Proof: [Lemma] sum(i=1..x) [degree n-1 polynomial] = [degree n polynomial] Proof of lemma: * Notice that x^n - (x-1)^n = x^n - x^n + nx^(n-1) - ... +- 1 = [degree n-1 polynomial, call it C(x)] * Hence by telescoping sums, sum(i=1..x) C(i) = x^n * Given any specific degree n-1 polynomial D(x), re-express it as D(x) = some k * C(x) + [deg <= n-2 stuff]. By induction on n, and by linearity, sum(1..x)D(i) = k * x^n + [deg <= n-1 stuff]. Therefore, sum(i=1...x) i^3 is degree 4, and (sum(i=1...x) i)^2 is degree 2*2 = 4. Now, explicitly evaluate (sum(i=1...x) i)^2 and sum(i=1...x) i^3 on x=1...5, answers are: 1, 9, 36, 100, 225 in both cases The two deg-4 polynomials are the same on five points, therefore they are equal Therefore sqrt(sum(i=1...x) i^3) = sum(i=1...x)

The parabolic potential well V(x) = ½kx² traps the particle. Unlike a classical ball that can sit still at the bottom, a quantum particle always has Zero-Point Energy. It can never be completely at rest (Heisenberg Uncertainty Principle). > Wavefunctions (ψₙ): Oscillating probability patterns with increasing “humps” and nodes as energy level n rises. > Quantization: Energy comes in discrete steps: Eₙ = ħω (n + ½) n = 0, 1, 2, ... > Dirac’s Ladder Operators elegantly raise and lower between these states. > Exact solutions use Hermite polynomials (Hₙ).

Proud of what our amazing team has accomplished. We spent the past few months pursuing one bet: that layouts are the right intermediate representation for generation and editing. [1/n]

Camera pose matters for video understanding! Today's MLLMs excel at recognizing activities, but still struggle with the underlying space and ego/object dynamics in video. We trace this gap to a missing piece: camera pose. Introducing Cambrian-P: a multimodal LLM natively grounded in camera pose. (1/n)

Cherenkov Radiation: Light’s Shock Wave. When a charged particle travels through a transparent medium (such as water or glass) faster than light can travel in that medium, it creates a beautiful electromagnetic shock wave: a cone of bluish light known as Cherenkov radiation. The physics is precise and elegant: > Threshold condition: v > c/n > Emission angle: cos θ = 1/(β n) where β = v/c and n is the refractive index of the medium. This striking blue glow is a direct visual demonstration of special relativity and classical electrodynamics working together in perfect harmony. It’s commonly seen in nuclear reactors and particle detectors.