Aniruddha Debhttps://aniruddhadeb.com/Recent content on Aniruddha DebHugo -- gohugo.ioen-usFri, 12 Jan 2024 23:00:00 +0530LLMs for parsing unstructured data (IIT Course Planner v2!)https://aniruddhadeb.com/articles/2024/iit-course-planner-v2/Fri, 12 Jan 2024 23:00:00 +0530https://aniruddhadeb.com/articles/2024/iit-course-planner-v2/Happy new year! IIT Course planner (the course dependency graphs) used to be (and probably still is) the only reason people used to visit my blog. So much that it accounts for more views than any other webpage, except the homepage, and outdoes my next most popular article by 2x
The total pageviews is more than I expected :) It’s also no secret that I was working on a successor for the course planner for a while.Lost in Japan: A Travelogue (Part 1)https://aniruddhadeb.com/articles/2023/japan-sakura/Fri, 15 Dec 2023 20:30:00 +0530https://aniruddhadeb.com/articles/2023/japan-sakura/Neon lights shining bright
Exploring bustling city streets
Meeting friendly minds
Knowledge and culture exchanged
Memories to last a lifetime
Introduction Sometime when I was interning in Summer, an innocent-looking email dropped into my Inbox
It’s premise was comically simple, in a way that reminded me of lottery ads on TV: form bharo aur jeeto Japan jaane ka mauka! (Fill this form and win a trip to Japan!)
Not thinking much, I filled it in.Rollerball Debriefhttps://aniruddhadeb.com/articles/2023/rollerball-debrief/Fri, 08 Dec 2023 17:00:00 +0530https://aniruddhadeb.com/articles/2023/rollerball-debrief/Since the semester is finally over, I think this is a good time to discuss my primary programming project over the last semester. This is unique enough to merit it’s own blog post: It’s something that impacted a lot of people at scale, and also something I believe is novel and required a metric ton of effort from my end to make a reality.
This is Rollerball, the COL333 competitive assignment for this semester.Luck, Skill or Both: Department Change at IITshttps://aniruddhadeb.com/articles/2023/luck-skill-both/Fri, 01 Sep 2023 17:00:00 +0000https://aniruddhadeb.com/articles/2023/luck-skill-both/A friend recently asked me to provide some advice on department change to some freshers. The usual fundaebaazi is something I didn’t want to do; it’s not impressionable, and won’t motivate any fresher to study for a department change. Instead, I chose to show and not tell
So is it luck or skill? Short answer: Both.
Let’s look at how first year courses are graded. For most theory courses (3 lecture hours + 1 tutorial hour / 2 lab hours), there’s a normal distribution of marks, and the top 7% of people on this distribution get an A.Intern Asheshttps://aniruddhadeb.com/articles/2023/intern-ashes/Sat, 29 Jul 2023 19:50:00 +0000https://aniruddhadeb.com/articles/2023/intern-ashes/[If you haven’t read the previous article in this series, check out Intern Ignition]
In the weekends of my internship, and in the days that followed, I got to tour Europe a bit. What follows is a small travelogue of my experiences in these two months1,
Stroopwaffel Optiver gave us a goodie bag when we arrived, and among other things, it contained a box of Stroopwaffel!
Stroopwaffel are really thin waffles filled with caramel, and they taste amazing!My (updated) DL workflow for 2023https://aniruddhadeb.com/articles/2023/updated-dl-workflow-2023/Sun, 04 Jun 2023 01:00:00 +0000https://aniruddhadeb.com/articles/2023/updated-dl-workflow-2023/Remember last semester, where I said this?
That went pretty well. Much better than I expected it to go :) I also used the HPC DL environment a lot, across 4 courses that I did (COL380, COL772, COL775 and COD310). It was so used that at a point I had four different GPU’s allocated just for running tasks for these different courses.
A lot could be done better, though. Since I have my summer ahead of me, I decided to clean house for the next semesterRCU comichttps://aniruddhadeb.com/articles/2023/rcu-comic/Sun, 19 Mar 2023 23:30:00 +0000https://aniruddhadeb.com/articles/2023/rcu-comic/Drawn from Wikipedia’s amazing RCU explanation.The Properly Illustrated Transformerhttps://aniruddhadeb.com/articles/2023/properly-illustrated-transformer/Wed, 08 Mar 2023 00:12:00 +0000https://aniruddhadeb.com/articles/2023/properly-illustrated-transformer/The title is (obviously) a shout out to Jay Alammar’s Blog, which is probably the best source to learn about transformers, followed by The Annotated Transformer. Both of them are unclear about the nuances though (where the dropouts are mostly), and so I made this to clear that up.
