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33-553: [REDACTED] Look up LM  or lm in Wiktionary, the free dictionary. The abbreviation LM or lm may refer to: Places [ edit ] County Leitrim , Ireland (vehicle plate code LM) Le Mans , a place in France Limburg-Weilburg , Germany (vehicle plate code LM) Liptovský Mikuláš , Slovakia (vehicle plate code LM) Lourenço Marques , Pearl of

66-400: A j {\displaystyle {\tfrac {a_{i}}{a_{j}}}} are irrational) then a tie can never occur, and the weight vector itself specifies a monomial ordering. In the contrary case, one could use another weight vector to break ties, and so on; after using n linearly independent weight vectors, there cannot be any remaining ties. One can in fact define any monomial ordering by

99-513: A linear combination of monomials, where S is a finite subset of M and the c u are all nonzero. When a monomial order has been chosen, the leading monomial is the largest u in S , the leading coefficient is the corresponding c u , and the leading term is the corresponding c u u . Head monomial/coefficient/term is sometimes used as a synonym of "leading". Some authors use "monomial" instead of "term" and "power product" instead of "monomial". In this article,

132-415: A Microsoft network operating system Language model , a mathematical model used in language processing and speech recognition Lebesgue measure , in measure theory Levenberg–Marquardt algorithm , used to solve non-linear least squares problems Leading monomial Linear Monolithic, a National Semiconductor prefix for integrated circuits; see List of LM-series integrated circuits LM hash ,

165-473: A Microsoft password hash function Long mode , a CPU mode of operation where 64-bit programs are executed ( lm is also set as a CPU flag) Science and technology [ edit ] Apollo Lunar Module spacecraft Leonard-Merritt mass estimator , a formula for estimating the mass of a spherical stellar system Light meter Light microscope Line maintenance, a type of Aircraft maintenance checks Listeria monocytogenes Lumen (unit) ,

198-521: A hymn-metre with four lines of 8 syllables Brands and enterprises [ edit ] L&M , a brand of cigarettes Ledgewood Mall , a shopping mall in New Jersey Legg Mason , a U.S. investment management firm; NYSE ticker symbol Lockheed Martin , a U.S. defense contractor In transportation [ edit ] ALM Antillean Airlines , a Netherlands Antillean airline; IATA airline designator code Lamborghini Militaria ,

231-495: A monomial is assumed to not include a coefficient. The defining property of monomial orderings implies that the order of the terms is kept when multiplying a polynomial by a monomial. Also, the leading term of a product of polynomials is the product of the leading terms of the factors. On the set { x n ∣ n ∈ N } {\displaystyle \left\{x^{n}\mid n\in \mathbb {N} \right\}} of powers of any one variable x ,

264-524: A sequence of weight vectors ( Cox et al. pp. 72–73), for instance (1,0,0,...,0), (0,1,0,...,0), ... (0,0,...,1) for lex, or (1,1,1,...,1), (1,1,..., 1,0), ... (1,0,...,0) for grevlex. For example, consider the monomials x y 2 z {\displaystyle xy^{2}z} , z 2 {\displaystyle z^{2}} , x 3 {\displaystyle x^{3}} , and x 2 z 2 {\displaystyle x^{2}z^{2}} ;

297-588: A series of light trucks, the Rambo Lambos Lexus LM , a luxury minivan Livingston Energy Flight , an Italian airline; IATA airline designator code Loganair , a Scottish airline; IATA airline designator code London Midland , a rail operator based in the West Midlands, England Business and finance [ edit ] IS–LM model in macroeconomics, where LM refers to Liquidity preference-Money supply Lean manufacturing Maltese lira ,

330-408: A specific monomial order. Besides respecting multiplication, monomial orders are often required to be well-orders , since this ensures the multivariate division procedure will terminate. There are however practical applications also for multiplication-respecting order relations on the set of monomials that are not well-orders. In the case of finitely many variables, well-ordering of a monomial order

363-413: A unit of luminous flux Sport [ edit ] 24 Hours of Le Mans race, and related car models Late model , a class of racing car Left midfielder , a defensive position in association football Other uses [ edit ] Legion of Merit , a United States military decoration See also [ edit ] 1M (disambiguation) IM (disambiguation) Topics referred to by

