diff --git a/src/content/algorithms/extrema-circular-buffer.mdx b/src/content/algorithms/extrema-circular-buffer.mdx
index 8250f0e..b2f628d 100644
--- a/src/content/algorithms/extrema-circular-buffer.mdx
+++ b/src/content/algorithms/extrema-circular-buffer.mdx
@@ -4,8 +4,7 @@ date: "30/07/2024"
useKatex: true
---
-import { Code } from 'astro:components';
-
+import { Code } from "astro:components";
# context
@@ -15,7 +14,6 @@ While working for [TRB Capital Management](https://trbcap.com/), certain strateg
Design a data structure supporting the following operations:
-
- `build(size_t capacity)` : initialize the data structure with capacity/window size `capacity`
The data structure must always hold $\leq$ `capacity` prices.
- `void push_back(double value)`
diff --git a/src/content/algorithms/leetcode-daily.mdx b/src/content/algorithms/leetcode-daily.mdx
index 7a53620..e22afc9 100644
--- a/src/content/algorithms/leetcode-daily.mdx
+++ b/src/content/algorithms/leetcode-daily.mdx
@@ -103,7 +103,7 @@ def countFairPairs(self, nums, lower, upper):
## optimizing the approach
-If we interpret the criteria this way, the above approach is relatively efficient. To improve this approach, we'll need to reinterpret the constraints. Forget about the indexing and consider the constraint in aggregate. We want to find all $i,j$ with $x=nums$i$+nums$j$$ such that $i\neq j,lower\leq x\leq upper$.
+If we interpret the criteria this way, the above approach is relatively efficient. To improve this approach, we'll need to reinterpret the constraints. Forget about the indexing and consider the constraint in aggregate. We want to find all $i,j$ with $x=nums[i]+nums[j]$ such that $i\neq j,lower\leq x\leq upper$.
We _still_ need to reduce the “dimensionality” of the problem—there are just too many moving parts to consider at once. This seems challening. Let's simplify the problem to identify helpful ideas: pretend `lower` does not exist (and, of course, that `nums` is sorted).
diff --git a/src/content/algorithms/models-of-production.mdx b/src/content/algorithms/models-of-production.mdx
index 5552192..e9c7b31 100644
--- a/src/content/algorithms/models-of-production.mdx
+++ b/src/content/algorithms/models-of-production.mdx
@@ -14,7 +14,9 @@ This post offers a basic introduction to the Solow, Romer, and Romer-Solow econo
The Solow Model is an economic model of production that incorporates the idea of capital accumulation. Based on the [Cobb-Douglas production function](https://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function), the Solow Model describes production as follows:
-$$Y_t=F(K_t,L_t)=\bar{A}K_t^\alpha L_t^{1-\alpha}$$
+$$
+Y_t=F(K_t,L_t)=\bar{A}K_t^\alpha L_t^{1-\alpha}
+$$
With:
@@ -41,7 +43,9 @@ Visualizing the model, namely output as a function of capital, provides helpful
Letting $(L_t,\alpha)=(\bar{L}, \frac{1}{3})$, it follows that:
-$$Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}} \bar{L}^{\frac{2}{3}}$$
+$$
+Y_t=F(K_t,L_t)=\bar{A}K_t^{\frac{1}{3}} \bar{L}^{\frac{2}{3}}
+$$
Utilizing this simplification and its graphical representation below, output is clearly characterized by the cube root of capital:
@@ -119,17 +123,20 @@ When investment is completely disincentivized by depreciation (in other words, $
Using this equilibrium condition, it follows that:
-$$Y_t^*=\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha}$$
-
-$$\rightarrow \bar{d}K_t^*=\bar{s}\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha}$$
-
-$$\rightarrow K^*=\bar{L}(\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}$$
-
-$$\rightarrow Y^*=\bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha}\bar{L}$$
+$$
+\begin{align*}
+Y_t^* &= \bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha} \\
+\bar{d}K_t^* &= \bar{s}\bar{A}{K_t^*}^\alpha\bar{L}^{1-\alpha} \\
+K^* &= \bar{L}(\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha} \\
+Y^* &= \bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha}\bar{L}
+\end{align*}
+$$
Thus, the equilibrium intensive form (output per worker) of both capital and output are summarized as follows:
-$$(k^*,y^*)=(\frac{K^*}{\bar{L}},\frac{Y^*}{\bar{L}}) =((\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}, \bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha})$$
+$$
+(k^*,y^*)=(\frac{K^*}{\bar{L}},\frac{Y^*}{\bar{L}}) =((\frac{\bar{s}\bar{A}}{\bar{d}})^\frac{1}{1-\alpha}, \bar{A}^\frac{1}{1-\alpha}(\frac{\bar{s}}{\bar{d}})^\frac{\alpha}{1-\alpha})
+$$
### analysis
@@ -138,7 +145,6 @@ Using both mathematical intuition and manipulating the visualization above, we f
- $\bar{A}$ has a positive relationship with steady-state output
- Capital is influenced by workforce size, TFP, and savings rate
- Capital output share's $\alpha$ impact on output is twofold:
-
1. Directly through capital quantity
2. Indirectly through TFP
@@ -147,7 +153,9 @@ Using both mathematical intuition and manipulating the visualization above, we f
Lastly (and perhaps most importantly), exogenous parameters $\bar{s}, \bar{d}$, and $\bar{A}$ all have immense ramifications on economic status. For example, comparing the difference in country $C_1$'s output versus $C_2$'s using the Solow Model, we find that a difference in economic performance can only be explained by these factors:
-$$\frac{Y_1}{Y_2}=\frac{\bar{A_1}}{\bar{A_2}}(\frac{\bar{s_1}}{\bar{s_2}})^\frac{\alpha}{1-\alpha}$$
+$$
+\frac{Y_1}{Y_2}=\frac{\bar{A_1}}{\bar{A_2}}(\frac{\bar{s_1}}{\bar{s_2}})^\frac{\alpha}{1-\alpha}
+$$
We see that TFP is more important in explaining the differences in per-capital output ($\frac{1}{1-\alpha}>\frac{\alpha}{1-\alpha},\alpha\in[0,1)$). Notably, the Solow Model does not give any insights into how to alter the most important predictor of output, TFP.
@@ -168,7 +176,9 @@ The Model divides the world into two parts:
The Romer Models' production function can be modelled as:
-$$Y_t=F(A_t,L_{yt})=A_tL_{yt}$$
+$$
+Y_t=F(A_t,L_{yt})=A_tL_{yt}
+$$
With:
@@ -179,7 +189,9 @@ Assuming $L_t=\bar{L}$ people work in the economy, a proportion $\bar{l}$ of the
Further, this economy garners ideas with time at rate $\bar{z}$: the "speed of ideas". Now, we can describe the quantity of ideas tomorrow as function of those of today: the Law of Ideal Motion (I made that up).
-$$A_{t+1}=A_t+\bar{z}A_tL_{at}\leftrightarrow\Delta A_{t+1}=\bar{z}A_tL_{at}$$
+$$
+A_{t+1}=A_t+\bar{z}A_tL_{at}\leftrightarrow\Delta A_{t+1}=\bar{z}A_tL_{at}
+$$
Analagously to capital in the solow model, ideas begin in the economy with some $\bar{A}_0\gt0$ and grow at an _exponential_ rate. At its core, this is because ideas are non-rivalrous; more ideas bring about more ideas.
