Steps to reproduce
- enter restricted mode in any vault
- write the following file:
## part d
- evaluate $\pi$ analytically using $N = 10, \beta = \gamma = 0.5, \delta = 1$:
- compute $B$:
$$
B = \sum_{n = 1}^N b_n
= \sum_{n = 1}^N \phi_n S_n
= 2 \sum_{n = 1}^{10} \sum_{j = 1}^n {10 \choose n} \frac1{j}
= \frac{1442426}{315} \approx 4579.13
$$
- compute $\pi_n$:
$$
\begin{align}
\pi_n
&= C a_n - J b_n \\
&= C a_n - \left(\delta \frac{C a_N}{1 - \delta b_N}\right) b_n \\
&= C \left(a_n - \delta \frac{a_N}{1 - \delta b_N} b_n\right) \\
&= \frac
{a_n - \delta \frac{a_N}{1 - \delta b_N} b_n}
{A - \delta \frac{a_N}{1 - \delta b_N} B}
\\
&= \frac
{a_n (1 - \delta b_N) - \delta a_N b_n}
{A (1 - \delta b_N) - \delta a_N B}
\\
&\approx \frac
{{10 \choose n} (1 - (1) (5.85794)) - (1) (1) \left(2 {10 \choose n} \sum_{j = 1}^n \frac1{j}\right)}
{(1024) (1 - (1) (5.85794)) - (1) (1) (4579.13)}
\\
\end{align}
$$
$$
\begin{align}
\Rightarrow \pi
\approx [
&0.0017191,
-0.046895,
-1.364593,
-10.30395,
-32.16308,
-46.72448, \\
&-32.61169,
-10.67016,
-1.488189,
-0.067240,
0.0002339,
]
\end{align}
$$
...that's not right T-T
TODO code
# problem 4
## part a
- show irreducibility:
- all states are connected in a ring
- -> can get from any one state to any other state by following connections around the ring
- -> there is some non-zero probability $p \geq \frac1{2^k} > 0$ of moving from any $i$ to any $j$ after $k_+ := j - i \mod N$ steps upwards, or $k_- := N - (j - i \mod N)$ steps downwards
- -> $\forall i, j: i \to j$
- -> $P$ is irreducible
- show aperiodicity:
- $P_{1, 1} = \delta > 0$
- -> state $1$ maps to itself
- -> $T_i = \gcd\{1, ...\} = 1$
- -> period of $P$ is $1$
- -> $P$ is aperiodic
- find stationary distribution via detailed balance:
- for $i \notin \{0, N\}$:
$$
\begin{align}
\pi_i P_{i, i+1} &= \pi_{i+1} P_{i+1, i} \\
\to \frac12 \pi_{i} &= \frac12 \pi_{i+1} \\
\to \pi_{i} &= \pi_{i+1} \\
\end{align}
$$
- for $i = 0$:
$$
\begin{align}
\pi_0 P_{0, 1} &= \pi_{1} P_{1, 0} \\
\to \frac12 (1 - \delta) \pi_0 &= \frac12 \pi_{1} \\
\to \pi_1 &= (1 - \delta) \pi_0 \\
\end{align}
$$
- for $i = N$: same as $i = 0$:
$$
\to \pi_N = (1 - \delta) \pi_0
$$
- normalize:
$$
\begin{align}
1 = \sum_{n = 1}^N \pi_n
&= \pi_0 (1 + (1 - \delta) (N-1)) \\
&= \pi_0 (N - \delta(N - 1)) \\
\end{align}
$$
$$
\begin{align}
\to \pi_0
&= \frac1{N - \delta(N - 1)} \\
\to \forall i \neq 0,
\pi_i &= \frac{1 - \delta}{N - \delta(N - 1)}
\\
\end{align}
$$
- generate a pdf of the above text with the “Export PDF” tool
- view the pdf
Did you follow the troubleshooting guide? YES
Expected result
the pdf should render correctly, identically to how it’s rendered in write mode:
- all bullets should look correct
- all latex blocks should be correctly rendered.
- no part of the file is rendered as a literal text code block (since that isn’t used in this file)
Actual result
- none of the second-indentation bullets in the first header “part d” aren’t rendered correctly
- the last latex block in “part d” gives error text
- everything below the latex error is shown as a literal code text block
Environment
SYSTEM INFO:
Obsidian version: v1.9.14
Installer version: v1.8.7
Operating system: Darwin Kernel Version 24.6.0: Mon Jul 14 11:30:34 PDT 2025; root:xnu-11417.140.69~1/RELEASE_ARM64_T8103 24.6.0
Login status: not logged in
Language: en
Insider build toggle: off
Live preview: on
Base theme: adapt to system
Community theme: none
Snippets enabled: 0
Restricted mode: on
RECOMMENDATIONS:
none
Additional information
a screenshot of the bug:
- bug workaround: if you add a newline before the offending latex block (i.e., between lines 30 and 31), the rendering bug goes away