Enjoying the thermodynamic limithttps://alessandromorita.com/https://alessandromorita.com/posts/odes/https://alessandromorita.com/posts/odes/How different methods for differential equations deal with notions like initial conditions and causalitySat, 14 Mar 2026 00:00:00 GMThttps://alessandromorita.com/posts/heat/https://alessandromorita.com/posts/heat/How constraining thermodynamic processes to one dimension makes heat into a proper state function, and heat capacity becomes well-defined.Sun, 08 Mar 2026 00:00:00 GMThttps://alessandromorita.com/posts/langevin/https://alessandromorita.com/posts/langevin/Connecting classical mechanics to statistical mechanics via the Langevin equation: giving a pendulum a temperature through stochastic forcing.Sat, 20 Sep 2025 00:00:00 GMThttps://alessandromorita.com/posts/adjoints/https://alessandromorita.com/posts/adjoints/From Lagrange multipliers to infinite-dimensional adjoint methods: an introduction to PDE-constrained optimization for engineering design.Mon, 06 Jan 2025 00:00:00 GMThttps://alessandromorita.com/posts/children_problem/https://alessandromorita.com/posts/children_problem/Three solutions to a classic probability problem: how many children does a couple have if they keep trying until they get a son?Thu, 26 Dec 2024 00:00:00 GMThttps://alessandromorita.com/posts/thermoelasticity/https://alessandromorita.com/posts/thermoelasticity/How heat and elasticity couple in solids, and why one can usually decouple the thermal and mechanical problems for sequential solving.Fri, 16 Aug 2024 00:00:00 GMThttps://alessandromorita.com/posts/collisions/https://alessandromorita.com/posts/collisions/Building a particle collision simulation from scratch, deriving the physics and implementing it without a game engine.Thu, 16 May 2024 00:00:00 GMThttps://alessandromorita.com/posts/attention/https://alessandromorita.com/posts/attention/Understanding the Transformer attention mechanism through the lens of three vector spaces: input embeddings, similarity space, and output space.Mon, 18 Dec 2023 00:00:00 GMThttps://alessandromorita.com/posts/cosmology/https://alessandromorita.com/posts/cosmology/A personal story about solving the Friedmann equations and plotting the entire history of the universe on a tablet screen.Fri, 04 Aug 2023 00:00:00 GMThttps://alessandromorita.com/posts/lie_derivatives/https://alessandromorita.com/posts/lie_derivatives/An algorithmic, notation-heavy approach to Lie derivatives in differential geometry, bridging abstract and index notation.Sun, 25 Jun 2023 00:00:00 GMThttps://alessandromorita.com/posts/local_coordinates/https://alessandromorita.com/posts/local_coordinates/How Riemann normal coordinates let us use Cartesian-like measurements on Earth's curved surface, and how good the approximation is.Thu, 11 May 2023 00:00:00 GMThttps://alessandromorita.com/posts/math_problem/https://alessandromorita.com/posts/math_problem/A beautiful high school geometry problem that can be solved in multiple ways, mixing algebra, geometry, and trigonometry.Sun, 16 Apr 2023 00:00:00 GMThttps://alessandromorita.com/posts/auc_optimiz/https://alessandromorita.com/posts/auc_optimiz/How to directly optimize the ROC AUC as a loss function using differentiable surrogates, with theory and implementation.Fri, 27 Jan 2023 00:00:00 GMThttps://alessandromorita.com/posts/smearing/https://alessandromorita.com/posts/smearing/How Jensen's inequality causes bias in log-transformed regressions, and how smearing retransformation fixes it.Sun, 25 Sep 2022 00:00:00 GMThttps://alessandromorita.com/posts/auc-variance/https://alessandromorita.com/posts/auc-variance/A full derivation of the variance of the ROC AUC statistic, including the special case of a random classifier under extreme class imbalance.Thu, 22 Sep 2022 00:00:00 GMThttps://alessandromorita.com/posts/linear_trees_in_LightGBM/https://alessandromorita.com/posts/linear_trees_in_LightGBM/A practical guide to using linear trees in LightGBM for piecewise linear models, with code examples and comparisons to standard GBDTs.Wed, 21 Sep 2022 00:00:00 GMThttps://alessandromorita.com/posts/cost_based_auc/https://alessandromorita.com/posts/cost_based_auc/How to incorporate misclassification costs into the ROC AUC metric using partial AUC, with derivations and visualizations.Tue, 20 Sep 2022 00:00:00 GMThttps://alessandromorita.com/posts/carr_madan/https://alessandromorita.com/posts/carr_madan/The Carr-Madan decomposition breaks any payoff into a combination of calls, puts, and a forward. We derive it and show numerical examples.Tue, 20 Sep 2022 00:00:00 GMThttps://alessandromorita.com/posts/probability_vector_space/https://alessandromorita.com/posts/probability_vector_space/Does it make sense to add two probabilities? We explore how probability distributions can inherit a vector space structure using the Aitchison geometry.Sun, 18 Sep 2022 00:00:00 GMT