In algebraic geometry, embedding tropical varieties into toric varieties relies on the process of tropicalization and the use of toroidal embeddings. Rather than embedding as geometric spaces, tropical varieties are embedded as polyhedral fans or cone complexes within the fans of target toric varieties to study compactifications and intersection theory.
Toric Varieties: Geometric varieties constructed from combinatorial data known as fans (collections of polyhedral cones). Tropicalization: A procedure that maps an algebraic variety X (typically a subvariety of an algebraic torus T) to a polyhedral complex in a real vector space via valuations. Toric Closures: When you embed a tropical variety in a compact toric variety, the boundary points correspond to limits of points in the variety going towards the boundary of the torus. Embedding Methodologies...Toric Compactification: Any tropical variety (a rational polyhedral fan) can be compactified by embedding it into a complete toric variety. The fan of the toric variety is chosen to contain the tropical variety as a subcomplex, which encodes the combinatorics of the compactification. Toroidal Embeddings: For varieties that are not strict toric varieties, they can often be mapped into toric varieties using the theory of toroidal embeddings. This involves equipping the tropicalized spaces (cone complexes) with balancing conditions and intersection theories.
To make revising LLM architectures and training methods faster, I created a deck of 180 visual flashcards. It started as a personal hobby, but slowly became cheat code for reviewing LLM concepts before technical interviews. People love it!
A dead 2013 Butterfly Labs "Jalapeno" SHA-256 mining ASIC sat in a drawer for a decade. It became the excuse for a small, careful question: how much structure can a tiny, cheap model learn in SHA-256, and how would I know if I were fooling myself? (The ML runs on CPU and a HF job, not the ASIC; the dead miner is just the origin story.)
Three findings, written up honestly:
1. A sharp round-4 cliff. Round-reduced SHA-256 is ~100% distinguishable through 3 rounds, then collapses to chance at round 4 and stays there out to the full 64. Reproduced across 5 seeds.
2. A controls-gated bounded null on full SHA-256: no learnable structure above a ~0.22% resolution floor at n=4,000,000. That is a bounded null at this budget, not a claim that SHA-256 is random.
3. A "signal" in the iterated-hash dynamics that a permuted-label control unmasked as a label-prior artifact. The instrument caught its own false positive. That was the point of building the controls.
Negative results, stated with their resolution. The dataset carries the controls on every row.
Turns out : if we predict π earth we can save a lot of time looking for interesting things and less time looking at things that we expect to see.
Sentinel-2 imagery π°οΈbasically takes a long time to download towards earth. so our "near real time" systems are quite far from that in practical terms.
meanwhile , if we "predict" what we will see , based on what we do see , we can send down much less data in a timely way , and prioritize π‘earth-bound response .
I'm talking about illegal fishing , logging , mining or building in nature reserves , the more of that we predict early the more we're able to stop it on time.
Just published: how we built production Sango (Central African Republic) translation without fine-tuning, parallel corpus, or training compute.
The method β vocabulary-augmented prompting with a 581-entry native-speaker-verified lexicon β generalizes to any of the ~2,000 African languages at the same data-poverty level. Recipe, dataset, and code template all included.