PS: Yes, this is just the encoder stack. I’ll probably make the decoder stack in a part 2?Mac Setup v2https://aniruddhadeb.com/articles/2023/mac-setup-2/Mon, 06 Mar 2023 12:00:00 +0000https://aniruddhadeb.com/articles/2023/mac-setup-2/My 2015 MBP 13" was giving me some pain over this entire semester, lagging a lot and being very slow overall. I recently upgraded to a 16" MBP, and there have been no complaints so far :) the notch is simply not there if you have a black desktop background, so +1 to apple for that. Although face ID would have been nice to have.
Setup Basic intro Screen
Dark ModeBatch Normalizationhttps://aniruddhadeb.com/articles/2023/batch-normalization/Thu, 16 Feb 2023 16:00:00 +0000https://aniruddhadeb.com/articles/2023/batch-normalization/$$ \require{physics} \newcommand{B}{\mathcal{B}}$$
Batch Normalization was proposed by Ioffe and Szegedy in 2015, and it spawned several normalization techniques that are used in SOTA models today (layer norm, weight norm, etc). Batch normalization normalizes the output of each layer based on the mean and variance of the examples in the current batch. Formally, if $\B = {x_1, \ldots, x_m}$ is our batch, then batch norm does the following transformation:
$$\begin{align} \mu_{\B} &= \frac{1}{m} \sum_i x_i \\ \sigma_{\B}^2 &= \left( \frac{1}{m} \sum_i x_i^2 \right) - \mu_{\B}^2 \\ \hat{x}_i &= \frac{x_i - \mu_{\B}}{\sqrt{\sigma^2_{\B} + \epsilon}} \\ y_i &= \gamma \hat{x}_i + \beta \end{align}$$L2 regularization intuitionhttps://aniruddhadeb.com/articles/2023/l2-regularization/Sun, 22 Jan 2023 10:15:00 +0000https://aniruddhadeb.com/articles/2023/l2-regularization/A nice intuition for L2 regularization comes from having a prior on the distribution of parameters: the prior assumes that the parameters are close to zero. Let’s assume that the prior is $\mathcal{N}(0, \Sigma)$. The MAP estimate of the parameters would then be
$$\begin{align} \theta_{\text{MAP}} &= \text{argmax}_{\theta} ; P(\theta | D) \\ &= \text{argmax}_{\theta} ; P(D | \theta) P(\theta) \\ &= \text{argmax}_{\theta} ; \log P(D | \theta) + \log P(\theta) \end{align}$$Optimizers, Part 1https://aniruddhadeb.com/articles/2023/optimizers-1/Mon, 02 Jan 2023 12:25:00 +0000https://aniruddhadeb.com/articles/2023/optimizers-1/Happy New Year! This is going to was supposed to be a long one, so sit back and grab a chocolate (and preferably view this on your laptop)
Some optimization algorithms. Click on a colour in the legend to hide/show it
Table of Contents Introduction Do Solutions Even Exist? How this guide is structured Gradient Descent Optimizers Stochastic Gradient Descent SGD with Momentum SGD with Nesterov Momentum Putting it all together References and Footnotes Introduction Most supervised learning tasks involve optimizing a loss function, in order to fit the model to the given training data.My DL workflow for 2023https://aniruddhadeb.com/articles/2022/dl-workflow-2023/Thu, 29 Dec 2022 23:40:00 +0000https://aniruddhadeb.com/articles/2022/dl-workflow-2023/I’ve kind of zeroed down on Deep Learning at this point, and putting my money where my mouth is, will be taking both COL772 (Natural Language Processing) and COL775 (Deep Learning) next semester.
Along with Operating Systems, Parallel Programming and Theory of Computation.
Why a workflow? I’ll need to train a lot of models, and while Kaggle and Colab are great at this, 30 hours a week won’t cut it. So I’ll need to move to using the IITD HPC to train some (most?HackerFX v2https://aniruddhadeb.com/articles/2022/hackerfx-v2/Thu, 08 Dec 2022 01:00:00 +0000https://aniruddhadeb.com/articles/2022/hackerfx-v2/Some of you might have come across HackerFX, an image binarizer. This was written a long time back, when the only language I knew was Java, and when I was still in school.
