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In formal expressions, the ordinal number used before the word order refers to the highest power in the series expansion used in the approximation. The expressions: a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth are used as fixed phrases. The expression a zero-order approximation is also ...
French also uses the indicators d and de for the alternative second ordinal number (masculine 2 d – second; feminine 2 de – seconde). In plural, all these indicators are suffixed with an s: ers (1 ers – premiers), res (1 res – premières), es (2 es – deuxièmes), ds (2 ds – seconds), des (2 des – secondes).
[6] [7] [a] The parentheses can be omitted if the input is a single numerical variable or constant, [2] as in the case of sin x = sin(x) and sin π = sin(π). [a] Traditionally this convention extends to monomials; thus, sin 3x = sin(3x) and even sin 1 / 2 xy = sin(xy/2), but sin x + y = sin(x) + y, because x + y is not a monomial ...
If α is any ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. This concept, a transfinite sequence (if α is infinite) or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α = ω, while a finite α corresponds to a tuple, a.k.a. string
A curve (top) is filled according to two rules: the even–odd rule (left), and the non-zero winding rule (right). In each case an arrow shows a ray from a point P heading out of the curve. In the even–odd case, the ray is intersected by two lines, an even number; therefore P is concluded to be 'outside' the curve.
English: Diagram showing how a curve (top) is filled according to two rules: the even-odd rule (left), and the non-zero winding rule (right). This is relevant to two-dimensional computer graphics. This is relevant to two-dimensional computer graphics.
If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus ℵ 1 is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set ω 1 : Any countable subset of ω 1 has an upper bound in ω 1 (this follows from the fact that the union of a ...
The (full) second-order induction scheme consists of all instances of this axiom, over all second-order formulas. One particularly important instance of the induction scheme is when φ is the formula " n ∈ X {\displaystyle n\in X} " expressing the fact that n is a member of X ( X being a free set variable): in this case, the induction axiom ...