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I write this blog because I enjoy writing. Some people enjoy reading what I write, which makes me feel really great! Recently, I took down a post and stopped writing for a few months because I didn't love the reaction I was getting on social media sites like Reddit and Hacker News. On these social networks, there seems to be an epidemic of "gotcha" commenters, contrarians, and know-it-alls. No matter what you post, you can be sure that folks will come with their sharpest pitchforks to try to skewer you. I'm not sure exactly what it is about those two websites in particular. I suspect it's the gamification of the comment system (more upvotes = more points = dopamine hit). Unfortunately, it seems the easiest way to win points on these sites is to tear down the original content. At any rate, I really don't enjoy bad faith Internet comments and I have a decent-enough following outside of these social networks that I don't really have to endure them. Some might argue I need thicker skin. I don't think that's really true: your experience on the Internet is what you make of it. You don't have to participate in parts of it if you don't want. Also, I know many of you reading this post (likely RSS subscribers at this point) came from Reddit or Hacker News in the first place. I don't mean to insult you or suggest by any means that everyone, or even the majority of users, on these sites are acting in bad faith. Still, I have taken a page from Tom MacWright's playbook and decided to add a bit of javascript to my website that helpfully redirects users from these two sites elsewhere: try { const bannedReferrers = [/news\.ycombinator\.com/i, /reddit\.com/i]; if (document.referrer) { const ref = new URL(document.referrer); if (bannedReferrers.some((r) => r.test(ref.host))) { window.location.href = "https://google.com/"; } } } catch (e) {} After implementing this redirect, I feel a lot more energized to write! I'm no longer worried about having to endlessly caveat my work for fear of getting bludgeoned on social media. I'm writing what I want to write and, if for those of you here to join me, I say thank you!
Here we go again: I'm so tired of crypto web3 LLMs. I'm positive there are wonderful applications for LLMs. The ChatGPT web UI seems great for summarizing information from various online sources (as long as you're willing to verify the things that you learn). But a lot fo the "AI businesses" coming out right now are just lightweight wrappers around ChatGPT. It's lazy and unhelpful. Probably the worst offenders are in the content marketing space. We didn't know how lucky we were back in the "This one weird trick for saving money" days. Now, rather than a human writing that junk, we have every article sounding like the writing voice equivalent of the dad from Cocomelon. Here's an approximate technical diagram of how these businesses work: Part 1 is what I like to call the "bilking process." Basically, you put up a flashy landing page promising content generation in exchange for a monthly subscription fee (or discounted annual fee, of course!). No more paying pesky writers! Once the husk of a company has secured the bag, part 2, the "bullshit process," kicks in. Customers provide their niches and the service happily passes queries over to the ChatGPT (or similar) API. Customers are rewarded with stinky garbage articles that sound like they're being narrated by HAL on Prozac in return. Success! I suppose we should have expected as much. With every new tech trend comes a deluge of tech investors trying to find the next great thing. And when this happens, it's a gold rush every time. I will say I'm more optimistic about "AI" (aka machine learning, aka statistics). There are going to be some pretty cool applications of this tech eventually—but your ChatGPT wrapper ain't it.
I have noticed a trend in a handful of products I've worked on at big tech companies. I have friends at other big tech companies that have noticed a similar trend: The products are kind of crummy. Here are some experiences that I have often encountered: the UI is flakey and/or unintuitive there is a lot of cruft in the codebase that has never been cleaned up bugs that have "acceptable" workarounds that never get fixed packages/dependencies are badly out of date the developer experience is crummy (bad build times, easily breakable processes) One of the reasons I have found for these issues is that we simply aren't investing enough time to increase product quality: we have poorly or nonexistent quality metrics, invest minimally in testing infrastructure (and actually writing tests), and don't invest in improving the inner loop. But why is this? My experience has been that quality is simply a hard sell in bigh tech. Let's first talk about something that's an easy sell right now: AI everything. Why is this an easy sell? Well, Microsoft could announce they put ChatGPT in a toaster and their stock price would jump $5/share. The sad truth is that big tech is hyper-focused on doing the things that make their stock prices go up in the short-term. It's hard to make this connection with quality initiatives. If your software is slightly less shitty, the stock price won't jump next week. So instead of being able to sell the obvious benefit of shiny new features, you need to have an Engineering Manager willing to risk having lower impact for the sake of having a better product. Even if there is broad consensus in your team, group, org that these quality improvements are necessary, there's a point up the corporate hierarchy where it simply doesn't matter to them. Certainly not as much as shipping some feature to great fanfare. Part of a bigger strategy? # Cory Doctorow has said some interesting things about enshittification in big tech: "enshittification is a three-stage process: first, surpluses are allocated to users until they are locked in. Then they are withdrawn and given to business-customers until they are locked in. Then all the value is harvested for the company's shareholders, leaving just enough residual value in the service to keep both end-users and business-customers glued to the platform." At a macro level, it's possible this is the strategy: hook users initially, make them dependent on your product, and then cram in superficial features that make the stock go up but don't offer real value, and keep the customers simply because they really have no choice but to use your product (an enterprise Office 365 customer probably isn't switching anytime soon). This does seem to have been a good strategy in the short-term: look at Microsoft's stock ever since they started cranking out AI everything. But how can the quality corner-cutting work long-term? I hope the hubris will backfire # Something will have to give. Big tech products can't just keep getting shittier—can they? I'd like to think some smaller competitors will come eat their lunch, but I'm not sure. Hopefully we're not all too entrenched in the big tech ecosystem for this to happen.
