Arya’s Algorithms

you can find the full paper here First Attempts, and the Limits of Energy When I first began thinking about angular occupancy, I wasn’t looking for a brand-new problem. I was chasing a theme that has long fascinated me in discrete geometry: how do we measure the geometric richness of a finite set of points? This fascination is not unique to me. Erdős, way back in 1946, posed the distinct distances problem : given n points in the plane, how many distinct distances can they determine? That question alone gave birth to entire decades of work in combinatorial geometry. Later, Fishburn and Füredi asked about distinct angles , while Pach and Sharir studied repeated angles . Each time, the same underlying itch was being scratched: when you place points in the plane, what kinds of diversity can they generate? Why angles, why bins? It was natural to think of angles — after all, distances and directions had already been well studied. But if you go straight for “distinct angles,” you run in...

  On Thinking I Might Be a Psychopath I’ve always carried a suspicion about myself — an itch in the back of my mind that whispers I might not be like other people. Sometimes, when I look at the way I act, the way I calculate, the way I detach, I see qualities that line up with one of the most chilling labels out there: psychopath. I don’t throw that word around lightly. Psychopaths are supposed to be cold, unfeeling, manipulative, even predatory. And when I examine myself, I do see threads of that: I’ve lied without guilt, manipulated people just to test my control, projected indifference as a shield. I’ve worn the face of someone who “doesn’t give a damn” and enjoyed the power that face seemed to generate. And yet, the story isn’t that simple. Because alongside those moments of icy detachment, I’ve also felt deep, surprising pangs of empathy. A beggar walks past me on the street, and something sharp pulls in my chest. A stray dog limps by, and I feel helpless, guilty, even ash...

  Getting Stuck: When Dijkstra Meets Negative Edges One of the joys (and pains) of computer science is when you take a tool you know, confidently apply it, and then watch everything break down in surprising ways. This happened to me recently when I decided to revisit shortest path algorithms — specifically, Dijkstra’s algorithm — only to get stuck in a swamp of negative edges. Setting the Scene The problem was classic: Given a directed, weighted graph and a source vertex, find the shortest path to all other vertices. This is the kind of problem that every CS student learns early, usually with Dijkstra’s algorithm. I already knew Dijkstra like an old friend: Keep a priority queue of distances. Always expand the node with the current smallest distance. Relax its edges. Simple, elegant, efficient ( O((V + E) log V) with a binary heap). So when I started coding a graph utility for my project — a tool to analyze transportation networks — I confidently plugged in D...

  The Theory of Everything - Unifying the Forces of Nature The dream of a single, elegant theory that explains all fundamental forces and particles has captivated physicists for over a century. My journey toward understanding the quest for a "theory of everything" began with the recognition that our current understanding, despite its remarkable successes, remains frustratingly incomplete. We have two pillars of modern physics - quantum mechanics and general relativity - that seem fundamentally incompatible, plus mysterious components like dark matter and dark energy that hint at physics beyond our current theories. The incompatibility between quantum mechanics and general relativity becomes apparent when trying to describe extreme conditions where both theories should apply. Near black hole singularities, in the first moments after the Big Bang, or at the Planck scale where quantum fluctuations of spacetime itself become important, our current theories break down. General r...

Problem Pit: The Matrix Eigenvalue Maze The Problem Find all possible values of $\det(A)$ where $A$ is a $3 \times 3$ matrix with integer entries such that $A^3 = 2A^2 + 3A - I$. This problem looked straightforward initially - just find the eigenvalues and compute their product. But as I dug deeper, I encountered surprising connections to algebraic number theory, minimal polynomials, and the subtle relationship between characteristic and minimal polynomials. First Approach: The Characteristic Polynomial Route If $A$ satisfies $A^3 = 2A^2 + 3A - I$, then rearranging: $$A^3 - 2A^2 - 3A + I = 0$$ This means the minimal polynomial of $A$ divides $m(x) = x^3 - 2x^2 - 3x + 1$. If $\lambda$ is an eigenvalue of $A$, then $m(\lambda) = 0$, so: $$\lambda^3 - 2\lambda^2 - 3\lambda + 1 = 0$$ Let me find the roots of this cubic. Using the rational root theorem, possible rational roots are $\pm 1$. Testing $\lambda = 1$: $$1 - 2 - 3 + 1 = -3 \neq 0$$ Testing $\lambda = -1...

