Monotonic

Definition: Monotonic

Monotonic

Adjective

1. (mathematics) of a sequence or function; consistently increasing and never decreasing or consistently decreasing and never increasing in value.

2. Sounded or spoken in a tone unvarying in pitch; "the owl's faint monotonous hooting".

Domain	Definition
Computing	Monotonic In domain theory, a function f : D -> C is monotonic (or monotone) if for all x,y in D, x <= y => f(x) <= f(y). ("<=" is written in LaTeX as \sqsubseteq). (1994-11-24). Source: The Free On-line Dictionary of Computing.
Source: compiled by the editor from various references; see credits.

The terms monotonic or monotone refer to functions between partially ordered sets. They first arose in calculus and were later generalized to the more abstract setting or order theory.

In calculus, a function f : X -> R (where X is a subset of the real numbers R) is monotonically increasing or simply increasing if, whenever x ≤ y, then f(x) ≤ f(y). An increasing function is also called order-preserving for obvious reasons.

Likewise, a function is decreasing if, whenever x ≤ y, then f(x) ≥ f(y). A decreasing function is also called order-reversing.

If the definitions hold with the inequalities (≤, ≥) replaced by strict inequalities (<, >) then the functions are called strictly increasing or strictly decreasing.

As was mentioned at the beginning, there is also a more general notion of monotonicity in case one is not concerned with the set of the real numbers (as in calculus) but with a function f between arbitrary partially ordered sets A and B. In this setting, a function f : A -> B is said to be order-preserving whenever a₁ ≤ a₂ implies f(a₁) ≤ f(a₂), and order-reversing if a₁ ≤ a₂ implies f(a₁) ≥ f(a₂). A function is monotonic if it is either order-preserving or order-reversing, and if the definitions hold when (≤, ≥) are replaced by (<, >) one adds the adverb strictly to the terms.

In calculus, each of the following properties of a function f : R -> R implies the next:

A function f is monotonic;
f has limits from the right and from the left at every point of its domain;
f can only have discontinuities of jump type;
f can only have countably many discontinuities in its domain.

These properties are the reason why monotonic functions are useful in technical work in analysis. Two facts about these functions are:

if f is a monotonic functions defined on an interval I, then f is differentiable almost everywhere on I, i.e. the set of numbers x in I such that f is not differentiable in x has Lebesgue measure zero.
if f is a monotonic function defined on an interval [a, b], then f is Riemann integrable.

An important application of monotonic functions is in probability theory. If X is a random variable, its cumulative distribution function

F_X(x) = Prob(X ≤ x)

is a monotonically increasing function.

Monotonicity criterion

(From Wikipedia, the free Encyclopedia)

In voting systems, a voting system is monotonic if it satisfies the following condition:

If an alternative X loses, and the ballots are changed only by placing X in lower positions, without changing the relative position of other candidates, then X must still lose.

It is considered a good thing if a voting system is monotonic. If a voting system is not monotonic, it can encourage tactical voting, as there will be situations in which a voter will be encouraged to bury their favorite (i.e. rank their favorite option lower than their sincere preference).

The Borda count is monotonic, while Coombs' method and Instant-runoff voting are not. Approval voting is monotonic, using a slightly different definition, because it is not a preferential system: you can never help a candidate by not voting for them.

Some parts of this article are derived from text at http://condorcet.org/emr/criteria.shtml

Compare Monotonicity.

Synonyms: Monotonic

Synonyms: monotone (adj), monotonous (adj). (additional references)

Antonym: nonmonotonic (adj). (additional references)

Crosswords: Monotonic

English words defined with "monotonic": decreasing monotonic ♦ increasing monotonic ♦ nonmonotonic. (references)

Specialty definitions using "monotonic": cangrejo cacerola, continuous function ♦ Noether's test for cyclical trend ♦ order-embedding ♦ Rate monotonic scheduling. (references)

Domain	Title
Books	A Practitioner's Handbook for Real-Time Analysis: Guide to Rate Monotonic Analysis for Real-Time Systems (The Kluwer International Series in Enginee) (reference) (more book examples)
Source: compiled by the editor from various references; see credits.

Play	Caption
	Count; counting; numbers; increasing; increase; numerals; numeric; number; one; two; three; four; series; arithmetic series; monotonic series.
Source: compiled by the editor from various references; see credits.

"Monotonic" is generally used as an adjective (general or positive) -- approximately 100.00% of the time. "Monotonic" is used about 50 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted)

Parts of Speech	Percent	Usage per 100 Million Words	Rank in English
Adjective (general or positive)	100%	50	48,117

Expressions using "monotonic": decreasing monotonic ♦ increasing monotonic ♦ rate monotonic scheduling. Additional references.
Hypenated Usage
	Ending with "monotonic": non-monotonic.
Source: compiled by the editor from various references; see credits.

The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com.

Language		Translations for "monotonic"; alternative meanings/domain in parentheses.
Chinese		单调 (humdrum, monotone, monotonous, monotony). (various references)

French		continent. (various references)

German		gleichbleibend (consistent, constant, persistent, steady, unchanging, uniform, unrelieved, unvariable, unvarying). (various references)

Hungarian		monoton (chant, jog-trot, monotone, monotonous, singsong, sing-song), egyhangú (dull, humdrum, jog trot, jog-trot, monotone, monotonous, muzzy, unanimous, unexciting). (various references)

Korean		단조음. (various references)

Pig Latin		onotonicmay

Russian		монотонный (monotone, monotonous, singsong, unrelieved). (various references)
Source: compiled by the editor from various translation references.

Derivations
	Words beginning with "monotonic": monotonically, monotonicities, monotonicity. (additional references)
Misspellings
	"Monotonic" is suggested in spellcheckers for the following: monatonic, monotomic, monotonicity, monotunis, montonic. (additional references)
Source: compiled by the editor, based on several corpora (additional references).

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "c-i-m-n-n-o-o-o-t"
	-2 letters: monocot.
	-3 letters: motion, noncom, notion.
	-4 letters: conin, conto, niton, nomoi, onion, ontic, tonic.
	-5 letters: cion, coin, coni, conn, coon, coot, icon, into, mint, mono, moon, moot, noon, omit, onto, otic, toom, toon.
	Words containing the letters "c-i-m-n-n-o-o-o-t"
	+3 letters: conformation, monotonicity, noneconomist.
	+4 letters: conformations, mitochondrion, monopsonistic, monotonically, noncompetitor, nonconformist, nonconformity, noneconomists, pathognomonic.
	+5 letters: conformational, conglomeration, mononucleotide, monotonicities, noncompetition, noncompetitors, nonconformists, nonconsumption.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.