@@ -280,19 +292,27 @@ Playing with the sliders, this graph may seem underwhelming in comparison to the
To find the output in terms of exogenous parameters, first note that
-$$L_t=\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}$$
+$$
+L_t=\bar{L}\rightarrow L_{yt}=(1-\bar{l})\bar{L}
+$$
Now, all that remains is to find ideas $A_t$. It is assumed that ideas grow at some rate $g_A$:
-$$A_t=A_0(1+g_A)^t$$
+$$
+A_t=A_0(1+g_A)^t
+$$
Using the growth rate formula, we find:
-$$g_A=\frac{\Delta A_{t+1}-A_t}{A_t}=\frac{A_t+\bar{z}A_tL_{at}-A_t}{A_t}=\bar{z}L_{at}=\bar{z}\bar{l}\bar{L}$$
+$$
+g_A=\frac{\Delta A_{t+1}-A_t}{A_t}=\frac{A_t+\bar{z}A_tL_{at}-A_t}{A_t}=\bar{z}L_{at}=\bar{z}\bar{l}\bar{L}
+$$
Thus, ideas $A_t=A_0(1+\bar{z}\bar{l}\bar{L})^t$. Finally, output can be solved the production function:
-$$Y_t=A_t L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})^t(1-\bar{l})\bar{L}$$
+$$
+Y_t=A_t L_{yt}=A_0(1+\bar{z}\bar{l}\bar{L})^t(1-\bar{l})\bar{L}
+$$
### analysis
@@ -385,7 +405,9 @@ From previous analysis it was found that $g_A=\bar{z}\bar{l}\bar{L}$.
Based on the Law of Capital Motion,
-$$g_K=\frac{\Delta K_{t+1}}{K_t}=\bar{s}\frac{Y_t}{K_t}-\bar{d}$$
+$$
+g_K=\frac{\Delta K_{t+1}}{K_t}=\bar{s}\frac{Y_t}{K_t}-\bar{d}
+$$
Because growth rates are constant on the Balanced Growth Path, $g_K$ must be constant as well. Thus, so is $\bar{s}\frac{Y_t}{K_t}-\bar{d}$; it must be that $g_K^*=g_Y^*$.
@@ -393,24 +415,35 @@ The model assumes population is constant, so $g_{\bar{L}}=0\rightarrow g_{\bar{L
Combining these terms, we find:
-$$g_Y^*=\bar{z}\bar{l}\bar{L}+\alpha g_Y^*+(1-\alpha)\cdot 0\rightarrow$$
-$$g_Y^*=\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}$$
+$$
+\begin{align*}
+g_Y^* &= \bar{z}\bar{l}\bar{L}+\alpha g_Y^*+(1-\alpha)\cdot 0 \\
+g_Y^* &= \frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}
+\end{align*}
+$$
Solving for $Y_t^*$ is trivial after discovering $g_K=g_Y$ must hold on a balanced growth path.
Invoking the Law of Capital Motion with magic chants,
-$$g_K^*=\bar{s}\frac{Y_t^*}{K_t^*}-\bar{d}=g_Y^*\rightarrow K_t^*=\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}}$$
+$$
+g_K^*=\bar{s}\frac{Y_t^*}{K_t^*}-\bar{d}=g_Y^*\rightarrow K_t^*=\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}}
+$$
Isolating $Y_t^*$,
-$$Y_t^*=A_t^* (\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}})^\alpha ({(1-\bar{l})\bar{L}})^{1-\alpha}$$
-
-$$\rightarrow {Y_t^*}^{1-\alpha}=A_t^*(\frac{\bar{s}}{g_Y^*+\bar{d}})^\alpha({(1-\bar{l})\bar{L}})^{1-\alpha}$$
+$$
+\begin{align*}
+Y_t^* &= A_t^* (\frac{\bar{s}Y_t^*}{g_Y^*+\bar{d}})^\alpha ({(1-\bar{l})\bar{L}})^{1-\alpha} \\
+{Y_t^*}^{1-\alpha} &= A_t^*(\frac{\bar{s}}{g_Y^*+\bar{d}})^\alpha({(1-\bar{l})\bar{L}})^{1-\alpha}
+\end{align*}
+$$
Plugging in the known expressions for $A_t^*$ and $g_Y^*$, a final expression for the Balanced Growth Path output as a function of the endogenous parameters and time is obtained:
-$$Y_t^*={(A_0(1+\bar{z}\bar{l}\bar{L})^t})^\frac{1}{1-\alpha}(\frac{\bar{s}}{\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}+\bar{d}})^\frac{\alpha}{1-\alpha}(1-\bar{l})\bar{L}$$
+$$
+Y_t^*={(A_0(1+\bar{z}\bar{l}\bar{L})^t})^\frac{1}{1-\alpha}(\frac{\bar{s}}{\frac{\bar{z}\bar{l}\bar{L}}{1-\alpha}+\bar{d}})^\frac{\alpha}{1-\alpha}(1-\bar{l})\bar{L}
+$$
### analysis
diff --git a/src/content/algorithms/proofs.mdx b/src/content/algorithms/proofs.mdx
index f867cea..bdbc839 100644
--- a/src/content/algorithms/proofs.mdx
+++ b/src/content/algorithms/proofs.mdx
@@ -5,7 +5,7 @@ useKatex: true
useD3: true
---
-import Pseudocode from '@components/Pseudocode.astro';
+import Pseudocode from "@components/Pseudocode.astro";
A computer science student attempting to learn proofs.
@@ -14,16 +14,19 @@ A computer science student attempting to learn proofs.
## E
Minimize-Digit-Diff($l, r$):
+
1. Initialize $ans=18$, the largest possible value of $f(l, x) + f(x, r)$ with $l \leq r\leq 10^9$
2. For $i:=1$ to $200$:
- - Let $x$ be a random sample from the open interval $[l, r]$
- - Let $cost:=f(l, x) + f(x, r)$, computed in $O(1)$ time
- - $ans:=min(ans,cost)$
+
+- Let $x$ be a random sample from the open interval $[l, r]$
+- Let $cost:=f(l, x) + f(x, r)$, computed in $O(1)$ time
+- $ans:=min(ans,cost)$
+
3. Return $ans$
---
-*Proof.*
+_Proof._
We are interested in minimizing the expected probability of failure over $t\leq10^4$ tests. Firstly, consider a single test (i.e. a single $(l,r)$ input):
@@ -32,7 +35,6 @@ Thus, a digit has approximately a $1-\frac{1}{5}=\frac{4}{5}$ probability of min
It follows that the candidate then has roughly a $(\frac{4}{5})^9\approx13\%$ chance of minimizing the entire score $f(l,x)+f(x,r)$.
So, one random sample $x$ has a $100-13\approx87\%$ chance of being a suboptimal candidate.
-
Consider sampling $n$ times--the probability of all random samples being suboptimal is $0.87^n$. Let's only settle for a probability of failure near $1/10^{10}$ and solve for our number of trials:
$$
@@ -57,14 +59,14 @@ Maximize-F(s):
---
-*Proof.*
+_Proof._
Let $X$ and $Y$ be the number of occurrences of zeroes and ones in the substring $p$ of $s$. Then, $f(p)=max(X, Y)=\frac{X+Y+|X-Y|}{2}$. The goal is to compute:
$$
\begin{align*}
-\sum_{1\leq r\leq n}\sum_{1\leq li}$, $|S_{j>i}-S_i|=S_{j>i}-S_i$--the sum is unchanged. Rewriting t
$$
\begin{align*}
-\sum_{1\leq r\leq n}\sum_{1\leq lI'm lazy: I grew tired of copying (and mis-copying) inputs, outputs, etc from online judges.
-- I'm lazy: I frequently submitted incorrect solutions after erroneously asserting that my outputs matched those of the sample test cases
-- External dependencies: it unsettles me that my bare-bones setup required copy-pasting an entire suite of scripts
-- Non-native NeoVim experience: while composition and the UNIX philosophy are great, there's only so much you can do with pipes and files.