Coding interviews are controversial. It can be unpleasant to code in front of someone else, knowing you're being judged. And who likes failing? Especially when it feels like you failed intellectually. But, coding interviews are effective. One big criticism of coding interviews is that they end up filtering out a lot of great candidates. It's true: plenty of great developers don't do well in coding interviews. Maybe they don't perform well under pressure. Or perhaps they don't have time (or desire) to cram leetcode. So if this is happening, then how can coding interviews be effective? Minimizing risk # Coding interviews are optimized towards minimizing risk and hiring a bad candidate is far worse than not hiring a good candidate. In other words, the hiring process is geared towards minimizing false positives, not false negatives. The truth is, there are typically a bunch of good candidates that apply for a job. There are also not-so-great candidates. As long as a company hires one of the good ones, they don't really care if they lose all the rest of the good ones. They just need to make sure they don't hire one of the no-so-great ones. Coding interviews are a decent way to screen out the false positives. Watching someone solve coding challenges gives you some assurance that they can, well, code. Why I myself like coding interviews # Beyond why coding interviews are beneficial for the company, I actually enjoy them as an interviewer. It's not that I like making people uncomfortable or judging them (I don't), but rather I like seeing how potential future colleagues think. How do they think about problems? Do they plan their solution or just jump in? This is a person with whom I'll be working closely. How do they respond to their code being scrutinized? Do I feel comfortable having to "own" their code? On automated online assessments (OAs) # The junior developer market right now is extremely competitive and therefore it is common to use automated coding challenges (OAs) as an initial screen. OAs kind of accomplish the false positive filtering mentioned above, but that assumes candidates aren't cheating. But some are. So you're filtering your candidate pool down to good candidates and dishonest candidates. Maybe that's worth it? Additionally, OAs don't give you any chance to interact with candidates. So you get no sense of what they'd really be like to work with. All in all, I'm not a fan of OAs. Far from perfect # Coding interviews are far from perfect. They're a terrible simulation of actual working conditions. They favor individuals who have time to do the prep work (e.g., grind leetcode). They're subject to myriad biases of the interviewer. But there's a reason companies still use them: they're effective in minimizing hiring risk for the company. And to them, that's the ball game.