An update was on the cards for a while, because I wanted to change my desktop background (which still looks surprisingly pretty for something made so long ago). I started playing around with PIL today evening to update this, and the end result is HackerFX v2, a python rewrite that supports multicoloured input images, hex, file dumps, variable colors and a manual 5x5 font for the output charactes (so the aspect ratio doesn’t have to be manually tweaked).Bootloader Deep Divehttps://aniruddhadeb.com/articles/2022/boot-deep-dive/Sat, 03 Dec 2022 05:00:00 +0000https://aniruddhadeb.com/articles/2022/boot-deep-dive/The Basics The most basic boot process is BIOS on a standard i386/x86-64 device. The BIOS first detects the CPU/RAM, then initializes the remainder of the hardware, and then begins the boot sequence. The boot sequence basically consists of finding a bootable device, loading the boot program in that bootable device into the processor cache and letting the processor do it’s thing.
How is a bootable device found? A device is considered bootable if it has the magic bytes 0x55 and 0xaa at offsets 511 and 512 from the start.Filtrationshttps://aniruddhadeb.com/articles/2022/filtrations/Mon, 28 Nov 2022 03:00:00 +0000https://aniruddhadeb.com/articles/2022/filtrations/Recap If you remember the note on random variables, then this picks up exactly where that left off. $\newcommand{\triple}{(\Omega, \mathcal{F}, \mathbf{P})}$$\newcommand{\P}{\mathbf{P}}$
Definition 1.1: A Measurable space $(X,\Sigma)$ consists of a set $X$ and a $\sigma$-algebra $\Sigma$ defined on $X$.
Recall that a $\sigma$-algebra is simply a collection of subsets over $X$ that is closed over unions, intersections and complements.
Definition 1.2: If $\mathcal{A}$ and $\mathcal{B}$ are $\sigma$-algebras defined on $\Omega$ and $\mathcal{B} \subseteq \mathcal{A}$, then $\mathcal{B}$ is a sub $\sigma$-algebra of $\mathcal{A}$Seven Days of Creationhttps://aniruddhadeb.com/articles/2022/seven-days-of-creation/Sun, 18 Sep 2022 12:00:00 +0000https://aniruddhadeb.com/articles/2022/seven-days-of-creation/When you learn more in the last week than in any other week in your life. A summary of how I built a high-performance peer-server-peer file sharing network in the span of a week.A Simple Sudoku Solverhttps://aniruddhadeb.com/articles/2022/simple-sudoku-solver/Sat, 03 Sep 2022 17:00:00 +0000https://aniruddhadeb.com/articles/2022/simple-sudoku-solver/The Problem The problem is a simple search problem: given a partially filled sudoku board, return a fully filled sudoku board with the initial numbers in place such that each digit features only once in every row, column and 3x3 subgrid.
The Solution These problems, in general, are called Constraint Satisfaction Problems. Given a set of constraints, find a valid solution which satisfies all the constraints.
A CSP is generally represented as a graph, with each variable as a node and a constraint between two variables as an edge.Variance Estimation in Heteroscedastic Regressionhttps://aniruddhadeb.com/articles/2022/heteroscedastic-variance-estimation/Mon, 29 Aug 2022 18:00:00 +0000https://aniruddhadeb.com/articles/2022/heteroscedastic-variance-estimation/Most of the times, the regression models we need to fit don’t have a constant standard deviation, but rather one that is some function of $x$. In this article, I try to learn a model with variance varying linearly as a function of x: kind of like a <em>regression on variance</em>The Rise of the Codefluencerhttps://aniruddhadeb.com/articles/2022/rise-of-codefluencer/Sat, 20 Aug 2022 17:00:00 +0000https://aniruddhadeb.com/articles/2022/rise-of-codefluencer/My Thoughts on Web Development, Programming Influencers/tutors and the portrayal of programming as a lavish dream rather than providing a careful solution to complex problems, bit by bit.Intern Infernohttps://aniruddhadeb.com/articles/2022/intern-inferno/Sun, 14 Aug 2022 15:30:00 +0000https://aniruddhadeb.com/articles/2022/intern-inferno/Not everything goes exactly as planned. That’s why it’s a plan, not a certainty.
Aiming in Before this entire joomla started, I asked myself: What is it I want from an intern?
