Arya’s Algorithms

  Functional Equations and Combinatorial Constructions of Pathological Functions in Real Analysis By Arya Dubey                                                                                                                                      Check out the full paper Abstract The existence of continuous nowhere differentiable (CND) functions remains one of the most striking and counterintuitive phenomena in real analysis. These objects are continuous everywhere but possess no derivative at any point. Once regarded as pathological monsters, they now arise naturally in probability, dynamics, fractal geometry, and harmonic analysis. This blog post ...

Pathological Beauty: Constructing Continuous but Nowhere Differentiable Functions: An Overview        When most of us first meet calculus, we’re told that a function is “nice” if it’s smooth — if its graph doesn’t have corners, jumps, or wild oscillations. We imagine curves that can be followed with the stroke of a pencil: circles, parabolas, sine waves. Differentiability, the existence of a tangent line at every point, feels almost automatic once continuity is in place. But mathematics has a way of defying intuition. In 1872, Karl Weierstrass shocked the mathematical world by constructing a function that is continuous everywhere but differentiable nowhere. A function you can draw without lifting your pencil — yet at no point can you lay down a tangent line. These so-called pathological functions became objects of fascination. At first, they seemed like monsters, counterexamples to the clean world analysts hoped to live in. But over time, they were revealed...

The Fragile Thread of Memory and the Self Sometimes I wonder: if all my memories were stripped away in a single moment, who would remain? Would I still be “me,” or would the person I call myself vanish with the past? Memory feels like a fragile thread tying together every moment of my existence. And yet, when I look closely, I am not sure how strong that thread really is. Memory as the Architect of Identity When I say “I,” it is usually a bundle of memories that speaks. I remember the face of my mother when I was a child, the classroom where I first solved a difficult math problem, the smell of rain during a walk home from school. These recollections are not just events—they are bricks in the house of identity. Without them, the house collapses. But then, memory is slippery. Neuroscientists remind us that each time we recall something, we do not retrieve a file from a cabinet—we reconstruct it, reshaping the past in the present. My childhood memory may not be what truly happened...

The Pit of Integer Partitions: When Numbers Break Apart There are math problems that are easy to state but impossible to tame. Integer partitions are one of the purest examples. The problem sounds like child’s play: In how many ways can you write n n as a sum of positive integers, order irrelevant? For example, with n = 4 n=4 : 4 = 4 , 3 + 1 , 2 + 2 , 2 + 1 + 1 , 1 + 1 + 1 + 1. 4 = 4, \quad 3+1, \quad 2+2, \quad 2+1+1, \quad 1+1+1+1. So there are 5 partitions of 4. Simple, right? That’s what I thought. Then I fell into the pit. Step 1: First Steps Let’s compute the partition numbers p ( n ) p(n) : p ( 1 ) = 1 p(1)=1 . p ( 2 ) = 2 p(2)=2 : 2 , 1 + 1 2,1+1 . p ( 3 ) = 3 p(3)=3 : 3 , 2 + 1 , 1 + 1 + 1 3,2+1,1+1+1 . p ( 4 ) = 5 p(4)=5 . p ( 5 ) = 7 p(5)=7 . p ( 6 ) = 11 p(6)=11 . The sequence is: 1 , 2 , 3 , 5 , 7 , 11 , 15 , 22 , 30 , 42 , … 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, \dots Already I was tempted to guess a formula. I failed. Step 2: The Wro...

  Black Hole Revelations - Where Physics Breaks Down and New Laws Emerge Black holes represent the universe's most extreme laboratories, where gravity becomes so strong that space and time themselves break down. My journey into these cosmic monsters began with a simple question: what happens when you fall into a black hole? The answer led me through some of the deepest paradoxes in modern physics and to the frontiers of our understanding about information, entropy, and the nature of spacetime itself. The classical picture seemed straightforward enough. Karl Schwarzschild found the exact solution to Einstein's field equations for a spherically symmetric mass just months after general relativity was published. The Schwarzschild metric describes spacetime geometry around any non-rotating mass, from planets to black holes. The critical difference lies in whether the object's surface lies inside or outside the Schwarzschild radius rs = 2GM/c². But black holes aren't just ...

  The Introvert-Extrovert Dance: Understanding Different Social Energy Styles The party is in full swing. In one corner, someone is holding court with an animated story, gesturing wildly while a growing crowd laughs and adds their own comments. Across the room, two people are having an intense one-on-one conversation about philosophy, completely absorbed in each other's ideas. Near the kitchen, someone is helping the host with dishes, grateful for a task that allows them to contribute while taking a break from socializing. Later that evening, the storyteller will feel energized and ready for more social connection, while others will be completely drained and need hours of solitude to recharge. This scene illustrates one of the most fundamental differences in how humans experience social interaction: the distinction between introversion and extroversion. Yet despite decades of research and popular psychology, these concepts remain widely misunderstood, often reduced to simplistic...

  The Alpha Complex: Understanding Dominance, Leadership, and the Drive to Be on Top They walk into rooms like they own them, speak with unwavering confidence, and somehow always end up in charge of group decisions. But behind the alpha's commanding presence lies a complex psychology of dominance, insecurity, and an relentless drive to maintain their position at the top. Here's what really drives those who seem naturally born to lead. You know them instantly. They're the ones who take charge when everyone else is standing around confused, who speak first in meetings and somehow get others to follow their lead, who seem to navigate social hierarchies with an instinctive understanding of power dynamics. They're the natural leaders, the decision-makers, the ones others look to when crisis hits and someone needs to take control. But the psychology of "alpha" behavior is far more complex than the confident exterior suggests. Behind that commanding presence often...