Why is it a known fact that the angle between two random vectors in a high dimensional space is atleast 45 degree?

TLDR: I was reading up on the Navier Stokes equations today and it is so elegant that it might be my favourite math equation. Equation 1: ∇u = 0 (conservation of mass) states that the divergence of the velocity vector u is zero, meaning there is no net change in fluid mass. Equation 2: ρ Du/Dt = -∇p + μ∇^2 u + ρF (conservation of momentum) expresses Newton's second law for fluid flow. It balances the acceleration of fluid particles (LHS) with internal forces (pressure and viscosity) and external forces (gravity or other external influences) on the RHS. This equation is foundational for modelling various fluid dynamics scenarios, from celestial bodies like stars and galaxies to F1 cars. Long Version: Here's how it works: Equation 1: ∇u = 0 (conservation of mass) So, u is velocity that can be represented as (u,v,w) vector, where u,v,w are x,y,z components of the vector. ∇u tells us that we need to do a partial derivative on u. So, ∇u = ∂u/∂x + ∂v/∂y + ∂w/∂z = 0 or, the partial derivative of every component wrt corresponding direction is 0. Equation 2: ρ Du/Dt = -∇p + μ∇^2 u + ρ F (conservation of momentum) LHS: Since, u is velocity, then Du/Dt is acceleration and ρ is density. Newton's second law, F = m x a, applies here. Wherein, Du/Dt is acceleration of fluid particles and m is the density of the fluid. RHS: -∇p + μ∇^2 are the internal forces of particles hitting into each other while F represents the external force. F in most cases is gravity, so one can replace it with g. However, if you put in electromagnetism then, you can combine Navier-Stokes with Maxwell's equations. This has over time led to the development of magnetohydrodynamics, ie how stars and galaxies form. You can model the growth of our sun with this. ∇p is our pressure gradient and represents the change in pressure. Essentially, fluids move from high pressure to low pressure. μ∇^2 represents viscous forces yielding from viscosity. Imagine this can model aerodynamics of F1 cars.

Take this work a pinch of salt, not to be triggered. I work in a company with a good number of folks working in product, growth/ strategy. Many of which are from tier 1 Indian business schools or some even from abroad. Paid hefty, I just don't see them being that smart. Putting way too much time into presenting something instead of getting work done. Jugaad method of calculating target metrics, ever changing requirements (low on clarity) is just the norm. I was someone who looked upto these colleges and wanted to be in one after having a good 5 yoe. Now I'm not sure. Hoping to be proved wrong, so there's no need to fire your guns at me :)

I will continue with this series for people who like this kind of content, drop "+1" in the chat and I will tag you the next time I post content. Imagine you have a very complex situation with varying degrees of randomness. How do you evaluate the probability of certain outcomes? One way is a deterministic way which is to sit down and try to compute probabilities of events. Another way is to simulate these interactions with various seed values and see how these outcomes vary. This is called a Monte Carlo simulation, where we use random sampling to model and analyze complex systems that involve uncertainty. Let's set up a game and apply a Monte Carlo simulation to it. You(Hero) and Me(Villain) engage in a strategic coin flipping competition over a series of rounds. Before each round, both of us independently decide whether to flip a coin or pass the turn to the opponent. The outcomes are as follows: 1. If both players decide to flip, a fair coin is tossed, and the player who called it correctly gains a point. 2. If one player decides to flip and the other passes, the flipping player gains a point without the need for a coin toss. 3. If both players pass, no points are gained or lost. We can simulate it by making each decision random: 1. Hero and Villain both independently choose between 0 and 1 for deciding whether to toss or pass. We use 0 as Pass and 1 as Toss. 2. If both pass then we let the scores as is. 3. If one passes and not the other, then we add +1 to whoever decided to toss. 4. If both decide to toss then, we do a random coin flip where 0 = Heads and 1 = Tails. Hero can randomly choose between Heads or Tails. The Villain takes the opposite position. First is the Monte Carlo Simulation and the difference between potential outcomes for 1000 rounds and 20 simulations, which is the graph shared.

Talking to my nephew nowadays seems like the AI bug has gotten to him. Even going back during the early parts of my own career and even undergrad, ML and stats was still a fledgling field. Most of the smart folks went into SDE roles at top tech. I still remember that even back then, hiring for roles like these were extremely theoretical. You would test even experienced candidates with undergrad topics which was slightly outdated tbh because most of the functions were abstracted away and it is obvious that you would not remember the matrix approach to finding weights for a simple linear regression model. Has this changed now? Because I don't think interviews like that actually evaluate the accumulated wisdom that experienced professionals bring to the table. I mean it is generally a waste of time for anyone to brush up manual derivations or remember SQL niche intricacies. Usually it is better to test for the 80% knowledge that is actually useful on a day to day basis. I mean if the main things that companies test in experienced candidates is something that can be rote-memorized from a book then they are not testing the right things no? What do you all think? Just wondering about this for some time now especially considering the hype around AI?

Very interesting that GPT3.5 has similar characteristics to human respondents.

Would be great if you can share somethings that you've learnt it the hard way especially on topics like stakeholder management (dealing with engineering/ execs/ founders/ P&L team), learning/ upskilling, switching internally/ externally, perfection vs hacky stuff