all articles
Freedom of Model Building
How do scientists decide which model to use when several can fit the same data? This post follows that question through the assumptions and creative choices behind scientific models, and looks at how physical reasoning and experiments help us decide which ones deserve our confidence.
Bertrand's Paradox
What’s the probability that a random chord in a circle is longer than the side of an inscribed triangle? Surprisingly, the answer is 1/4… or 1/3… or 1/2. All are correct. Bertrand’s Paradox reveals why “random” is not a mathematical concept until you define the underlying probability measure — and why hidden modeling assumptions shape every simulation you run.
Lagrangian Mechanics
This article provides an in-depth exploration of Lagrangian mechanics, detailing its foundation in the principle of stationary action, its connection to Newtonian mechanics, and its application to complex systems like oscillating pendulums and double pendulums through elegant energy-based equations of motion
Logistic Map
This article explores the logistic map, a nonlinear difference equation capturing dynamic behaviors, including chaos, with insights into bifurcation and complex systems modeling
Logistic Growth Model
This article introduces the logistic growth model, exploring population dynamics with a carrying capacity, and compares it to a modified version with an optimal growth ratio using mathematical modeling and simulations
Tensor Intro
This article provides an introduction to tensors, exploring their geometric and mathematical definitions, transformation rules, notation, and applications, with detailed explanations of vectors, covectors, and coordinate system changes
Network Theory
This article provides an extensive overview of network theory, covering graph representations, directed and bipartite graphs, cocitation, bibliographic coupling, hypergraphs, paths, geodesics, and specialized concepts like Eulerian and Hamiltonian paths, supported by mathematical definitions and visual examples