I wanted to tour: I didn’t pick up a foreign research intern this summer, and was tired of hearing/seeing how others were roaming the world while I was scooping shuttles off the floor of my room. I wanted something that paid well: Apart from the feel-good factor, a good paying intern would put money for my masters’ in the bank Above both of these:The Arch Linux Journeyhttps://aniruddhadeb.com/articles/2022/arch-linux-1/Sun, 07 Aug 2022 15:30:00 +0000https://aniruddhadeb.com/articles/2022/arch-linux-1/For the upcoming semesters, we have a ton of systems courses. So, I needed a linux box that I could hack around, preferably at the kernel level. I finally got around to installing Arch Linux on a Raspberry Pi1: Here’s how I did it.
Context For starters, I already use linux/unix systems quite a bit. The real need for a linux system became apparent last semester, when one of our courses needed a program that would run only on Windows/Ubuntu.Data Wrangling in Practice: Dataset Creation from Poorly Structured Sourceshttps://aniruddhadeb.com/articles/2022/iit-course-planner-1/Wed, 02 Mar 2022 01:36:00 +0000https://aniruddhadeb.com/articles/2022/iit-course-planner-1/I seldom get time to do anything other than course related activities nowadays, but in this midsem break, I did nothing other than roam around campus and laze in bed. And publish the first release of the IIT Delhi Course Planner Dataset (Also available on GitHub). This is a scraping of the IIT Delhi Courses of Study 2020-21, a behemoth 373-page PDF, which contains the course structure and information of all the Undergraduate and Postgraduate courses offered at IIT Delhi, and then some more.A Note on Conditional Probabilityhttps://aniruddhadeb.com/articles/2021/note-on-conditional-probability/Sat, 25 Dec 2021 15:20:00 +0000https://aniruddhadeb.com/articles/2021/note-on-conditional-probability/$\newcommand{\cE}[2]{\mathbf{E}(#1\ |\ #2)}$$\newcommand{\cP}[2]{\mathbf{P}(#1\ |\ #2)}$$\renewcommand{\P}[1]{\mathbf{P}(#1)}$$\newcommand{\E}[1]{\mathbf{E}(#1)}$$\newcommand{\F}{\mathcal{F}}$$\newcommand{\G}{\mathcal{G}}$$\newcommand{\ind}[1]{\mathbf{1}_{#1}}$ To motivate this note, I’ll pose the following problem:
Consider $X \sim \text{Uniform}([0,1])$. What is $\cP{X > \frac{1}{2}}{X \in \mathbb{Q}}$ ?
At first glance, the answer seems simple: it’s 1/2! Closer inspection reveals that the event $X \in \mathbb{Q}$ is not measurable: the rationals have measure zero, hence from our high-school (perhaps even undergraduate) definition of conditional probability, the given probability should be undefined. However, the conditional probability does exist, and it equals 1/2.A Note on Random Variableshttps://aniruddhadeb.com/articles/2021/note-on-random-variables/Fri, 03 Dec 2021 21:00:00 +0000https://aniruddhadeb.com/articles/2021/note-on-random-variables/$\newcommand{\triple}{(\Omega, \mathcal{F}, \mathbf{P})}$$\newcommand{\P}{\mathbf{P}}$ This note on random variables follows as a result of confusing notation in several math textbooks. I’ll explain random variables (in measure theoretic terms) as verbosely as I can, and then prove some results. This article assumes that the reader is familiar with probability triples $\triple$, as well as a basic idea of what random variables are, in non-measure theory terms.
1. Random Variable Prerequisites We start with defining measurable spaces and measurable functionsFunction transforms (providing a broader picture of Laplace Transforms)https://aniruddhadeb.com/articles/2021/function-transforms/Wed, 12 May 2021 18:00:00 +0000https://aniruddhadeb.com/articles/2021/function-transforms/I’ll begin this article by brushing up a few definitions:
A function is a mapping between two sets: the domain D and the codomain C.