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396-455: Is different from Wikidata All article disambiguation pages All disambiguation pages LM">LM The requested page title contains unsupported characters : ">". Return to Main Page . Leading monomial Monomial orderings are most commonly used with Gröbner bases and multivariate division . In particular, the property of being a Gröbner basis is always relative to

429-403: Is equivalent to the conjunction of the following two conditions: Since these conditions may be easier to verify for a monomial order defined through an explicit rule, than to directly prove it is a well-ordering, they are sometimes preferred in definitions of monomial order. The choice of a total order on the monomials allows sorting the terms of a polynomial. The leading term of a polynomial

462-405: Is important as it allows elimination , an operation which corresponds to projection in algebraic geometry. Weight order depends on a vector ( a 1 , … , a n ) ∈ R ≥ 0 n {\displaystyle (a_{1},\ldots ,a_{n})\in \mathbb {R} _{\geq 0}^{n}} called the weight vector. It first compares

495-465: Is preceded only by a finite number of other monomials; this is not the case for lexicographic order, where all (infinitely many) powers of y are less than x (that lexicographic order is nevertheless a well ordering is related to the impossibility of constructing an infinite decreasing chain of monomials). For monomials of degree at most two in two indeterminates x 1 , x 2 {\displaystyle x_{1},x_{2}} ,

528-496: Is rarely considered.) In the example below we use x , y and z instead of x 1 , x 2 and x 3 . With this convention there are still many examples of different monomial orders. Lexicographic order (lex) first compares exponents of x 1 in the monomials, and in case of equality compares exponents of x 2 , and so forth. The name is derived from the similarity with the usual alphabetical order used in lexicography for dictionaries, if monomials are represented by

561-449: Is rarely used: the Gröbner basis for the graded reverse lexicographic order, which follows, is easier to compute and provides the same information on the input set of polynomials. Graded reverse lexicographic order (grevlex, or degrevlex for degree reverse lexicographic order ) compares the total degree first, then uses a lexicographic order as tie-breaker, but it reverses the outcome of

594-405: Is that the reverse order exhibits all variables among the small monomials of any given degree, whereas with the non-reverse order the intervals of smallest monomials of any given degree will only be formed from the smallest variables. Block order or elimination order (lexdeg) may be defined for any number of blocks but, for sake of simplicity, we consider only the case of two blocks (however, if

627-551: Is thus the term of the largest monomial (for the chosen monomial ordering). Concretely, let R be any ring of polynomials. Then the set M of the (monic) monomials in R is a basis of R , considered as a vector space over the field of the coefficients. Thus, any nonzero polynomial p in R has a unique expression p = ∑ u ∈ S c u u {\displaystyle p=\textstyle \sum _{u\in S}c_{u}u} as

660-392: The dot product of the exponent sequences of the monomials with this weight vector, and in case of a tie uses some other fixed monomial order. For instance, the graded orders above are weight orders for the "total degree" weight vector (1,1,...,1). If the a i are rationally independent numbers (so in particular none of them are zero and all fractions a i

693-543: The Indian Ocean, Mozambique Lower Mainland , a region in British Columbia, Canada Lower Manhattan , Southern part of Manhattan, New York Arts, entertainment, and media [ edit ] Little Mix , a British four-piece girl group LM (magazine) , a defunct British computer game magazine Living Marxism magazine, published under the name LM between 1997 and 2000 Long metre or Long Measure,

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726-462: The former currency of Malta Organizations [ edit ] Liberal Movement (Australia) , a defunct Australian political party Lower Merion High School , a Pennsylvania secondary school Mathematics, science, and technology [ edit ] Mathematics and computing [ edit ] Linear model , a type of statistical model Lagrange multiplier , a method for finding maxima and minima subject to constraints LAN Manager ,

759-451: The graded lexicographic order (with x 1 > x 2 {\displaystyle x_{1}>x_{2}} ) is x 1 2 > x 1 x 2 > x 2 2 > x 1 > x 2 > 1. {\displaystyle x_{1}^{2}>x_{1}x_{2}>x_{2}^{2}>x_{1}>x_{2}>1.} Although very natural, this ordering

