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A function that is not monotonic. In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [ 1 ][ 2 ][ 3 ] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non- increasing, or non- decreasing. In its simplest form, it says that a non-decreasing bounded -above sequence of real numbers converges to ...
Discontinuities of monotone functions. In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and there are at most countably many of them.
Then F and G form a monotone Galois connection between the power set of X and the power set of Y, both ordered by inclusion ⊆. There is a further adjoint pair in this situation: for a subset M of X, define H(M) = {y ∈ Y | f −1 {y} ⊆ M}. Then G and H form a monotone Galois connection between the power set of Y and the power set of X.
Cyclical monotonicity. In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function. [1][2]
A generalized quantifier GQ is said to be monotone increasing (also called upward entailing) if, for every pair of sets X and Y, the following holds: if , then GQ(X) entails GQ(Y). The GQ every boy is monotone increasing. For example, the set of things that run fast is a subset of the set of things that run.
Synchrony and diachrony are two complementary viewpoints in linguistic analysis. A synchronic approach (from Ancient Greek: συν- "together" and χρόνος "time") considers a language at a moment in time without taking its history into account. Synchronic linguistics aims at describing a language at a specific point of time, often the present.
From 1992 to 1993, Fogg was "one of the founders of the Student Review, Brigham Young University's independent student newspaper" and "taught English and design at BYU." [13] While at BYU, Fogg published eight short stories and poems in Dialogue: A Journal of Mormon Thought; [14] [15] [16] Sunstone, "a quarterly journal of Mormon experience, scholarship, issues, and art"; [17] [18] and other ...