It’s very important to note here that the function is the mapping itself, and not an element in the codomain or the domain. The function operates on an element in the domain to give an element in the codomain. Formally, we would write this function as $f:D \to C$C development on a Machttps://aniruddhadeb.com/articles/2021/c-mac/Thu, 18 Mar 2021 11:46:00 +0000https://aniruddhadeb.com/articles/2021/c-mac/C development on a Mac has it’s fair share of traps and pitfalls, even though Mac is supposed to be POSIX compliant. I’ll cover a few of them here, with the hope that nobody repeats the same mistakes :)
XCode CLT Pitfalls The first thing one does while starting out with C development is to install a compiler. This is slightly more complicated than it looks on Mac: the simple way is to go the way of installing XCode command line tools, but there’s a few things here to note:Understanding Jacobianshttps://aniruddhadeb.com/articles/2021/understanding-jacobians/Fri, 29 Jan 2021 15:11:00 +0000https://aniruddhadeb.com/articles/2021/understanding-jacobians/$\newcommand{\pdv}[2]{\frac{\partial{#1}}{\partial{#2}}}$ $\newcommand{\ah}{\pmb{a} + \pmb{h}}$ $\newcommand{\a}{\pmb{a}}$ $\newcommand{\h}{\pmb{h}}$
The Jacobian Matrix Consider a function that maps reals to reals, $f:\Bbb{R} \to \Bbb{R}$. The linear approximation of this function is given by $$f(a+h) \approx f(a) + hf’(a)$$ This is pretty simple to do, and follows from taylor’s expansion upto the first order.
Let’s try expanding this concept to vector spaces. For a function $f:\Bbb{R}^n \to \Bbb{R}$, it’s linear approximation is given by $$f(\pmb{a} + \pmb{h}) \approx f(\pmb{a}) + \pmb{h}\cdot\nabla{f}(\a)$$ (bold type indicates vectors).Sorting Testbench in Chttps://aniruddhadeb.com/articles/2020/algorithms-in-c/Wed, 11 Nov 2020 19:20:00 +0000https://aniruddhadeb.com/articles/2020/algorithms-in-c/I started reading CS theory in anticipation of my upcoming curriculum and I found that CLRS starts the reader off with sort algorithms (technically they start with math and basic CS, but I’ve already covered that). I decided to make a Sorting testbench, whose requirements were pretty simple:
It should be able to generate arrays of a given entropy: the entropy of an array is defined as the number of swaps required to sort the array.IIT Department Finder Launch (plus personal website domain)https://aniruddhadeb.com/articles/2020/iit-dep-finder/Wed, 28 Oct 2020 20:39:00 +0000https://aniruddhadeb.com/articles/2020/iit-dep-finder/For IIT admissions, I had created a small command line tool called iit_dep_finder.py to check the departments I would get, given a particular rank. Once the round one allocation results came out, I decided to create a web version, for others to use as well. This article chronicles the steps I took, and while I did get the department finder up and running, I also did a lot of other stuff in the process (including changing the domain name, in case you haven’t noticed :).To IIT Delhi and beyondhttps://aniruddhadeb.com/articles/2020/iit-delhi-mnc/Mon, 19 Oct 2020 14:23:00 +0000https://aniruddhadeb.com/articles/2020/iit-delhi-mnc/I think an update on the JEE Situation has been due for a long time, and I’ll try to compress the past few months into this single post. Long story short, my JEE(M) was held on 2nd September, in the evening shift. After giving my best and getting a 260/300, I was disappointed (putting it mildly) to see the CRL of 1861 on my screen. AITS JEE(A) Mocks followed, and after giving three of them, I had to give my BITSAT.Setting up my Mac (again)https://aniruddhadeb.com/articles/2020/mac-system-setup/Wed, 14 Oct 2020 22:48:00 +0000https://aniruddhadeb.com/articles/2020/mac-system-setup/This was also published on GitHub. The GitHub version is more up-to-date, and I won’t be updating this one as frequently.
System Setup Log My MacBook battery started to swell around a year back, and the machine became unusable 6 months ago and was put into temporary hibernation. Post my exams, I did a battery replacement and the computer was out of the hospital today, with a fresh operating system and a formatted disk.Math Notes: Lagrange interpolationhttps://aniruddhadeb.com/articles/2020/math-notes-1/Fri, 14 Aug 2020 17:37:00 +0000https://aniruddhadeb.com/articles/2020/math-notes-1/This is a small set of posts that come into the category of “notes”: self-explanations of concepts that I have recently picked up and found interesting.
