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Geometry of Seeing Part 2 47 min

Euler's Elastica and Jacobi Elliptic Functions

The Pontryagin Maximum Principle turns the SE(2) geodesic problem into a pendulum ODE. Its solutions are the three families of Euler's elastica, parametrised exactly by Jacobi's elliptic functions sn, cn, dn — and their spatial period is 4K(k²), the complete elliptic integral of the first kind.

Deriving Euler's elastica from the Pontryagin Maximum Principle on SE(2): the pendulum equation, three curve families (inflectional, non-inflectional, Euler spiral), their Jacobi elliptic parametrisations, and the period 4K(k²).

Mathematics
sub-riemannian SE2 elastica jacobi-elliptic
Euler's Elastica and Jacobi Elliptic Functions
Geometry of Seeing Part 1 90 min

The Visual Cortex as a Contact Manifold

Your brain fills in contours that do not exist. Jean Petitot showed in 2003 that this visual completion is equivalent to finding shortest paths on the Lie group SE(2) — the group of rigid motions of the plane — equipped with a sub-Riemannian metric whose geodesics are Euler's elastica, parametrised by Jacobi elliptic functions.

Petitot's 2003 model of primary visual cortex V1 as a contact manifold on SE(2), the sub-Riemannian metric, and how optimal contour completion reduces to geodesics parametrised by Jacobi elliptic functions.

Mathematics
sub-riemannian SE2 visual-cortex elliptic-functions
The Visual Cortex as a Contact Manifold
Software 9 min

Foundation Models, The AI Frontier

A four-year-old child processes roughly one petabyte of visual information — fifty times more than GPT-4 has ever read — and learns effortlessly. Closing that gap is what's pushing AI past sequential token prediction toward predictive world models, retrieval-augmented memory, and Karpathy's Software 3.0 paradigm: the LLM as operating system.

Artificial Intelligence has evolved rapidly. Central to this evolution are Foundation Models — advanced AI systems trained on vast datasets, capable of adapting to numerous tasks without extensive retraining.

machinelearning deeplearning
Foundation Models, The AI Frontier
Software 24 min

Train Resnet50 on ImageNet with PyTorch

Eighty lines of PyTorch — RandomCrop and ColorJitter for augmentation, DataParallel for multi-GPU, cross-entropy with Adam — fine-tune a pretrained ResNet-50 on the ImageNet Object Localization Challenge. Wrap it in a Dockerfile that pulls the dataset from Kaggle and you have a self-contained, reproducible 1000-class training pipeline.

A complete single-file training script for ResNet-50 on ImageNet using PyTorch — covering data loading, mixed-precision training, learning-rate scheduling, and top-1/top-5 accuracy tracking.

machinelearning deeplearning
Train Resnet50 on ImageNet with PyTorch
Programming 3 min

Find all tables without primary key in PostgreSQL

A single LEFT JOIN against information_schema.table_constraints exposes every BASE TABLE missing a PRIMARY KEY — the silent latency tax that creeps in through schema migrations and ad-hoc DDL. Audit the whole cluster, then narrow to a single schema with one extra WHERE clause.

Query that helps to find tables without primary key, simple and efficient way to find performance bottlenecks

postgresql database linux