- - Raw I/O files meant I couldn't see colored stdin/stdout
- - Fine-grained per-testcase I/O was suspect--isolating and running a subset of test cases required manual intervention
+- I'm lazy: I grew tired of copying (and mis-copying) inputs, outputs,
+ etc from online judges.
+- I'm lazy: I frequently submitted incorrect solutions after erroneously
+ asserting that my outputs matched those of the sample test cases
+- External dependencies: it unsettles me that my bare-bones setup
+ required copy-pasting an entire suite of scripts
+- Non-native NeoVim experience: while composition and the UNIX philosophy
+ are great, there's only so much you can do with pipes and files. - Raw I/O
+ files meant I couldn't see colored stdin/stdout - Fine-grained per-testcase
+ I/O was suspect--isolating and running a subset of test cases required manual
+ intervention
The solution was to leverage Neovim's great APIs to give me granular control over every aspect of my problem-solving experience.
@@ -25,22 +30,42 @@ The solution was to leverage Neovim's great APIs to give me granular control ove
The GitHub page documents the plugin well enough so I'll avoid re-hashing it here. Instead, what's more interesting to document is why I thought this was a worthwhile experience.
-1. Making Something Useful for Others: cp.nvim is an opportunity for me to make my first open-source project "right"--not some side project or demo, but a *real*, usable tool that I'll be rolling out to the public soon. I consider the following in my active development of the plugin:
+1. Making Something Useful for Others: cp.nvim is an opportunity for me
+ to make my first open-source project "right"--not some side project or demo,
+ but a *real*, usable tool that I'll be rolling out to the public soon. I
+ consider the following in my active development of the plugin:
-- Comprehensive continuous integration (*real* testing, linting, and more)
+- Comprehensive continuous integration (_real_ testing, linting, and more)
- [LuaRocks](https://luarocks.org/) integration (the future of NeoVim package management)
- Concise and thorough Vimdoc documentation that communicates effectively
- Modern lua tooling: use of [busted](https://lunarmodules.github.io/busted/), [selene](https://kampfkarren.github.io/selene/) and more integrated with the NeoVim lua interpreter
- Sensible user defaults & extreme customization
- Proper versioning, tagging, and releases
-2. The Neovim Community: I'm elated to finally give back to the community (even if no one uses this plugin). [folke](https://github.com/folke), [bfredl](https://github.com/bfredl), and [echasnovski](https://github.com/echasnovski) are my greatest inspirations as an open-source developer and I've had enough of taking without giving back.
+2. The Neovim Community: I'm elated to finally give back to the community
+ (even if no one uses this plugin). [folke](https://github.com/folke),
+ [bfredl](https://github.com/bfredl), and
+ [echasnovski](https://github.com/echasnovski) are my greatest inspirations as
+ an open-source developer and I've had enough of taking without giving back.
+
- In the coming months I plan to contribute to [NeoVim core](https://github.com/neovim/neovim), including making `:checkhealth` asynchronous and integrating an [mdx](https://mdxjs.com/) parser.
-3. Learning Random things: I think this plugin is *really* cool by virtue of its efficacy and the miscellany of knowledge I accrued in the 15k+ LOC as of version v0.3.0. Some things I learned include:
+3. Learning Random things: I think this plugin is *really* cool by virtue
+ of its efficacy and the miscellany of knowledge I accrued in the 15k+ LOC as
+ of version v0.3.0. Some things I learned include:
-- ANSI terminal colors and escape codes: I wrote my own stateful ANSI escape sequence parser to map raw bytes to native NeoVim highlighted text
-- Extmarks: NeoVim extmarks (`:h extmarks`) are extremely powerful. Here, I used them to apply dynamic highlighting across various components of the plugin but I also plan to leverage virtual text to catch compile errors in real-time
-- VIM filetypes and diffing: Vim is strange and the event-based system is fragile. I faced filetype detection race conditions and odd side effects of functions (such as `:diffthis` resetting `foldcolumn`).