792-1384: The graded reverse lexicographic order is thus to compare by the total degree first, then compare exponents of the last indeterminate x n but reversing the outcome (so the monomial with smaller exponent is larger in the ordering), followed (as always only in case of a tie) by a similar comparison of x n −1 , and so forth ending with x 1 . The differences between graded lexicographic and graded reverse lexicographic orders are subtle, since they in fact coincide for 1 and 2 indeterminates. The first difference comes for degree 2 monomials in 3 indeterminates, which are graded lexicographic ordered as x 1 2 > x 1 x 2 > x 1 x 3 > x 2 2 > x 2 x 3 > x 3 2 {\displaystyle x_{1}^{2}>x_{1}x_{2}>x_{1}x_{3}>x_{2}^{2}>x_{2}x_{3}>x_{3}^{2}} but graded reverse lexicographic ordered as x 1 2 > x 1 x 2 > x 2 2 > x 1 x 3 > x 2 x 3 > x 3 2 {\displaystyle x_{1}^{2}>x_{1}x_{2}>x_{2}^{2}>x_{1}x_{3}>x_{2}x_{3}>x_{3}^{2}} . The general trend

825-425: The indeterminates are named x 1 , x 2 , x 3 , ... in decreasing order for the monomial order considered, so that always x 1 > x 2 > x 3 > ... . (If there should be infinitely many indeterminates, this convention is incompatible with the condition of being a well ordering, and one would be forced to use the opposite ordering; however the case of polynomials in infinitely many variables

858-479: The lexicographic comparison so that lexicographically larger monomials of the same degree are considered to be degrevlex smaller. For the final order to exhibit the conventional ordering x 1 > x 2 > ... > x n of the indeterminates, it is furthermore necessary that the tie-breaker lexicographic order before reversal considers the last indeterminate x n to be the largest, which means it must start with that indeterminate. A concrete recipe for

891-441: The lexicographic order (with x 1 > x 2 {\displaystyle x_{1}>x_{2}} ) is x 1 2 > x 1 x 2 > x 1 > x 2 2 > x 2 > 1. {\displaystyle x_{1}^{2}>x_{1}x_{2}>x_{1}>x_{2}^{2}>x_{2}>1.} For Gröbner basis computations,

924-414: The lexicographic ordering tends to be the most costly; thus it should be avoided, as far as possible, except for very simple computations. Graded lexicographic order (grlex, or deglex for degree lexicographic order ) first compares the total degree (sum of all exponents), and in case of a tie applies lexicographic order. This ordering is not only a well ordering, it also has the property that any monomial

957-454: The monomial orders above would order these four monomials as follows: When using monomial orderings to compute Gröbner bases, different orders can lead to different results, and the difficulty of the computation may vary dramatically. For example, graded reverse lexicographic order has a reputation for producing, almost always, the Gröbner bases that are the easiest to compute (this is enforced by

990-432: The number of blocks equals the number of variables, this order is simply the lexicographic order). For this ordering, the variables are divided in two blocks x 1 ,..., x h and y 1 ,..., y k and a monomial ordering is chosen for each block, usually the graded reverse lexicographical order. Two monomials are compared by comparing their x part, and in case of a tie, by comparing their y part. This ordering

1023-431: The only monomial orders are the natural ordering 1 <  x  < x  < x  < ... and its converse, the latter of which is not a well-ordering. Therefore, the notion of monomial order becomes interesting only in the case of multiple variables. The monomial order implies an order on the individual indeterminates. One can simplify the classification of monomial orders by assuming that

LM - Misplaced Pages Continue

1056-401: The same term [REDACTED] This disambiguation page lists articles associated with the title LM . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=LM&oldid=1234260966 " Category : Disambiguation pages Hidden categories: Short description

1089-499: The sequence of the exponents of the indeterminates. If the number of indeterminates is fixed (as it is usually the case), the lexicographical order is a well-order , although this is not the case for the lexicographical order applied to sequences of various lengths (see Lexicographic order § Ordering of sequences of various lengths ). For monomials of degree at most two in two indeterminates x 1 , x 2 {\displaystyle x_{1},x_{2}} ,

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