text837.htm (4216 words)

	Element Operations
		It is possible to rearrange the linear combination of basis elements so that the basis elements indexed by the longest partition print first and also so that the basis elements indexed by the partition with greatest maximal part (first entry) print first.
		If b is expressed with respect to a Schur basis and a is expressed with respect to a power sum, elementary or monomial basis then the algorithm based on Muirs rule [Mui60] is used.
		If a is expressed with respect to a homogeneous basis the algorithm based on the Pieri rule [Mac95] is used and if a is also expressed with respect to a Schur basis then the method of Schubert polynomials [LS85] is used.
	wwwmaths.anu.edu.au /research.programs/aat/htmlhelp/text1333.htm (1646 words)

	The Nephroid Lab: Gröbner Bases
		A Gröbner basis for an ideal I is a set of generators for I having a certain property with respect to an ordering < on the monomials, but we first need to explain what a monomial ordering is. A monomial ordering is a total ordering < of all the monomials {1, x, y, z, x
		This ordering says that to compare two monomials m, m' we first see if one of them has smaller power of x and say this monomial is smaller in < if so.
		For a given monomial order <, any polynomial f will have a unique term whose monomial is largest in the < order, called the <-leading term of f (NB: we are ignoring the coefficient in front of each term to obtain a monomial, e.g.
	www.geom.uiuc.edu /~fjw/calc-init/nephroid/grobner.html (790 words)

	A Gröbner Basis Theorem on Elimination Ideals
		The NCProcess command generates output which is displayed as a list of V-categories (see §) and the V-categories are defined in such a way that if one of the polynomials in a category is in the elimination ideal, then the entire category is a subset of the elimination ideal.
		If one has a Gröbner Basis with respect to certain types of monomial orders, then, if one considers the subset of the Gröbner Basis which lies in the j-th elimination ideal, then this set is itself a Gröbner Basis and generates the j-th elimination ideal.
		If a pure lex order is used, if one runs NCProcess until the GBA being used by NCProcess generates a Gröbner Basis and all of the shrinking (see §) parts of the NCProcess commands are turned off, then the categories which this outputs can be used to determine generating sets for the elimination ideals.
	math.ucsd.edu /~ncalg/StrategyPaper/node60.html (1054 words)

	Transition Matrices
		Computes the matrix for the expansion of a monomial symmetric function indexed by a partition of weight n as a sum of elementary symmetric functions.
		The matrix converting from homogeneous basis to monomial basis is symmetric.
		The matrix converting from elementary basis to monomial basis is symmetric.
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text1334.htm (1107 words)

	MATH 531: Introduction to Singular
		The definition of a ring consists of three parts: the first part determines the ground field, the second part determines the names of the ring variables, and the third part determines the monomial ordering to be used.
		When we are not interested in a basis of the quotient vector space, but only in the resulting dimension we may even use the command
		A standard basis with respect to a local ordering is an anologue of a grobner basis for global monomial orderings.
	www.math.uic.edu /~leykin/math531/intro2singular.html (1130 words)

	SMID project
		When the rational model is written down in terms of the classical monomial basis, in one or more dimensions, the problem is highly structured and in general ill conditioned.
		In one variable, the basis of Bernstein polynomials is known to minimize the condition number for the solution of interpolation problems.
		Once the optimal basis has been indentified and computed, the rational model, which is the desired output of the modelling process, has to be computed as the solution of a structured system of equations that express the modelling conditions.
	www.cs.kuleuven.ac.be /~nalag/research/projects/RAM.shtml (1850 words)

	Cornell Math - Thesis Abstracts (Combinatorics)
		For example, we show that the fundamental basis of peak functions arising from enriched P-partitions of chains is precisely the basis that is needed to properly encode the cd-index, a common invariant in the study of convex polytopes and Eulerian posets.
		Much of our analysis builds on basic properties of a monomial-like basis for the peak algebra, as well as a recursively defined basis of eigenvectors for the descents-to-peaks map.
		We give a simple criterion, in terms of this monomial basis, for determining when an element of Pi is symmetric.
	www.math.cornell.edu /~www/Research/Abstracts/combinatorics.html (1886 words)

	[No title]
		A recursive change of basis formula is produced to convert between the Milnor basis and each of the bases for which the change of basis matrix in every grading is upper triangular.
		In particular, it is shown that the basis of admissible monomials, the new $P_t^s$ bases, and two bases due to D. Arnon, are all bases having this property, and the corresponding change of basis formula is produced for each of them.
		Some monomial relations for the mod 2 Steenrod algebra are then obtained by exploring the change of basis transformations.
	claude.math.wesleyan.edu /~mhovey/archive/letter10 (578 words)

	AMCA: Strong monomial basis property and canonical basis for a cyclic quiver by Jie Du (Site not responding. Last check: 2007-10-13)
		AMCA: Strong monomial basis property and canonical basis for a cyclic quiver by Jie Du Atlas Mathematical Conference Abstracts
		Strong monomial basis property and canonical basis for a cyclic quiver
		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/t/25.htm (269 words)

	On the Complexity of Sparse Elimination \| EECS at UC Berkeley
		We examine polynomial systems that generate 0-dimensional ideals: a generic monomial basis for the coordinate ring of such a system is defined from a mixed subdivision.
		We offer a simple proof of this known fact and relate the computation of a monomial basis to the calculation of Mixed Volume.
		It is shown that the size of monomial bases and multiplication maps in the context of sparse elimination theory is a function of the Mixed Volume of the Newton polytopes, whereas classical elimination considers simply total degree.
	www.eecs.berkeley.edu /Pubs/TechRpts/1994/5746.html (256 words)