Lagrange Interpolation Lagrange Interpolation is a concept that allows us to find a polynomial of least degree passing through a given set of points. Specifically:
If $S = {(x_i, y_i) : x_i, y_i \in R, 1 \lt i \le n}$, then the polynomial of least degree passing through all the points in $S$ is given by $$P(x) = \sum_{i=1}^n L_i(x) \cdot y_i$$ where $$L_i(x) = \frac{(x-x_1)(x-x_2)…(x-x_{i-1})(x-x_{i+1})…(x-x_n)} {(x_i-x_1)(x_i-x_2)…(x_i-x_{i-1})(x_i-x_{i+1})…(x_i-x_n)}$$ or simply, $$P(x) = \sum_{i=1}^n \frac{\prod_{r \ne i}(x-x_r)}{\prod_{r \ne i}{(x_i-x_r)}} \cdot y_i$$ note that the degree of polynomial $P < n$.Is the shortest solution to a problem the best?https://aniruddhadeb.com/articles/2020/shortest-solution-best-or-not/Sun, 26 Jul 2020 21:00:00 +0000https://aniruddhadeb.com/articles/2020/shortest-solution-best-or-not/My old man always used to say that there are two ways to solve a problem: there’s the horse way, and there’s the donkey way. The donkey way involves tedious calculations, whereas the horse way ‘cuts’ through the problem. Take this problem as an example:
For each positive integer $n$, let $$y_n = ((n+1)(n+2)…(n+n))^\frac1n$$ for $x \in \mathbb{R}$. Let $[x]$ be the greatest integer less than or equal to $x$. If $\lim_{n \to \infty} y_n = L$, then the value of $[L]$ is ______ ?ISC 2020 Analysis: An application of Data Science and Statisticshttps://aniruddhadeb.com/articles/2020/isc-2020-analysis/Sat, 11 Jul 2020 16:30:00 +0000https://aniruddhadeb.com/articles/2020/isc-2020-analysis/The ISC Exam results were released on 10th June, 3 PM IST. In this article, I’ll be analyzing the results of the 117 students in the science stream of my school and showing how they performed. I’ll also weave a story with the data and point out things that could have been improved, which would be helpful for future students.
Index: Elementary inferences: How have people fared overall? Frequency Distribution of Marks: How many people got the same marks in a subject?Solving the African Integral (from YG file cover)https://aniruddhadeb.com/articles/2020/african-integral/Thu, 09 Jul 2020 20:30:00 +0000https://aniruddhadeb.com/articles/2020/african-integral/A long time ago, I gave my friend (Let’s call him C) an integral to solve. He came back to me a few days later and the conversation went something like this:
C: Debu, I couldn’t solve that African integral you gave me.
Me: Ok, but why are you calling it African?
C: Because Africa is a really big continent, y’know?
Me: Ok, but isn’t Asia bigger?
C: Debu, please stop making me feel dumbIcosahedron Resistor Networkhttps://aniruddhadeb.com/articles/2020/icosahedron-resistor-network/Tue, 09 Jun 2020 12:30:00 +0000https://aniruddhadeb.com/articles/2020/icosahedron-resistor-network/Consider the icosahedral resistance network shown in the following figure. Each edge is made of a rod of resistance $r$. Select the option(s) that is(are) correct:
A) The resistance between points A and B is $\frac{r}{2}$
B) The resistance between points A and B is $\frac{r}{3}$
C) The resistance between any two adjacent vertices is $\frac{11r}{30}$
D) The resistance between any two adjacent vertices is $\frac{11r}{20}$
This question looks really tricky.A Compilation of hard limitshttps://aniruddhadeb.com/articles/2020/hard-limits/Fri, 05 Jun 2020 11:30:00 +0000https://aniruddhadeb.com/articles/2020/hard-limits/This list consists of the limits that I found most challenging.
