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Topic: Monomial

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	PlanetMath: monomial matrix
		Obviously, a monomial matrix is a square matrix and there exists a rearrangement of rows and columns such that the result is a diagonal matrix.
		A monomial matrix is invertible but, unlike a permutation matrix, not necessarily orthogonal.
		This is version 2 of monomial matrix, born on 2005-05-13, modified 2005-05-16.
	planetmath.org /encyclopedia/MonomialMatrix.html (125 words)

	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 (309 words)

	PlanetMath: monomial
		First, the degree of a monomial is the sum of the elements of its vector representation.
		If a polynomial is represented as a sum over a set of monomials, then the degree of a polynomial can be defined as the degree of the monomial of largest degree belonging to that polynomial.
		This is version 2 of monomial, born on 2002-04-10, modified 2003-07-03.
	planetmath.org /encyclopedia/Monomial.html (202 words)

	Monomial Encyclopedia (Site not responding. Last check: 2007-08-16)
		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.
	www.hallencyclopedia.com /topic/Monomial.html (426 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 (0 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.
		The order is called "graded reverse lexicographical" since it first grades the monomials by total degree, and then decides ties by the negation of the lexicographical order applied to the variables in reverse order.
		The order is called "univariate" since when monomials are compared, any monomial not containing the i-th variable is greater than any monomial containing the i-th variable.
	www.umich.edu /~gpcc/scs/magma/text835.htm (1472 words)

	Monomial Representations for Gröbner Bases Computations
		Monomial representations and operations for Gröbner bases computations are investigated from an implementation point of view.
		The technique of vectorized monomial operations is introduced and it is shown how it expedites computations of Gröbner bases.
		Furthermore, a rank-based monomial representation and comparison technique is examined and it is concluded that this technique does not yield an additional speedup over vectorized comparisons.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/18/paper_html/paper.html (70 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.
		The order is called "graded reverse lexicographical" since it first grades the monomials by total degree, and then decides ties by the negation of the lexicographical order applied to the variables in reverse order.
		The order is called "univariate" since when monomials are compared, any monomial not containing the i-th variable is greater than any monomial containing the i-th variable.
	www.math.niu.edu /help/math/magmahelp/text615.html (0 words)

	Automatic Generation of Transform Algorithms
		The approach extends the well known concept of discrete Fourier transforms over finite groups and became feasible through two advances: A method to decompose monomial representations of solvable groups by structural recursion and a method to compute certain symmetries of matrices by combinatorial search.
		The methods have been implemented in the library AREP and used to generate fast algorithms for a class of transforms including the discrete Fourier, cosine, sine, and Hartley transform, automatically.
		We present an efficient algorithm which decomposes a monomial representation of a solvable group G into its irreducible components.
	www.ece.cmu.edu /~smart/papers/autgen.html (865 words)

	Monomial Orders
		The order is called "univariate" since monomials are compared so that any monomial not containing the i-th variable is greater than any monomial containing the i-th variable.
		Thus all variables but the i-th are "eliminated" so that a Gröbner basis of an 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.
		The order is specified by the arguments ("weight", Q) where Q is a sequence of n^2 non-negative integers or rationals describing the n weight vectors of length n (in row major order).
	www.math.wisc.edu /help/magma/text384.html (0 words)

	Monomial orders for free modules (Site not responding. Last check: 2007-08-16)
		The effect of this response is that the higher the component, or row, number, the greater the monomial.
		c ; largest degree of a monomial : 217 % ; the component number is now considered before the order on the monomials % ; of the ring.
		For column vectors this is not possible, since the lead monomial can occur in any row.
	www.math.columbia.edu /online/Macaulay1-rel0994-html/node52.html (0 words)

	Monomial orderings
		As you will discover using the Groebner applet, different monomial orderings not only lead to different Groebner bases for the same ideal, but they also affect substantially the computation.
		Since a polynomial is a linear combination of monomials, we would like to rearrange unambiguously the terms in a polynomial in descending (or ascending) order.
		The lex order is useful when we need to eliminate the variables in solving systems of equations, but when we only want to eliminate certain variables and we do not care about the others, we can use the more efficient k-th elimination order.
	www.geocities.com /CapeCanaveral/Hall/3131/orderings.html (0 words)

	Monomial orders
		If their total degrees are not equal, then the monomial with the smaller total degree is considered smaller in the graded lex order.
		Also two monic monomials containing b are ranked by first comparing the number of occurrences of the variable b in the monomial.
		are monic monomials in a and c (or in a and d or in b and c or in b and d), then they are ordered by the pure lex order a « c (or a « d or b « c or b « d).
	math.ucsd.edu /~ncalg/StrategyPaper/node51.html (0 words)