Problem Pit: The Stubborn Series Convergence The Problem Determine the convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n^p}$ for various values of $p$. This series fascinated me with its combination of the rapidly growing factorial in the sine function and polynomial decay. Standard convergence tests seemed useless against the erratic behavior of $\sin(n!)$, leading me into the world of equidistribution theory and deep properties of factorials. The Obvious Case First Since $|\sin(n!)| \leq 1$ for all $n$, we have: $$\left|\frac{\sin(n!)}{n^p}\right| \leq \frac{1}{n^p}$$ By comparison test, if $p > 1$, the series converges absolutely since $\sum \frac{1}{n^p}$ converges. But what about $p \leq 1$? This is where things get interesting and where standard methods fail. Dead End #1: Ratio and Root Tests The ratio test gives: $$\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \left|\frac{\sin((n+1)!)}{(n+1)^p} \cdot \frac{n^p}{\sin...

  Problem Pit: The Persistent Polynomial Mystery The Problem Find all polynomials $P(x)$ such that $P(x^2) = P(x) \cdot P(x+1)$ for all real $x$. When I first saw this functional equation, it looked straightforward. Just find a polynomial satisfying one condition - how hard could it be? Turns out, this innocent equation would drag me through multiple dead ends before revealing its secrets. First Attempts: Plugging in Values My instinct was to substitute simple values and see what constraints emerged. Starting with $x = 0$: $$P(0^2) = P(0) \cdot P(0+1)$$ $$P(0) = P(0) \cdot P(1)$$ If $P(0) \neq 0$, then $P(1) = 1$. Next, $x = 1$: $$P(1^2) = P(1) \cdot P(1+1)$$ $$P(1) = P(1) \cdot P(2)$$ Since $P(1) = 1$, we get $P(2) = 1$. With $x = 2$: $$P(4) = P(2) \cdot P(3) = 1 \cdot P(3) = P(3)$$ And $x = -1$: $$P(1) = P(-1) \cdot P(0)$$ $$1 = P(-1) \cdot P(0)$$ So I was getting specific values at integer points, but needed a systematic approach. Dead End #1: The...

  The Multiverse Question - Are We Living in One Universe Among Many? The possibility that our universe might be just one of infinitely many universes fundamentally challenges our understanding of reality, existence, and our place in the cosmic order. My exploration of multiverse theories began with seemingly innocent questions about fine-tuning and evolved into grappling with the deepest philosophical questions in cosmology: What makes our universe special? Are the laws of physics unique? And if multiple universes exist, how could we ever know? The fine-tuning problem provided my entry point into multiverse thinking. Our universe appears remarkably well-suited for the existence of complexity, life, and consciousness. If the strong nuclear force were slightly weaker, nuclei wouldn't form. If it were slightly stronger, hydrogen would be rare and long-lived stars impossible. The electromagnetic force, gravitational strength, and even the mass differences between particles all seem ...

  Trust and Vulnerability: The Currency of Deep Relationships She hesitates for a moment, looking down at her coffee before meeting your eyes. "I haven't told anyone this," she begins, and you can feel the weight of what's coming. In the next few minutes, she shares something deeply personal – a fear, a failure, a dream she's afraid to voice. Your response to this moment will determine whether your relationship deepens into genuine intimacy or retreats back to the safety of surface-level connection. This scene plays out countless times in human relationships, though we rarely recognize its significance in the moment. These instances of vulnerability – when someone risks emotional exposure by sharing their authentic self – are the building blocks of meaningful connection. But vulnerability without trust is simply dangerous exposure, and trust without vulnerability remains shallow and limited. Together, they form the currency that purchases our deepest and most s...