# User:Prime.mover/Constructs

## Useful Links to pages that use the Transclusion Extension

- ... and some proof structures

## Useful constructs

Useful constructs for anyone to cut and paste:

{{TableauLine | n = | pool = | f = | rlnk = | rtxt = | dep = | c = }}

`{{EndSequence}}`

Structure of simple conditional within template:

`{{#if: {{{param|}}} |{{{param}}}|}}`

Blackboard characters: $\N \Z \Q \R \C \P \S$

$fred := bert$

The sequence of ... begins:

subsuming $...$ into arbitrary constant

For how to use substack:

- $\displaystyle \sum_{\map \Phi j} a_j = \paren {\lim_{n \mathop \to \infty} \sum_{\substack {\map \Phi j \\ -n \mathop \le j \mathop < 0} } a_j} + \paren {\lim_{n \mathop \to \infty} \sum_{\substack {\map \Phi j \\ 0 \mathop \le j \mathop \le n} } a_j}$

Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric progression of integers.

Let $M_1 = \struct {A_1, d_1}, M_2 = \struct {A_2, d_2}, \ldots, M_n = \struct {A_n, d_n}$ be metric spaces.

Let $\displaystyle \AA = \prod_{i \mathop = 1}^n A_i$ be the cartesian product of $A_1, A_2, \ldots, A_n$.

Let $d_\infty: \AA \times \AA \to \R$ be the Chebyshev distance on $\AA$:

- $\displaystyle \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, y_i} }$

where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \AA$.

Let $A$ be an algebra over the field $\R$ whose bilinear map $m: A^2 \to A$ is called **multiplication**.

Let the unity of $A$ be $1$ such that $\forall a \in A: \map m {1, a} = a = \map m {a, 1}$.

We can abbreviate $\map m {a, b}$ as $a b$.

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $X$ be transcendental over $F$.

Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$.

Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $\struct {D, +, \circ}$ be an integral subdomain of $R$.

For arbitrary $x \in R$, let $D \sqbrk x$ be the ring of polynomials in $x$ over $D$.

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {S, +, \circ}$ be a subring of $R$.

For arbitrary $x \in R$, let $S \sqbrk x$ be the set of polynomials in $x$ over $S$.

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $\floor x$ denote the floor of $x$.

Let $\ceiling x$ denote the ceiling of $x$.

Let $T = \struct {S, \tau}$ be a topological space.

Let $M = \struct {A, d}$ be a metric space.

Let $a \in A$.

Let $\map {B_\epsilon} {a; d}$ be an open $\epsilon$-ball of $a$ in $M$.

Let $\xi \in \R$ be a real number.

Let $\displaystyle \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$.

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.

Let $f$ have a primitive $F$ on $\closedint a b$.

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a convergent series in $\R$.

Let $\sequence {s_n}$ be the sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

Let $\sequence {x_n}=$ be a sequence in $\R$.

Let $\sequence {x_n}=$ be a Cauchy sequence.

Let $\displaystyle \lim_{n \mathop \to \infty} x_n = l$.

Let $x_n \to l$ as $n \to \infty$.

Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$.

Let $\mathbf A = \left[{a}\right]_{m n}$ be an $m \times n$ matrix.

Let $\mathbf A = \left[{a}\right]_n$ be a square matrix of order $n$.

Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.

Let $\map {\MM_S} {m, n}$ be the $m \times n$ matrix space over $S$.

Let $\set {x, y, z}$ be a set.

Let $\powerset S$ be the power set of the set $S$.

Let $\struct {S, \circ}$ be an algebraic structure or a semigroup.

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $\struct {S, \circ, *}$ be a Huntington algebra whose identity for $\circ$ is $e^\circ$ and whose identity for $*$ is $e^*$.

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $\struct {K, +, \circ}$ be a division ring whose zero is $0_K$ and whose unity is $1_K$.

Let $\gen S$ be the subgroup generated by $S$.

Let $\gen g = \struct {G, \circ}$ be a cyclic group.

Let $\struct {G, +_G, \circ}_R$ be an $R$-module.

Let $\struct {G, +_G, \circ}_K$ be a $K$-vector space.

Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module whose dimension is finite.

Let $\map {\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$.

Let $\map {\LL_R} G$ be the set of all linear operators on $G$.

Let $\sqbrk {u; \sequence {b_m}, \sequence {a_n} }$ be the matrix of $u$ relative to $\sequence {a_n}$ and $\sequence {b_m}$.

