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content/notes/math/algebra.mdx

@@ -18,9 +18,9 @@ showToc: true
 \begin{document}
 
 \begin{tikzcd}[row sep=large, column sep=large, every label/.append style = {font = \scriptsize}]
-  A \arrow[r, "f"] \arrow[dr, "f\circ g" swap] \arrow[in=120, out=150, loop, "\mathrm{id}_A"] &
-  B \arrow[d, "g"] \arrow[in=30, out=60, loop, "\mathrm{id}_B"] \\
-  & C \arrow[in=300, out=330, loop, "\mathrm{id}_C"]
+  A \arrow[r, "f"] \arrow[dr, "f\circ g" swap] \arrow[in=120, out=150, loop, "\operatorname{id}_A"] &
+  B \arrow[d, "g"] \arrow[in=30, out=60, loop, "\operatorname{id}_B"] \\
+  & C \arrow[in=300, out=330, loop, "\operatorname{id}_C"]
 \end{tikzcd}
 
 \end{document}
@@ -33,19 +33,19 @@ A category $\mathcal{C}$ consists of
 - a class $\hom(\mathcal{C})$ of *morphism*
 - for each morphism $f\in\hom(\mathbb{C})$, two objects $A,B\in\mathrm{ob}(\mathbb{C})$ called the *source* and *target* of $f$, respectively
 - for each $A,B,C\in\mathrm{ob}(\mathcal{C})$ a function $\circ: \hom_\mathcal{C}(B,C)\times\hom_\mathcal{C}(A,B)\to\hom_\mathcal{C}(A,C)$ defined by $(f,g)\mapsto g\circ f$ called *composition*
-- for each $A\in\mathrm{ob}(\mathcal{C})$ an element $\mathrm{id}_A\in \hom_\mathcal{C}(A,A)$ called the *identity* on $A$
+- for each $A\in\mathrm{ob}(\mathcal{C})$ an element $\operatorname{id}_A\in \hom_\mathcal{C}(A,A)$ called the *identity* on $A$
 
 satisfying
 - $(h\circ g)\circ f = h \circ (g \circ f)$ for each $f\in\mathcal{C}(A,B)$, $g\in\mathcal{C}(B,C)$ and $h\in\mathcal{C}(C,D)$ **(associativity)**
-- $f\circ \mathrm{id}_A = f = \mathrm{id}_B \circ f$ for each $f\in\mathcal{A}$ **(identity law)**
+- $f\circ \operatorname{id}_A = f = \operatorname{id}_B \circ f$ for each $f\in\mathcal{A}$ **(identity law)**
 </MathBox>
 
 The following notation is commonly used
 - $A\in\mathcal{C}$ instead of $A\in\mathrm{ob}(\mathcal{C})$
 - $\mathcal{C}(A,B)$ instead of $\hom_\mathcal{C}(A,B)$
 - $f:A\to B$ or $A \xrightarrow{f} B$ instead of $f\in\hom_\mathcal{C}(A,B)$
 - $gf$ instead of $g\circ f$
-- $1_A$ instead of $\mathrm{id}_A$
+- $1_A$ instead of $\operatorname{id}_A$
 
 A category $\mathcal{C}$ in which both $\mathrm{ob}(\mathbf{C})$ and $\hom(\mathcal{C})$ are sets is called a *small category*. A category in which each class of morphisms $\hom_\mathcal{C}(A,B)$ is a set is called *locally small*. 
 

content/notes/math/dynamics.mdx

@@ -22,7 +22,7 @@ The evolution operator satisfies the group properties
 
 For a fixed $x \in \mathcal{M}$, called the initial state, the evolution operator forms a map $t \mapsto\Phi^t (x) = \Phi(x, t)$ that defines a flow in the phase space. The map $t \mapsto\Phi^t$ is a group morphism of $\R$, as an additive group, to the group of bijections of $\mathcal{M}$, where the group operation is composition of maps. Thus $\Phi^t$ has the following properties
 
-1. Identity: $\Phi^0 = \mathrm{id}(\mathcal{M})$
+1. Identity: $\Phi^0 = \operatorname{id}(\mathcal{M})$
 2. Closure: $\Phi^s \circ \Phi^t = \Phi^{s + t}$
 3. Associativity: $\Phi^r \circ \left(\Phi^s \circ \Phi^t \right) = \left(\Phi^r \circ \Phi^s \right) \circ \Phi^t $
 4. Inverse: $\Phi^t \circ \Phi^{-t} = \Phi^{-t} \circ \Phi^t = \Phi^0$
@@ -163,13 +163,13 @@ where $[\cdot]$ indicates the $\sim$-equivalence class. If $\mathcal{M}$ is conn
 
