Crossword clues for divisor
Longman Dictionary of Contemporary English
The Collaborative International Dictionary
Divisor \Di*vi"sor\, n. [L., fr. dividere. See Divide.] (Math.) The number by which the dividend is divided.
Common divisor. (Math.) See under Common, a.
Douglas Harper's Etymology Dictionary
early 15c., Latin agent noun from dividere (see divide (v.)).
n. 1 (context arithmetic English) A number or expression that another is to be divided by. 2 An integer that divides another integer an integral number of times.
n. one of two or more integers that can be exactly divided into another integer; "what are the 4 factors of 6?" [syn: factor]
the number by which a dividend is divided
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields.
The background is that codimension-1 subvarieties are understood much better than higher-codimension subvarieties. This happens in both global and local ways. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-r subvariety need not be definable by only r equations when r is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this good property, much of algebraic geometry studies an arbitrary variety by analyzing its codimension-1 subvarieties and the corresponding line bundles.
On singular varieties, this good property can fail, and so one has to distinguish between codimension-1 subvarieties and varieties which can locally be defined by one equation. The former are Weil divisors while the latter are Cartier divisors. Topologically, Weil divisors play the role of homology classes, while Cartier divisors represent cohomology classes. On a smooth variety (or more generally a regular scheme), a result analogous to Poincare duality says that Weil and Cartier divisors are the same.
The name "divisor" goes back to the work of Dedekind and Weber, who showed the relevance of Dedekind domains to the study of algebraic curves. The group of divisors on a curve (the free abelian group on its set of points) is closely related to the group of fractional ideals for a Dedekind domain.
An algebraic cycle is a higher-dimensional generalization of a divisor; by definition, a Weil divisor is a cycle of codimension 1.
Divisor may refer to:
- Divisor (algebraic geometry)
- Divisor (ring theory)
Usage examples of "divisor".
The teacher was covered with chalkdust and the blackboards were covered with simple additions and subtractions, by multiplicands, multipliers, and products, by dividends, divisors, and quotients.
This same number, viewed a bit differently, was a special element in the set of positive integers, being a mathematically perfect number, equal to the sum of its divisors.
Notation by sixty also had the advantage over the decimal system in all work involving fractions because sixty has more divisors than ten.
For example, six is the first "perfect" number, equal to the sum of its divisors (1 + 2 + 3).
This common divisor he found to be 1 - 1440 of a dollar, or 1 - 1600 of the crown sterling.
What their ignorance left out of account -- and mine too, which saw no reply to their objection -- was a technical breakthrough he'd recently achieved and was about to put to use: a precision honing device he called the Infinite Divisor.
In that equation, Vretha, you are a divisor, a liability, because the Visionists have managed to put your face on their attacks.