Let $D \sqbrk X$ be the ring of polynomial forms in $X$ over $D$.

Let $\map P D$ be the ring of polynomial functions over $D$.

Let $G^*$ be the algebraic dual of $G$.

Let $G^{**}$ be the algebraic dual of $G^*$.

Let $M^\circ$ be the annihilator of $M$.

Let $\gen {x, t'}$ be as defined in Definition:Evaluation Linear Transformation.

Let $J$ be an ideal of $R$.

Let $\struct {R / J, +, \circ}$ be the quotient ring defined by $J$.

Let $\struct {D, +, \circ}$ be an integral domain or a principal ideal domain whose zero is $0_D$ and whose unity is $1_D$.

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $\struct {K, +, \circ}$ be a quotient field of an integral domain $\struct {D, +, \circ}$.

Let $\struct {D, +, \circ, \le}$ be a totally ordered integral domain whose zero is $0_D$ and whose unity is $1_D$.

Let $\struct {S, \preceq}$ be a totally ordered set.

Let $\struct {S, \circ, \preceq}$ be an ordered structure.

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

$\closedint m n$ is the closed interval between $m$ and $n$.

$\N$, $\N_{> 0}$, $\N_k$, $\N^*_k$

$\Z$, $\Z_{\ne 0}$, $\Z_{\ge 0}$, $\Z_{> 0}$,

Let $\Z_m$ be the set of integers modulo $m$.

Let $\Z'_m$ be the reduced residue system modulo $m$.

Let $\struct {\Z, +}$ be the additive group of integers.

Let $\struct {\Z, +, \times}$ be the integral domain of integers.

Let $\struct {\Z_m, +_m, \times_m}$ be the ring of integers modulo $m$.

Let $\struct {\Z_m, +_m}$ be the additive group of integers modulo $m$.

Let $n \Z$ be the set of integer multiples of $n$.

Let $\ideal x$ be the principal ideal of $\struct {\Z, +, \times}$ generated by $x$.

Let $\Char R$ be the characteristic of $R$.

The cardinality of a set $S$ is written $\card S$.

Let $\sequence {s_k}_{k \mathop \in A}$ be a sequence in $S$.

Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.

Let $\lcm \set {a, b}$ be the lowest common multiple of $a$ and $b$.

Let $\size a$ be the absolute value of $a$.

$a \equiv b \pmod m$ "$a$ is *congruent to $b$ modulo $m$*."

$\eqclass a m$ is the residue class of $a$ (modulo $m$).

Let $\index G H$ be the index of $H$ in $G$.

Let $C_G \paren H$ be the centralizer of $H$ in $G$.

Let $N_G \paren S$ be the normalizer of $S$ in $G$.

Let $G / N$ be the quotient group of $G$ by $N$.

Let $Z \paren G$ be the center of $G$.

Let $x \in G$.

Let $N_G \paren x$ be the normalizer of $x$ in $G$.

Let $\index G {N_G \paren x}$ be the index of $N_G \paren x$ in $G$.

Let $S_n$ denote the set of permutations on $n$ letters.

Let $S_n$ denote the symmetric group on $n$ letters.

Let $\Fix \pi$ denote the set of elements fixed by $\pi$.

Matrix (square brackets): $\begin{bmatrix} x & y \\ z & v \end{bmatrix}$

Matrix (round brackets): $\begin{pmatrix} x & y \\ z & v \end{pmatrix}$

two-row notation: $\begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{bmatrix}$

cycle notation: $\begin{bmatrix} x & y \end{bmatrix}$

Let $\Orb x$ be the orbit of $x$.

Let $\Stab x$ be the stabilizer of $x$ by $G$.

Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ be an $R$-algebraic structure.

## URM Programs

Line | Command | Comment | ||
---|---|---|---|---|

$1$ | $\map Z n$ | |||

$2$ | $\map S n$ | |||

$3$ | $\map C {m, n}$ | |||

$4$ | $\map J {m, n, q}$ |

...etc.

Let $P$ be a URM program.

Let $P$ be a normalized URM program.

Let $l = \map \lambda P$ be the number of basic instructions in $P$.

Let $u = \map \rho Q$ be the number of registers used by $Q$.

Trace Table:

Stage | Instruction | $R_1$ | $R_2$ | $R_3$ |
---|---|---|---|---|

$0$ | $1$ | $r_1$ | $r_2$ | $r_3$ |

$1$ | $2$ | $r_1$ | $r_2$ | $r_3$ |

...etc.