Wiki Syntax

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# Theorems and Definitions

On this wiki in particular, it is possible to create vanity boxes for popular theorems and definitions. These are most useful if created as their own pages starting with theorem:<theorem-name> and definition:<definition-name>.

For instance, here is a page containing a box for the Van Kampen theorem: theorem:van-kampen which is then included on the page about Van Kampen Theorem proper.

Once a unique theorem:<theorem-name> or definition:<definition-name> page is created it can be included with a regular [[include theorem<theorem-name>]] escape. For example:

[[include theorem:van-kampen]]

Produces:

 Van Kampen Theorem Theorem. Let $A, B \subset X$, with intersection $A \overset{\iota_A}{\hookleftarrow} A \cap B \overset{\iota_B}{\hookrightarrow} B$ be open and path-connected with finitely-presented fundamental groups: $$\pi_1(A) \cong \langle a_1, \cdots, a_i\; \vert \; \alpha_1, \cdots, a_n \rangle$$ $$\pi_1(B) \cong \langle b_1, \cdots, b_j\; \vert \; \beta_1, \cdots, \beta_m\rangle$$$$\pi_1(A \cap B) \cong \langle g_1, \cdots, g_k\; \vert \; \gamma_1, \cdots, \gamma_l\rangle$$ Then union $A \cup B$ has fundamental group with presentation: $$\pi_1(A \cup B) = \langle a_1, \cdots, a_i, b_1, \cdots, b_j\; \vert \; \alpha_1, \cdots, \alpha_n, \beta_1, \cdots, \beta_m, \iota_A(g_1)\iota_B(g_1)^{-1}, \cdots, \iota_A(g_k)\iota_B(g_k)^{-1} \rangle$$

## Creating new Theorem Box

[[include inc:theorem
title=Equipartition Theorem|
body=For a system with a [[[Hamiltonian]]] [[$H(\mathbf{p}, \mathbf{q})$]] with generalized coordinates [[$\mathbf{q}$]] and momenta [[$\mathbf{p}$]]:[[$$\left\langle q_i \frac{\partial H}{\partial p_i} \right\rangle = k T$$]] for each quadratic degree of freedom, where [[$\langle \cdot \rangle$]] denotes the canonical [[[ensemble average]]].|
]]

Produces:

 Equipartition Theorem Theorem. For a system with a Hamiltonian $H(\mathbf{p}, \mathbf{q})$ with generalized coordinates $\mathbf{q}$ and momenta $\mathbf{p}$:$$\left\langle q_i \frac{\partial H}{\partial p_i} \right\rangle = k T$$ for each quadratic degree of freedom, where $\langle \cdot \rangle$ denotes the canonical ensemble average.

## Creating new Definition Box

Similarly,

[[include inc:definition
title=[[[Regular Cell Complex]]] |
body=A //regular cell complex// is a topological space [[$X$]] partitioned into subspaces [[$\{X_\alpha\}_{\alpha\in PX}$]] satisfying the following conditions:
# For each [[$x \in X$]], every sufficiently small neighborhood of [[$x$]] intersects finitely many [[$X_\alpha$]].
# For all [[$\alpha, \beta$]], [[$X_\alpha \cap X_\beta \neq \emptyset$]] only if [[$X_\beta \subseteq X_\alpha$]].
# Every [[$X_\alpha$]] is homeomorphic to [[$\mathbb{R}^{n_\alpha}$]] for some [[$n_\alpha$]].
# For every [[$\alpha$]], there is a homeomorphism of a closed ball in [[$\mathbb{R}^{n_\alpha}$]] to [[$X_\alpha$]] that maps the interior of the ball homeomorphically onto [[$X_\alpha$]].|]]

Will produce:

 Definition. Regular Cell Complex: A regular cell complex is a topological space $X$ partitioned into subspaces $\{X_\alpha\}_{\alpha\in PX}$ satisfying the following conditions: For each $x \in X$, every sufficiently small neighborhood of $x$ intersects finitely many $X_\alpha$. For all $\alpha, \beta$, $X_\alpha \cap X_\beta \neq \emptyset$ only if $X_\beta \subseteq X_\alpha$. Every $X_\alpha$ is homeomorphic to $\mathbb{R}^{n_\alpha}$ for some $n_\alpha$. For every $\alpha$, there is a homeomorphism of a closed ball in $\mathbb{R}^{n_\alpha}$ to $X_\alpha$ that maps the interior of the ball homeomorphically onto $X_\alpha$.
page revision: 4, last edited: 22 Aug 2023 06:46