User:Inthar/Style guide: Difference between revisions
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This | == Math notation == | ||
== Variables == | This section documents my xen math notation and its differences from conventional xen notation or conventional math notation. | ||
=== Variables === | |||
<!--* Capital italicized Latin letters may denote scales written cumulatively: i.e. with ''S''(0) = '''0''' and ''S''(''i'' + ''p'') = '''E''' + ''S''(''i'') (''p'' = length, '''E''' = equave) for every ''i''. | <!--* Capital italicized Latin letters may denote scales written cumulatively: i.e. with ''S''(0) = '''0''' and ''S''(''i'' + ''p'') = '''E''' + ''S''(''i'') (''p'' = length, '''E''' = equave) for every ''i''. | ||
** ''S''(''n'') = 100''n'' cents | ** ''S''(''n'') = 100''n'' cents | ||
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** Blackdye is <math>\mathsf{Flought}(\mathrm{Pyth}[5]; 10/9)</math> | ** Blackdye is <math>\mathsf{Flought}(\mathrm{Pyth}[5]; 10/9)</math> | ||
== Discrete sets == | === Discrete sets === | ||
* For <math>k \in \mathbb{R}</math> and <math>n\in \mathbb{Z}_{>0},</math> <math>[n]_k</math> denotes <math>\{k, k+1, ..., k+n-1\}.</math> I may also use <math>[i:j]</math> for <math>[j-i]_i.</math> For ''n'' = 0, [0]<sub>k</sub> is the empty set. | * For <math>k \in \mathbb{R}</math> and <math>n\in \mathbb{Z}_{>0},</math> <math>[n]_k</math> denotes <math>\{k, k+1, ..., k+n-1\}.</math> I may also use <math>[i:j]</math> for <math>[j-i]_i.</math> For ''n'' = 0, [0]<sub>k</sub> is the empty set. | ||
== Words == | === Words === | ||
* Zero-indexing is used for indices. | * Zero-indexing is used for indices. | ||
* A ''(linear) word'' is a function <math>w : [n]_0 \to \mathcal{A}</math> where <math>\mathcal{A}</math> is a set of letters and <math>n \in \mathbb{Z}_{\ge 0}.</math> ''n'' is called the ''length'' of ''w''. The letter of ''w'' at index ''i'' is denoted ''w''[''i'']. If 0 ≤ ''i'' < ''j'' ≤ |''w''| − 1, the slice notation ''w''[''i'':''j''] denotes the (''j'' − ''i'')-letter word ''w''[''i'']''w''[''i''+1]...''w''[''j''−1]. | * A ''(linear) word'' is a function <math>w : [n]_0 \to \mathcal{A}</math> where <math>\mathcal{A}</math> is a set of letters and <math>n \in \mathbb{Z}_{\ge 0}.</math> ''n'' is called the ''length'' of ''w''. The letter of ''w'' at index ''i'' is denoted ''w''[''i'']. If 0 ≤ ''i'' < ''j'' ≤ |''w''| − 1, the slice notation ''w''[''i'':''j''] denotes the (''j'' − ''i'')-letter word ''w''[''i'']''w''[''i''+1]...''w''[''j''−1]. | ||
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* Substitution: If ''w'' is a linear or based circular word in '''X''' and possibly other letters, and ''u'' is a based circular word, then <math>\mathsf{subst}(w, \mathbf{X}, u)</math> denotes the word ''w'' but with the ''i''th occurrence of '''X''' replaced with ''u''[''i''] (for ''i'' ≥ 0). | * Substitution: If ''w'' is a linear or based circular word in '''X''' and possibly other letters, and ''u'' is a based circular word, then <math>\mathsf{subst}(w, \mathbf{X}, u)</math> denotes the word ''w'' but with the ''i''th occurrence of '''X''' replaced with ''u''[''i''] (for ''i'' ≥ 0). | ||
== Algebraic structures == | === Algebraic structures === | ||
* <math>\mathrm{JI}\langle p_1, ..., p_r \rangle</math> is the ''p''<sub>1</sub>.[...].''p''<sub>''r''</sub> subgroup, the subgroup of <math>(\mathbb{Q}_{>0}, \cdot)</math> generated by rationals <math>p_1, ..., p_r.</math> For not-necessarily-JI generators, <math>\mathrm{M}\langle p_1, ..., p_r \rangle</math> is used. | * <math>\mathrm{JI}\langle p_1, ..., p_r \rangle</math> is the ''p''<sub>1</sub>.[...].''p''<sub>''r''</sub> subgroup, the subgroup of <math>(\mathbb{Q}_{>0}, \cdot)</math> generated by rationals <math>p_1, ..., p_r.</math> For not-necessarily-JI generators, <math>\mathrm{M}\langle p_1, ..., p_r \rangle</math> is used. | ||
* If ''R'' is a commutative ring with 1, <math>R^r\langle a_1, ..., a_r\rangle</math> is the rank-''r'' free ''R''-module generated by basis elements <math>a_1, ..., a_r.</math> Ordered tuples in such modules are assumed to be in the given basis. Example: <math>\mathbf{m} + 3\mathbf{s} = (0,1,3) \in \mathbb{Z}^3\langle \mathbf{L}, \mathbf{m}, \mathbf{s}\rangle</math> | * If ''R'' is a commutative ring with 1, <math>R^r\langle a_1, ..., a_r\rangle</math> is the rank-''r'' free ''R''-module generated by basis elements <math>a_1, ..., a_r.</math> Ordered tuples in such modules are assumed to be in the given basis. Example: <math>\mathbf{m} + 3\mathbf{s} = (0,1,3) \in \mathbb{Z}^3\langle \mathbf{L}, \mathbf{m}, \mathbf{s}\rangle</math> | ||
== Miscellaneous | === Miscellaneous === | ||
* <math>\log</math> with no subscript is base e. | * <math>\log</math> with no subscript is base e. | ||
== Other style guidelines == | == Other style guidelines == | ||