	Greatest monomial factor level1
		Now you know that monomials can be broken down into their various factors, and this is called "factoring monomials".
		Find the greatest common monomial factor of this polynomial, divide it out, and rewrite the polynomial as a multiplication problem using the factors you found
		Try these using the same technique as the one variable, greatest common monomial factoring.
	www.themathlab.com /Algebra/factoring/grtmono1.htm (0 words)

	Monomial (Site not responding. Last check: 2007-08-16)
		Default constructor for the class, initalizes coefficient to 0, the variables to null, and the number of variables to 0.
		Checks to see if this Monomial has like terms with another.
		This makes a copy of the array so you can change your data without changing the monomial (useful for multiple assignments).
	faculty.ncwc.edu /jeperry/javadocs/ClassicMath/Monomial.html (154 words)

	Buchberger (Site not responding. Last check: 2007-08-16)
		In my application, the polynomials are monomial differences, that arise from the following problem.
		binomial(const monomial and lm, const monomial and sm);
		We could of course extract the leading monomial from a polynomial each time we needed it, but it's faster to do it this way, since Spoly and the division routines are such that they only require the leading terms.
	www.contrib.andrew.cmu.edu /~roehrig/CompFin/Week5/Buchberger.html (0 words)

	Monomial Ideals - Ho, Smith (ResearchIndex) (Site not responding. Last check: 2007-08-16)
		In particular, we include the Macaulay 2 code for generating interesting families of monomial ideals.
		The rst section introduces the basic functions on monomial ideals in Macaulay 2.
		...a monomial ideal described in terms of its minimal generators can be calculated using Macaulay 2.
	citeseer.ist.psu.edu /453987.html (0 words)

	Integral Closure of Monomial Ideals on Regular Sequences, Karlheinz Kiyek, Jürgen Stückrad
		It is well known that the integral closure of a monomial ideal in a polynomial ring in a finite number of indeterminates over a field is a monomial ideal, again.
		An ideal $\mathfrak a$ of $R$ is called a monomial ideal with respect to $(x_1,\ldots,x_d)$ if it can be generated by monomials $x_1^{i_1}\cdots x_d^{i_d}$.
		We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.rmi/1063050165 (0 words)

	Citebase - Lifting Monomial Ideals (Site not responding. Last check: 2007-08-16)
		We show how to lift any monomial ideal J in n variables to a saturated ideal I of the same codimension in n+t variables.
		We show that I has the same graded Betti numbers as J and we show how to obtain the matrices for the resolution of I.
		LIFTING MONOMIAL IDEALS 25 [11] A.V. Geramita, T. Harima and Y.S. Shin, An Alternative to the Hilbert Function for the Ideal of a Finite Set of Points in Pn, preprint 1998.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9907045 (0 words)

	Algebraic Combinatorics -- Complete monomial groups
		In this case we shorten the notation by putting
		, the complete monomial group of degree n over H.
		Many important groups are of this form, examples will be given in a moment.
	www.mathe2.uni-bayreuth.de /frib/html/book/hyl00_35.html (159 words)

	MSN Encarta - Search Results - Monomial
		Monomial, expression in algebra consisting of a single term, such as 3y or x
		dividing polynomial terms by the monomial, factoring monomials in polynomials, multiplying a polynomial by a monomial, multiplying and dividing...
		See all search results in Homework Help (16)
	ca.encarta.msn.com /Monomial.html (0 words)

	[No title] (Site not responding. Last check: 2007-08-16)
		The main contribution of this paper is that an earlier decomposition method from Larsen for the De Simone format is extended to the more general ntyft/ntyxt format without lookahead.
		The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cut-free monomial proof nets can correspond to the same cut-free proof.
		Thus the problem of finding a satisfactory notion of proof net for unit-free multiplicative-additive linear logic (MALL) has remained open since the inception of linear logic in 1986.
	boole.stanford.edu /pub/ABSTRACTS (0 words)

	Algebra Worksheets: Fundamentals of Equations and Formulas!
		Division of Monomials (Multiplication and Power-Of-A-Power or Power-Of-A-Product)
		Multiplying Monomials (Negatives and Power-Of-A-Power or Power-Of-A-Product) Crossword Puzzle
		Division of Monomials (Multiplication and Power-Of-A-Power or Power-Of-A-Product) Crossword Puzzle
	www.edhelper.com /algebra.htm (1712 words)

	Hidden Field Equations public key cryptosystem home page (HFE)
		Then for all the monomials of the form
		I have never seen a one-to-one polynomial over a finite field with big degree and with many monomials and in addition having small non-linear degree as
		Big degree polynomials with little monomials behave the same way as small degree polynomials for the reasons that become obvious if we use normal bases described in [MOV] p.642.
	www.minrank.org /hfe (0 words)

	Polynomials Worksheets
		Dividing a Monomial by a Monomial Crossword Puzzle
		Dividing a Polynomial by a Monomial Crossword Puzzle
		Mixing Addition, Multiplication, and Division in one problem Crossword Puzzle
	www.edhelper.com /polynomials.htm (174 words)

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