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A nonnegative fraction with positive denominator and numerator smaller than its denominator, so its value lies in $[0,1)$. The zero value has the reduced fraction representation $0/1$. The term is sometimes extended to negative fractions by comparing absolute values; the nonnegative convention is the one used here.

The diamond-shaped floor of the fourth ventricle, formed by the dorsal pons and upper medulla oblongata.

For a finite complex measure on the additive circle group $\mathbb{R}/(2\pi \mathbb{Z})$, its Fourier coefficients are integrals against the group characters $e^{-inx}$, $n\in \mathbb{Z}$. They obey $|\hat{\mu }(n)|\le \Vert \mu \Vert$, where the norm is the total variation norm of a measure. A probability measure has zeroth coefficient one. The normalized Haar measure $m(dx)=dx/(2\pi )$ has all nonzero-frequency coefficients zero. A bounded family of finite Radon measures converges in the weak-star topology exactly when its coefficients converge to those of the proposed limit: extend linear combinations to all continuous tests using the density of trigonometric polynomials and the common variation bound.

Suppose every continuous periodic function $f$ has a unique continuous periodic $Tf$ with Fourier coefficients $\widehat{Tf}(n)={\alpha }_n\hat{f}(n)$. Uniqueness first makes $T$ linear. If $f_j\to f$ and $T{f}_j\to g$ uniformly, continuity of each coefficient functional gives $\hat{g}(n)={\alpha }_n\hat{f}(n)$ for every fixed $n$, hence $g=Tf$. Thus $T$ has a closed graph and the closed graph theorem makes it bounded. Applying it to normalized exponentials gives $|{\alpha }_n|\le \Vert T\Vert$.

For $e_n(t)=(2\pi {)}^{-1/2}{e}^{int}$, the bounded functional $\varphi (f)=\sqrt{2\pi }(Tf)(0)$ satisfies $\varphi (e_n)={\alpha }_n$. The normalization factor follows from $e_n(0)=(2\pi {)}^{-1/2}$.

For $F(N)=(N-1/2)(N-1)(2-N)+1/2$, every positive initial value with $F(N_0)>0$ has positive subsequent iterates. Indeed $F(N)<3/2$ for every $N>0$, since $F(N)-3/2=-N(2N^2-7N+7)/2<0$. On $[0,3/2]$, the negative part of the cubic product has magnitude at most $3/8$, so $F(N)\ge 1/8$. Thus the first positive image enters the forward-invariant interval $(0,3/2)$.

An invariant interval for a scalar map is an interval whose points remain in that interval under every forward iterate. Here invariance means forward invariance and does not assert surjectivity onto the interval.

The deprotonated counterpart of formic acid, a product of alkaline chloral cleavage.

Describe operations through laws relating their observable results, without selecting an internal representation. For an abstract data type, the laws must state the operation domains and error results as well as ordinary cases. A disjoint union can distinguish a successful value from an error with the same textual spelling.

Specify behavior by constructing a mathematical model and defining the operations on that model. Finite sequences model a FIFO queue: insertion appends, the head is the first element, and successful deletion removes it. An axiomatic specification can then be checked against the model without requiring the implementation to use that representation.

A conclusion is formally derivable from premises when a permitted formal proof exists with those premises as its undischarged assumptions. This is a syntactic relation, distinct from semantic entailment.

A formation rule specifies how well-formed expressions can be constructed from simpler expressions in a formal language. The formal grammar of propositional formulas has rules for propositional variables, negation and binary logical implication.

Blue fluorescence of suitably acidified quinine solutions, used in identifying quinine and illustrating fluorescence.

A small flower, especially one forming part of a dense inflorescence such as a flower head.