-- [LuaCATS](https://github.com/LuaCATS): apparently writing comments is the best way to typecheck in lua...
-- The (Neo)Vim event loop: Scraper subprocesses spawned with `vim.system`. Though a powerful API, I often had to obey the event loop and wrap side effects with `vim.schedule` to ensure they ran after jobs finished. This was useful to defer UI updates.
+- ANSI terminal colors and escape codes: I wrote my own stateful ANSI
+ escape sequence parser to map raw bytes to native NeoVim highlighted text
+- Extmarks: NeoVim extmarks (`:h extmarks`) are extremely powerful. Here,
+ I used them to apply dynamic highlighting across various components of the
+ plugin but I also plan to leverage virtual text to catch compile errors in
+ real-time
+- VIM filetypes and diffing: Vim is strange and the event-based system is
+ fragile. I faced filetype detection race conditions and odd side effects of
+ functions (such as `:diffthis` resetting `foldcolumn`).
+- [LuaCATS](https://github.com/LuaCATS): apparently writing comments is
+ the best way to typecheck in lua...
+- The (Neo)Vim event loop: Scraper subprocesses spawned with
+ `vim.system`. Though a powerful API, I often had to obey the event loop and
+ wrap side effects with `vim.schedule` to ensure they ran after jobs finished.
+ This was useful to defer UI updates.
diff --git a/src/content/software/hosting-a-git-server.mdx b/src/content/software/hosting-a-git-server.mdx
index 4383769..c68bf85 100644
--- a/src/content/software/hosting-a-git-server.mdx
+++ b/src/content/software/hosting-a-git-server.mdx
@@ -83,7 +83,7 @@ Restart=on-failure
[Install]
WantedBy=multi-user.target
-````
+```
12. I use the name `main` instead of `master` as my default branch (it's less letters, ok?)--set this up on the server too:
diff --git a/src/layouts/BaseLayout.astro b/src/layouts/BaseLayout.astro
index f311a85..fd925ef 100644
--- a/src/layouts/BaseLayout.astro
+++ b/src/layouts/BaseLayout.astro
@@ -52,7 +52,9 @@ const {
case "git":
return "#cc5500";
default:
- return "#000000";
+ return getComputedStyle(document.documentElement)
+ .getPropertyValue("--text")
+ .trim();
}
};
diff --git a/src/layouts/GitLayout.astro b/src/layouts/GitLayout.astro
index 46c0ba1..411560f 100644
--- a/src/layouts/GitLayout.astro
+++ b/src/layouts/GitLayout.astro
@@ -1,5 +1,5 @@
---
-const { frontmatter, post } = Astro.props;
+const { frontmatter } = Astro.props;
---
@@ -10,7 +10,7 @@ const { frontmatter, post } = Astro.props;
-
+
diff --git a/src/layouts/PostLayout.astro b/src/layouts/PostLayout.astro
index 9fbddd1..37af82e 100644
--- a/src/layouts/PostLayout.astro
+++ b/src/layouts/PostLayout.astro
@@ -22,14 +22,6 @@ interface Props {
const { frontmatter, post } = Astro.props as Props;
const { title, description, useKatex = false, useD3 = false } = frontmatter;
-const filePath = post?.id ?? "";
-const categoryFromId = (filePath.split("/")[0] ?? "").replace(
- /\.(mdx?|MDX?)$/,
- "",
-);
-const category =
- (frontmatter.category ?? post?.collection ?? categoryFromId) || "";
-
let documentTitle = title;
if (post?.collection === "git" && post?.slug) {
documentTitle = `${post.slug}.git`;
@@ -40,7 +32,7 @@ if (post?.collection === "git" && post?.slug) {
documentTitle = `${post.slug}`;
}
-const topicColor = getTopicColor(post.collection);
+const topicColor = getTopicColor(post?.collection);
---
@@ -57,10 +49,14 @@ const topicColor = getTopicColor(post.collection);
/>
)
}
+ {useD3 && }
-