$$\lim_{n \to \infty} \left( \frac{n!}{n^n} \right) ^\frac 1n$$ $$\lim_{x \to 0} \left( \frac{1}{\ln(x + \sqrt{x^2+1})} - \frac 1{\ln(x+1)} \right)$$ $$\lim_{n \to \infty} \frac{n + n^2 + n^3 + … + n^n}{1^n + 2^n + 3^n + … + n^n}$$ $$\lim_{n \to \infty} \left( \frac{n^n(x+n)\left(x+\frac n2\right)…\left(x+\frac nn\right)}{n!(x^2+n^2)\left(x^2+\frac {n^2}{4}\right)…\left( x^2 + \frac{n^2}{n^2}\right)}\right)^{\frac x n}$$ $$\lim_{x \to 0} \left( 1^{\sin^{-2}x} + 2^{\sin^{-2}x} + … + n^{\sin^{-2}x}\right)^{\sin^2 x}$$ $$\lim_{n \to \infty} \sqrt[\leftroot{-2}\uproot{2}n+1]{(n+1)!Limit involving higher order infinitesimalshttps://aniruddhadeb.com/articles/2020/limit-higher-order-infinitesimals/Fri, 05 Jun 2020 07:30:00 +0000https://aniruddhadeb.com/articles/2020/limit-higher-order-infinitesimals/Simple limit problems consist of the form $\lim_{x \to 0}\frac{O_1(x)O_2(x)..}{O_a(x)O_b(x)..}$, such as $\lim_{x \to 0} \frac{\sin 3x \tan 2x \tan^{-1} 5x}{x^2 \ln(1+x)}$. Here, the infinitesimals are well defined and cancel out easily. Some trickier limit problems involve the difference of two infinitesimals. A good example is $\lim_{t \to 0} \frac{\sin t - \tan t}{t^3}$. These kind of problems require expansions to solve them cleanly. L’Hopital rule is tedious as it will need to be repeated atleast thrice for this given limit, which involves infinitesimals of order 3.No square ends in 3https://aniruddhadeb.com/articles/2020/no-square-ends-in-3/Fri, 15 May 2020 11:30:00 +0000https://aniruddhadeb.com/articles/2020/no-square-ends-in-3/This is an interesting number theory fact that seems strange when taken at face value. Here’s a small proof of it:Inorganic Heterocyclic compoundshttps://aniruddhadeb.com/articles/2020/inorganic-heterocycles/Sun, 03 May 2020 10:40:00 +0000https://aniruddhadeb.com/articles/2020/inorganic-heterocycles/The 45th IChO preparatory problem list had this as it’s fifth problem:
5.1 The interaction of thionyl chloride and sodium azide at $-30^\circ \text{C}$ gives colorless crystals X, containing 36.4 wt.% of Cl. The crystals consist of cyclic trimers. Find the composition of X and give the reaction equation.
5.2 Draw two stereoisomers of X.
5.3 A colorless liquid Y was prepared by a reaction between X and antimony(III) fluoride. Addition of 1.Roots of f(f(..f(x)..)), f(x) = ax² + bx + chttps://aniruddhadeb.com/articles/2020/roots-of-fffx/Thu, 30 Apr 2020 20:30:00 +0000https://aniruddhadeb.com/articles/2020/roots-of-fffx/Define $ f(x) = ax^2 + bx + c , a,b,c \in \mathbb{R}$ and $ f^n(x) = f(f^{n-1}(x)), n>1 $. Prove that the real roots of $ f^n(x) $ are symmetric about the vertical line passing through vertex i.e. $ x = \frac{-b}{2a} $
This seems like a complicated problem. Let’s start by first making intuitional sense of this problem. We can say that $ f^n(x) $ will be a polynomial of the order $ 2^n $ and will have as many roots.An integral involving ζ(2) (And Euler's first proof for the Basel Problem)https://aniruddhadeb.com/articles/2020/zeta-2-integral/Wed, 29 Apr 2020 22:30:00 +0000https://aniruddhadeb.com/articles/2020/zeta-2-integral/Evaluate the integral $$\int_0^1 \frac{\log(x)}{x-1}dx$$
There are a few methods of doing this: the first one uses the taylor series expansion of \( \log(1-x) \): $$I = -\int_0^1 \frac{\log(1-x)}{x}dx$$ $$I = \int_0^1 \frac 1x \left( x + \frac{x^2}{2} + \frac{x^3}{3} + … \right) $$
This expression nicely integrates to give $$I = 1 + \frac{1}{2^2} + \frac{1}{3^2} + … = \boxed{\frac{\pi^2}{6}}$$
The last step here involves the summation \( \zeta(2) = \sum_{k=1}^{\infty} \frac 1 {k^2} = \frac{\pi^2}{6}\).An Interesting Electronics Problemhttps://aniruddhadeb.com/articles/2020/interesting-electronics-problem/Wed, 29 Apr 2020 22:20:00 +0000https://aniruddhadeb.com/articles/2020/interesting-electronics-problem/An interesting electronics problem I encountered recently was this: Find the resistance between points A and B in the following resistor network
This looks like a simple wheatstone bridge problem, but note that the top right and bottom left resistances are R and not r. To solve this problem, we’ll use the concept of symmetry and a few tricks. Let’s start by connecting a battery to the circuit and labeling the currents and loopsMathJax Testhttps://aniruddhadeb.com/articles/2020/mathjax-test/Wed, 29 Apr 2020 10:20:00 +0000https://aniruddhadeb.com/articles/2020/mathjax-test/This is a test to see if MathJax works on blogger.
