0Persepctrum.com @0Perspectrum

Universe is a pretty big place. If it's just us, seems like an awful waste of space. - Carl Sagan,

We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or how you had the wrong idea first... So there isn't any place to publish, in a dignified manner, what you actually did in order to get to the do the work. -Richard Feynman

Usually books on geometry have only a very few if no figures.... ...But personally, I like figures to understand quickly and grasp concepts! IMHO, this ***great book*** is a masterpiece at conveying notions and intuitions of geometry with many figures! Highly recommended!!!

The Anatomy of Bayes’ Theorem.

🌀 From simple curvature toward dynamic curling. A modified framework primarily designed to explains the universal mechanics of spin and rotation at all scales. As the Einstein's Field Equation is insufficient to fulfill comprehensive Unification Theories, here’s proposed a solution that formalize the ubiquitous presence of circular motion, by incorporating torque and Coriolis force into the spacetime manifold as a viscous, rotating medium, where the angular momentum of the continuum serve as the fundamental driver of spin in organized matter. When a gradient of energy-density moves toward centripetal equilibrium, it induces a natural vortex or spin. The framework expands the formulation into a higher-dimensional space referred to as the U₄ spacetime manifold. This accommodates the components directly into the curvature and stress-energy tensors of general relativity. Contrarily to standard cosmological models the new approach posits that spin is an inherent geometric property of the spacetime vacuum itself, framing the resulting solution as a modified Kerr-Newman metric. 🔗 researchgate.net/publication/22…

This 17 page pdf reveals the same technique Hedge Funds like Jim Simons' Renaissance Technologies use to find signal through noise. Stanford released the complete Hidden Markov Model framework for everyone to use it. Bookmark it before someone takes it down:

The best textbook I ever read was by @stevenstrogatz

Michael Nielsen @michael_nielsen

3 years ago

What’s the most wonderful textbook you’ve ever read?

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GBrain is your company brain

Shann³ @shannholmberg

4 days ago

how I’m building an agent company inside my agency. the structure looks like this: Agency gBrain → Orchestrator Hermes Agent → Department verticals → Specialist agents → Scoped sub-agents gBrain is the company brain. It gets ingested with the data and experience we

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Curvatures shape how any 3D surface bends locally. This diagram visualizes a parametrized surface r⃗(x) equipped with unit normal N and principal directions X₁ and X₂. It presents the principal curvatures κ₁ = 1/r₁ and κ₂ = 1/r₂ along with extrinsic curvature H = κ₁ + κ₂ = 1/r₁ + 1/r₂ and Gaussian curvature K = κ₁κ₂ = 1/(r₁r₂). The top insets highlight examples of local curvature geometry. These measures underpin applications from architectural shell design and vehicle aerodynamics to realistic physics simulations in digital animation.

Did you know that one of the most beautiful ideas in physics may not be F = ma, but action? We first learn motion through forces: push a box and it moves, gravity pulls a planet and it orbits, acceleration follows from cause and effect. But deeper down, physics offers a more elegant view. Through the principle of stationary action, the universe can be described as if it is selecting one path from many possible ones, the path where a hidden quantity called action comes out just right. For many classical systems, action is built from kinetic and potential energy over time. A falling apple, a planet around the Sun, and even light bending through glass can be understood through this kind of variational thinking. This idea shaped Lagrangian mechanics, Hamiltonian mechanics, quantum mechanics, and Feynman’s path integrals. It does not mean nature is literally thinking, but the mathematics makes motion feel almost poetic. Physics becomes not just the study of what happens, but of what could have happened, and why reality follows this path instead.

Google has published a paper that might end the transformer era. For the last 7 years, every major AI, ChatGPT, Claude, Gemini, has been built on the exact same architecture: The Transformer. But Transformers have a fatal flaw. To remember context, they have to process every single word against every other word. It’s called quadratic complexity. As your prompt gets longer, the compute cost explodes. The alternative is the old-school RNN (Recurrent Neural Network). RNNs are incredibly cheap and fast, but they have a fixed memory size. If you give them a long document, they get amnesia. Until today. Google researchers published Memory Caching: RNNs with Growing Memory. And it fixes the biggest bottleneck in AI. Instead of an RNN having a fixed, rigid memory that constantly overwrites itself, Google gave it a "save" button. The technique allows the RNN to cache checkpoints of its hidden states as it reads. The memory capacity of the RNN can now dynamically grow as the sequence gets longer. They built four different variants, including sparse selective mechanisms where the AI actively chooses exactly which checkpoints matter most. The results rewrite the rules of efficiency. On long-context understanding and recall-intensive tasks, these new Memory-Cached RNNs closed the gap with Transformers. They achieved competitive accuracy without the explosive, quadratic compute cost. It perfectly bridges the gap between the cheap efficiency of an RNN and the massive capability of a Transformer. We have spent billions scaling Transformers because we thought they were the only way an AI could remember a long conversation. But Google just proved we don't need to process the whole history every single time. We just needed a smarter cache.

Introducing a New Series ✍️ Electrostatics ~ (Electrodynamics) >Electric ⚡ Charge >Coulomb's Law >Principle of Superposition

The Superformula. A single generalisation of the equation for a circle, with a handful of parameters. Sweep them and the same formula becomes a flower, a starfish, a crystal, a cell. Nature did not need a thousand blueprints, it just needed one with good knobs. #GenerativeArt #math #simulation

"I looked round the trees. The thin net of reality. These trees, this sun. I was infinitely far from home. The profoundest distances are never geographical." ~John Fowles, The Magus

The periodic table of elements scaled to show the actual abundance of the elements on Earth by Prof. Wm. F. Sheehan in 1976.

Karl Weierstrass (31 Oct 1815 – 19 Feb 1897) constructed what a function looks like that’s continuous everywhere but has no derivative at any point: f(x) = ∑_{n=0}^∞ aⁿ cos(bⁿ π x). The magnified inset reveals its fractal self-similarity at every scale. “A mathematician who is not also something of a poet will never be a complete mathematician.” Today this function underpins fractal geometry and models irregular real-world phenomena; from coastlines and turbulent flows in physics to stock-price volatility in finance and rough signals in engineering.

MIT Entrance Examination

This is the single most beautiful moment in all of calculus. The limit as h → 0... Watch the secant line collapse into the exact tangent at that point.

3D visualization of Euler’s formula: e^{jθ} = cos(θ) + j sin(θ)