Arya’s Algorithms

  The Statistical Nature of Reality - How Thermodynamics Emerges from Chaos The transition from studying individual particles obeying deterministic laws to understanding the collective behavior of vast numbers of particles opened my eyes to an entirely different way of thinking about physics. Statistical mechanics revealed that many of the most fundamental concepts we take for granted - temperature, pressure, entropy, and even the arrow of time - are not properties of individual atoms but emergent features arising from the statistical behavior of enormous ensembles. This realization fundamentally changed how I think about the relationship between microscopic laws and macroscopic reality. The journey began with a simple question that had puzzled me since first learning about atoms: if individual atoms follow precise, deterministic laws of motion, why does the macroscopic world seem filled with irreversible processes? When I drop a cup and it shatters, countless atoms suddenly rear...

  The Sacred Bond: Understanding Friendship in All Its Dimensions Dedicated to Hari Sheth, Sai Anish Reddy, Rajit Gupta, Terence George and all of the TEAS community. Thank you for being a friend when I needed one. In the grand tapestry of human experience, few threads are as vibrant, enduring, and essential as friendship. These voluntary bonds we forge with others transcend blood relations, geographical boundaries, and social constructs to create some of life's most meaningful connections. Yet despite friendship's central role in human flourishing, we often take these relationships for granted, failing to fully appreciate their complexity, power, and profound moral dimensions. This exploration delves deep into the multifaceted nature of friendship – examining how these relationships sustain us through life's challenges, the moral responsibilities they entail, and the myriad ways they shape who we become. From the playground bonds of childhood to the chosen families of a...

  The Elegant Machinery of Classical Mechanics - Lagrangians, Hamiltonians, and the Principle of Least Action Before diving deep into quantum mysteries and cosmic phenomena, I had to master the beautiful machinery of classical mechanics. What started as Newton's simple F = ma evolved into some of the most elegant mathematics in physics - a framework so powerful that it not only describes planetary motion and pendulums, but also provides the foundation for quantum field theory and general relativity. Understanding Lagrangian and Hamiltonian mechanics felt like discovering the hidden mathematical architecture underlying all of physics. Newton's second law seemed straightforward enough when I first encountered it. Force equals mass times acceleration - a simple relationship that explains why objects fall, why rockets launch, and how planets orbit. But as I tackled more complex systems - coupled oscillators, spinning tops, systems with constraints - the vector approach became inc...

  Measuring Angular Diversity: ε-Angle Classes in Planar Point Sets One of the joys of combinatorial geometry is how simple questions about points in the plane open doors to deep mathematics. From Erdős’s famous problem on distinct distances, to the countless studies of directions, incidences, and angles, each setting turns a finite set of points into a rich structure of combinatorial possibilities. This blog explores a new addition to that landscape: the ε-angular occupancy function , introduced in the paper “On ε-Angle Classes Determined by Planar Point Sets.” At heart, it asks: Given a set of points, how many “angle classes” do they determine if we only measure angles up to a resolution ε? This question blends discrete geometry with an “approximate” perspective: not every angle needs to be distinct, just distinguishable at scale ε. From Distances to Angles Let’s recall some classical milestones in the study of geometric diversity: Distinct distances problem (Erdős...

  Cosmic Dawn - The First Stars and Galaxies Light Up the Universe The period between the cosmic microwave background's last scattering and the formation of the first stars represents the universe's "dark age" - hundreds of millions of years when no stars shone and no galaxies existed. Understanding how the universe transitioned from this dark, simple state to the luminous, complex cosmos we see today became my window into cosmic evolution and the emergence of complexity from simplicity. The cosmic microwave background shows us the universe at age 380,000 years, when temperatures dropped enough for electrons and protons to combine into neutral hydrogen atoms. This recombination event made the universe transparent for the first time, releasing the thermal radiation we observe today as the CMB. But the temperature fluctuations in this ancient light are incredibly small - only about one part in 100,000. These tiny fluctuations were the seeds of all cosmic structure. Q...

  Nowhere Differentiability at the Edge: The Critical Takagi Function When we think of a “pathological function” in analysis, the usual suspects come to mind: Weierstrass’s nowhere differentiable monster, Cantor’s dust of measure zero, or perhaps the Koch snowflake with its infinitely jagged boundary. Each shows us that functions and shapes can behave in ways our geometric intuition never expected. Among these classics lives the Takagi function , a beautifully simple series that hides fractal complexity. In recent years, mathematicians have generalized it, producing an entire family of functions whose regularity depends on two parameters. And the paper we’re discussing today — “Nowhere Differentiability of the Generalized Takagi Function at the Critical Threshold” — nails down the last missing piece of the puzzle: what happens exactly at the delicate “critical” parameter setting. The Takagi Function and Its Generalization The original Takagi function was introduced in 1903...