For \(\alpha, \beta \gt -1\), find the value of \(\lim_{n \to \infty} n^{\beta - \alpha} \frac{1^\alpha + 2^\alpha + 3^\alpha + … + n^\alpha}{1^\beta + 2^\beta + 3^\beta + … + n^\beta}\)
So we have
$$L = \lim_{n \to \infty} \frac{n^\beta}{n^\alpha} \times \frac{\sum_{r=0}^n r^\alpha}{\sum_{r=0}^n r^\beta}\ = \lim_{n \to \infty} \frac{\sum_{r=0}^n (\frac rn)^\alpha}{\sum_{r=0}^n (\frac rn)^\beta}$$
The summations can now be evaluated as Riemann sums by multiplying numerator and denominator by (\frac 1n)Electronic Clock II - Generating the Clock Signal pt. 2https://aniruddhadeb.com/articles/2018/electronic-clock-2-ii/Sat, 16 Jun 2018 10:30:00 +0000https://aniruddhadeb.com/articles/2018/electronic-clock-2-ii/In the previous post, we saw how to make a 2Hz clock signal. However, for the seconds counter, we will need a 1Hz clock signal. In order to achieve this, I will clock down the frequency using a JK Flip Flop:
Flip-Flop: Concept A Flip flop is a one-bit storage device which looks somewhat like this:
The truth table of this device looks somewhat like this:
Note that the flip flop toggles only on the rising edge of the clock.Electronic Clock II - Generating the Clock Signal pt.1https://aniruddhadeb.com/articles/2018/electronic-clock-2-i/Thu, 14 Jun 2018 22:30:00 +0000https://aniruddhadeb.com/articles/2018/electronic-clock-2-i/In the previous installment of this series, I spoke about the requirements and the architecture of the clock. In this installment, I’m going to engineer the clock signal generator, the most crucial part of our clock
Importance of the Signal in a clock Every clock is only as accurate as it’s signal. An atomic clock uses radiations emitted by cesium-133 as it’s signal, a quartz clock uses a 32768 Hz quartz crystal as it’s signal whereas a mechanical clock uses an Escapement powered by a hairspring (in wristwatches) or by a pendulum (in clocks).Electronic Clock I - Getting Startedhttps://aniruddhadeb.com/articles/2018/electronic-clock-1/Thu, 14 Jun 2018 10:30:00 +0000https://aniruddhadeb.com/articles/2018/electronic-clock-1/This is a new series in which I will describe all the steps I went through while creating an electronic clock. Make sure you’re subscribed to the RSS feed so that you don’t miss out on anything!
Getting Started An electronic clock is a good project for intermediate to advanced electronic hobbyists, which is why I decided to build one.
The first task is enumerating the requirements of the clock. A humble requirement list looks like this:ICSE Mathematics - Last 23 years analysis and 2018 forecasthttps://aniruddhadeb.com/articles/2018/icse-mathematics-analysis/Tue, 09 Jan 2018 10:30:00 +0000https://aniruddhadeb.com/articles/2018/icse-mathematics-analysis/A speculative format for the 2018 ICSE Mathematics paper, created by analysis of the mathematics question papers of the past 23 years.
<img src="https://aniruddhadeb.com/articles/2018/res/icse_math_analysis.png" alt="Analysis results">TCS IT Wiz - an Experiencehttps://aniruddhadeb.com/articles/2017/tcsitwiz-experience/Thu, 14 Dec 2017 10:30:00 +0000https://aniruddhadeb.com/articles/2017/tcsitwiz-experience/A Firsthand account of the TCS IT Wiz 2017 JourneyPublicationshttps://aniruddhadeb.com/publications/Mon, 01 Jan 0001 00:00:00 +0000https://aniruddhadeb.com/publications/ Aniruddha Deb, Neeva Oza, Sarthak Singla, Dinesh Khandelwal, Dinesh Garg, Parag Singla. “Fill in the Blank: Exploring and Enhancing LLM Capabilities for Backward Reasoning in Math Word Problems”. arXiv preprint arXiv:2310.01991 (2023).