 #### Suspension to cylinder or Möbius strip
 
-Consider two dynamical systems with state space $M = (-1, 1)$, time set $T = \Z$ and evolution operators $\Phi_\pm^1 = \pm\mathrm{id}$. To construct the suspension we consider $M \times [0, 1] = (-1, 1) \times [0, 1]$, which forms a rectangle.
+Consider two dynamical systems with state space $M = (-1, 1)$, time set $T = \Z$ and evolution operators $\Phi_\pm^1 = \pm\operatorname{id}$. To construct the suspension we consider $M \times [0, 1] = (-1, 1) \times [0, 1]$, which forms a rectangle.
 
-In the case with $\Phi_+^1 = \mathrm{id}$, the boundary points are $(x, 0)$ and $(x, 1)$, so that $\tilde{\mathcal{M}}$ forms a cylinder. In the case with $\Phi_-^1 = -\mathrm{id}$, the boundary points are $(x, 0)$ and $(-x, 1)$, which results in $\tilde{\mathcal{M}}$ forming the Möbius strip.
+In the case with $\Phi_+^1 = \operatorname{id}$, the boundary points are $(x, 0)$ and $(x, 1)$, so that $\tilde{\mathcal{M}}$ forms a cylinder. In the case with $\Phi_-^1 = -\operatorname{id}$, the boundary points are $(x, 0)$ and $(-x, 1)$, which results in $\tilde{\mathcal{M}}$ forming the Möbius strip.
 
 
 
-In both cases, the suspended evolutions are periodic. In the first case where $\Phi_+^1 = \mathrm{id}$, we have $\Phi^n (x) = x$. For any evolution starting at $[x, 0]$, the pairs $(x, t)$ and $(x', t')$ require a single revolution around the cylinder to become equivalent. This implies that the evolution has period $1$. In the second case where $\Phi_-^1 = -\mathrm{id}$, we have $\Phi^n (x) = (-1)^n x$. For an evolution starting at $[0, 0]$, the pairs $(0, t)$ and $(0, t')$ require a single revolution along the Möbius strip to become equivalent. However, for any evolution starting at $[x, 0]$, with $x \neq 0$, the pairs $(x, t)$ and $(x', t')$ require two revolutions along the Möbius strip to become equivalent. This implies that the evolution has period $2$.
+In both cases, the suspended evolutions are periodic. In the first case where $\Phi_+^1 = \operatorname{id}$, we have $\Phi^n (x) = x$. For any evolution starting at $[x, 0]$, the pairs $(x, t)$ and $(x', t')$ require a single revolution around the cylinder to become equivalent. This implies that the evolution has period $1$. In the second case where $\Phi_-^1 = -\operatorname{id}$, we have $\Phi^n (x) = (-1)^n x$. For an evolution starting at $[0, 0]$, the pairs $(0, t)$ and $(0, t')$ require a single revolution along the Möbius strip to become equivalent. However, for any evolution starting at $[x, 0]$, with $x \neq 0$, the pairs $(x, t)$ and $(x', t')$ require two revolutions along the Möbius strip to become equivalent. This implies that the evolution has period $2$.
 
 ### Poincaré map
 

content/notes/math/functional_analysis.mdx

@@ -354,7 +354,7 @@ Isometry is an equivalence relation on metric spaces that satisifies, for metric
 <details>
 <summary>Proof</summary> 
 
-1. The identity function $\mathrm{id}:X\to X$ is an isometry from $X$ to $X$.
+1. The identity function $\operatorname{id}:X\to X$ is an isometry from $X$ to $X$.
 2. if $X$ is isometric to $Y$, there is a bijective isometry $f:X\to Y$ with an isometric inverse $f^{-1}: Y\to X$. Hence $Y$ is isometric to $X$.
 3. if $X$ is isometric to $Y$ and $Y$ is isometric to $Z$, there are two bijective isometries $f: X\to Y$ and $g:Y\to Z$. Then the composition $g\circ f: X\to Z$ is an isometry from $X$ to $Z$.
 </details>