	[No title]
		Lambda(n+t) subset Lambda(n) and as Singer observes in his preprint (where he proves his result), this immediately implies the Lambda EHP sequence (given the Lambda admissible monomial basis).
		Singer and Bousfield seem to agree with me that the Lambda EHP sequence is nontrivial, and a good explanation of non-triviality is given by the complexity of the Curtis excess formula.
		********** definition of Lambda(n) & H ******** Define P(n) to be the vector subspace of the tensor algebra with basis** the n-pseudo-admissible sequences.
	www.lehigh.edu /~dmd1/br614.txt (920 words)

	ACM TOMS: Bibliographic record for `Majaess:1992:SAB' (Site not responding. Last check: 2007-10-13)
		There are several widely available computer codes based on this approach, all of which have at their core a particular choice of basis representation for the piecewise polynomials used to approximate the solutions.
		Until recently, the most popular approach was to use a B-spline representation, but it has been shown that the B-spline basis is inferior, both in operation counts and conditioning, to a certain monomial basis, and the latter has come more into favor.
		In this paper, we describe a linear algebraic equations which arise in spline collocation at Gaussian points with such a monomial basis.
	www.acm.org /toms/cgi-bin/TOMSbibget?Majaess:1992:SAB (156 words)

	CISC-271
		It is is also used as a programming environment for implementing some of the algorithms that are studied.
		The Lagrange basis is not likely to cause numerical errors as is the case when using the monomial basis for polynomial interpolation.
		There is no mention in Ell, of using Vandermonde matrices to interpolate with the monomial basis.
	www.cs.queensu.ca /home/daver/B271/index.html (1327 words)

	Singular 2-0-4 Manual: option
		disables the bucket representation of polynomials during standard basis computations.
		reduction of the tails of polynomials during standard basis computations.
		for inhomogeneous input, polynomial reductions during standard basis computations are never postponed, but always finished through.
	www.math.sunysb.edu /~sorin/online-docs/Singular/sing_244.htm (524 words)

	interpolation
		Instead of the monomial and Lagrange basis we can construct our interpolant with the use of a Newton basis.
		The computation of the interpolant with the use of a monomial basis costs a lot of computations, which must be redone for different function
		Remark 6 The use of a Lagrange basis for the construction of the interpolant has its advantage: No system of equations has to be solved.
	www.win.tue.nl /~maubach/university/education/lectures/2n330/lectures/interpolation/interpolation/index.html (1107 words)

	University of Michigan Combinatorics Seminar (Site not responding. Last check: 2007-10-13)
		We introduce a new basis of monomial peak functions for Pi which behaves much like the monomial basis for Qsym.
		For example, the stucture constants relative to this new basis count quasi-shuffles of peak compositions.
		Here we find that the monomial basis for Pi is dual to N. Reading's Charney-Davis index, which appears to be an Eulerian analog of the flag f-vector.
	www.math.lsa.umich.edu /seminars/combin/fall02/sep13.html (189 words)

	SYMMETRICA-MANUAL -- Multivariate polynomials (Site not responding. Last check: 2007-10-13)
		Therefore you can switch from the monomial basis to the basis of Schubert polynomials.
		Here are the main lines of a corresponding program which first evaluates the Schubert polynomial associated with the scanned permutation, then it gives the polynomial in a single indeterminate q which arises by replacing the i-th indeterminate of the Schubert polynomial by q
		, while the final line replaces each monomial summand by 1 so that the result is the sum of the coefficients of the monomials (a complete program is stored in ex11.c):
	www.mathe2.uni-bayreuth.de /frib/html/symman/manual_18.html (250 words)

	STEENROD Program
		The version below also includes an extremely efficient algorithm by Vince Giambalvo for computing the Milnor basis elements in a given grading of A or An having a prescribed maximum excess.
		Convert a sum of monomials in Sq(i) to the admissible basis using the Adem relations
		Find all of the elements of A in a given grading in either the Milnor or admissible monomial basis
	math.scranton.edu /monks/software/Steenrod/steen.html (309 words)

	AMCA: Monomial Bases for q-Schur algebras by Jie Du (Site not responding. Last check: 2007-10-13)
		In turn, we obtain a new basis for the integral q-Schur algebra S
		(n, r) which consists of certain monomials in the original generators.
		Finally, when n >= r, we interpret the Hecke algebra part of the monomial basis for S
	at.yorku.ca /c/a/h/w/03.htm (146 words)

	Citebase - Monomial Cycle Basis on Koszul Homology Modules (Site not responding. Last check: 2007-10-13)
		It gives a class of p-Borel principal ideals of a polynomial algebra over a field K for which the graded Betti numbers do not depend on the characteristic of K and the Koszul homology modules have monomial cyclic basis.
		Also it shows that all principal p-Borel ideals have binomial cycle basis on Koszul homology modules.
		Users are cautioned not to use it for academic evaluation yet.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0505656 (117 words)

	Saragreadme.html (Site not responding. Last check: 2007-10-13)
		in the Bernstein basis for l,(l+r)/2 of 2^deg(P) P
		in the Bernstein basis for (l+r)/2,r of 2^deg(P) P
		-- monomial [root isolation in the monomial basis] (**)
	www.math.gatech.edu /~saugata/saragreadme.html (1463 words)

	Abstract: Alexander Yong - University of Michigan (Site not responding. Last check: 2007-10-13)
		We introduce a new basis of {\em monomial peak functions} for $\Pi$ which
		For example, the stucture constants relative to this new basis count
		the monomial basis for $\Pi$ is dual to N. Reading's Charney-Davis index,
	garsia.math.yorku.ca /seminar/Previous/F2002/abstractsam.html (161 words)

Try your search on: Qwika (all wikis)