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The physicist Sidney Nagel delights in solving mysteries of the universe that are hiding in plain sight. The post Finding Beauty and Truth in Mundane Occurrences first appeared on Quanta Magazine
Federal enforcement of environmental laws has slowed significantly under President Trump. Read more on E360 →
Earlier today I gave a talk in the graduate student seminar titled “Counting is Hard. Complex Analysis is Easy.” based in part on my recent blog post about analytic combinatorics and based in part on Varilly’s notes on Dirichlet’s Theorem, showing how to count the number of trees of a certain shape and the number of “primes of a certain shape” by doing complex analysis. While prepping for this talk, I realized there’s a very pretty geometric way to see what’s going on when counting rooted ordered ternary trees! I don’t usually write blog posts about my local seminar talks anymore, but I think this is more than worthy of an exception! For instance, I think this could serve as a great visiual example of branches and mild singularities in complex analysis. So, let’s remember the problem! We want to count the number of rooted ordered ternary trees with $n$ nodes. Call this number $t_n$. We note that every such tree is either a root node a root with one child a root with two children a root with three children where each child is recursively a rooted ordered ternary tree. If we define $T(z) = \sum_n t_n z^n$ to be the (ordinary) generating function for the $t_n$ we see from this recurrence relation that it must satisfy the functional equation That is, if $P(z,w) = -w + z + zw + zw^2 + zw^3$ then $P(z,T(z)) = 0$. If we draw (the real locus of) $P$ it looks like and the implicit function theorem says that we can find functions $f(z)$ so that $P(z,f(z)) = 0$ through any starting point $P(z_0,f_0)$ on the curve for as long as $\frac{\partial P}{\partial w} \neq 0$. In particular, we know that our $T(0) = t_0 = 0$ since there are no trees with zero vertices, so that our $T(z)$ takes $z$ to the unique part of this curve passing through the origin, for as long as this is defined. Here we’ll plot our curve $P$ in blue, and the implicit function $T(z)$ in orange. Note we have to stop at $z \approx 0.27695$ since here $\frac{\partial P}{\partial w} = 0$ so that the slope is infinite and we can’t continue. Said another way, we bend backwards and if we tried to continue we would fail the vertical line test. But since we have to stop here, we see that $0.27695$ is the radius of convergence for $T$, and is the dominant singularity! Next, can we find a good approximation for $T$ near this singularity? In the main post we used puiseux series for this, but it’s instructive to do it by hand! Note that $\frac{\partial P}{\partial z} \neq 0$ at $(0.27695, 0.65730)$ so there’s nothing stopping us from approximating $z$ as a function of $w$ at this point (then inverting it). Indeed, we can solve for $z = \frac{w}{1+w+w^2+w^3}$ and then just taylor expand this at our point of interest: Of course, we chose this point because $\frac{\mathrm{d}z}{\mathrm{d}w} = \frac{- \frac{\partial P}{\partial w}}{\frac{\partial P}{\partial z}}$ is $0$ here. So we know this $2.97 \times 10^{-8}$ is a rounding error and our leading order expansion is Here’s a graph zoomed into near the singularity. The actual graph of $P$ is shown in blue, and our approximating parabola is shown in orange: But of course if $z - 0.27695 \approx -.34680 (w - 0.6573)^2$ then we can solve for and if we want this to agree with our $w = T(z)$ branch through the origin we obviously have to choose the negative square root (since we want the lower half of this sideways parabola1). Here we draw our $T(z)$ in blue, and we draw the approximation $0.6573 - 0.8936 \sqrt{1 - \frac{z}{0.27695}}$ in orange so you can see how well they line up! But now from here we can run exactly the same argument as in the previous post! We compute $t_n = \frac{1}{2\pi i} \oint \frac{T}{z^{n+1}} \ \mathrm{d}z$, along a keyhole contour around the singular point $z=0.27695$ with a branch cut along the real axis. The brunt of this integral comes from the cutout near the singular point, where $T$ is well approximated by $0.6573 - 0.8936 \sqrt{1 - \frac{z}{0.27695}}$ so that Again, for more details about exactly what this keyhole contour is and how you estimate this integral along it, see the main post. Let’s go ahead and call it here! My actual talk was slightly longer than this, since I also sketched a proof of Dirichlet’s Theorem following Varilly’s notes on the subject, but there’s no sense in me writing that up since those notes are already so good! Plus it’s getting late and I want to go to bed, haha. As always, here’s a copy of my title and abstract. Unfortunately I don’t have any slides or recordings today, but I’m giving another talk at WiSCons in Madison next week and I should have slides for that. I’m also hoping to finish a sister blog post for that talk where I do more example computations in more detail than I could possibly do in a 20 minute talk. Lots of writing to do this week! Take care all, stay safe, and I can’t wait to talk soon ^_^ Counting is Hard. Complex Analysis is Easy. Don’t you miss being a kid, when your mom would ask you to count how many of your alphabet fridge magnets were both red and vowels? When you saw the answer was “2”, you felt such accomplishment…. But counting has only gotten harder since then, and now if you want to count how many objects satisfy some properties (say, being both red and vowels) it can be borderline impossible! In this talk we’ll show how you can solve counting problems by doing complex analysis instead. First we’ll count the number of trees of a certain shape, then (given time) we’ll count how many prime numbers are of a (different) certain shape. In the main post we used puiseux series for this, and we had to essentially guess which branch was correct. Now we see how the geometry of the situation tells us that we have to choose the negative branch of the square root! After all, this is the one that locally agrees with the rest of the graph of $T$! ↩
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