Laboratory Assistant

[Thread] What is your favourite math equation?

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.

UI/UX Design Concepts for Power BI Dashboard

Hi folks, I am in charge of designing dashboards in power bi and would like to follow any UI/UX design concepts, especially while choosing color gradients. Primarly I am a Data Analyst and hence would love to know more about what colours to choose and all. Condition given that I have the right to set the standard irrespective of what colour client likes. Thanks

This Research Paper changed my life forever.

It was one of the papers that was discussed in my interview at Goldman. I came to know about this research paper a few years back after consulting a friend doing an ML PhD at University of Maryland, College Park. The explanation of the paper: 1. Initialize the neural network with small random values typically (-0.1,0.1) to avoid symmetry issues. 2. Now get ready to do Forward propagation: you pass thetraining data through the multilayer perceptron and compute the output. For each neuron in the MLP, calculate the weighted sum of its inputs and apply the activation function. (my favourite is tanh for LSTM applications) 3. Now compute the loss using a loss function like mean squared error, between output computed and the actual value. 4. Now get ready to do backpropagation, where you need to calculate the gradient of the loss function with respect to each weight by propagating the error backward through the network. 5. So, compute partial derivatives of the loss with respect to each weight, starting from the output layer and moving back to the input layer. 6. Here is the fun part: update the weights using the gradients obtained from the backward pass. here people usually use adam optimizer, which allows for accelerated stochastic gradient descent. Fun trivia: Adam stands for "Adaptive Moment Estimation". 7. Now repeat the forward and backward propagation process for numerous tries until theperformance of the model stabilizes.