Miguel Alex Rom Jar ∴

Engineer, pragmatic humanist, pantheist, scientific divulger, geek & badass

[Epic Maps 🗺️ @theepicmap]

THE GEOID — Earth’s true shape defined by gravity

[Life Doctor @LifeDoctor247]

Sitting a secret in making your sciatica

[Pablo Alcalde @HProductiva]

Un minuto parece poco hasta que lo haces todos los días y notas cómo cambia tu cabeza.

[Cosmos Archive @cosmosarcive]

“The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe.”

— Philip W. Anderson

[美女達の話題。 @bijo_wadai]

横から見た時のシルエットが綺麗な人って体重よりも姿勢が違う。

体を伸ばすだけで
首・肩・背中の見え方かなり変わる↓

[The Scientific Lens @LensScientific]

"What we observe as material bodies and forces are nothing but shapes and variations in the structure of space and time."

– Erwin Schrödinger

[Lex @lexx_aura]

Bro, this Goku vs Saitama fight goes harder than any official episode I've seen lately 🔥

The way they just keep breaking their limits is absolutely insane to watch

Huge props to the creators for putting together this epic Fan Made masterpiece

[Imladris @__Imladris__]

La Casa de la Moneda de Reino Unido ha lanzado una colección especial para celebrar el 25 aniversario de El Señor de los Anillos. El diseño está basado en el Anillo Único y cuando la luz atraviesa la moneda se proyecta el ojo de Sauron.

[WhatdoIknow @Earstohearyou]

From SOL light to SOUL life
Reality is a wheel -
a spiral through time.

Form divides.
Experience transforms.
CONSCIOUSNESS returns.

[Tech with Mak @techNmak]

This math sits underneath every AI model being trained right now.

Gradient. Jacobian. Hessian.

Three words that look intimidating at first.

But they are really just three ways of measuring change.

𝟭. 𝗚𝗿𝗮𝗱𝗶𝗲𝗻𝘁 ∇f
Takes a scalar function:

f : ℝⁿ → ℝ

Returns a vector of first-order partial derivatives.

It answers:

"Which direction makes f increase fastest?"

That is why gradients are central to optimization.

Gradient descent moves in the opposite direction because the gradient points uphill.

Backpropagation efficiently computes gradients during training.

𝟮. 𝗝𝗮𝗰𝗼𝗯𝗶𝗮𝗻 J_F
Takes a vector-valued function:

F : ℝⁿ → ℝᵐ

Returns an m × n matrix of first-order partial derivatives.

It answers:

"How does each output change with each input?"

The Jacobian is the local linear map of a vector-valued function.

It shows up in:
→ sensitivity analysis
→ change of variables
→ automatic differentiation
→ forward-mode AD
→ reverse-mode AD / backpropagation

In simple terms:

forward-mode AD uses Jacobian-vector products.

reverse-mode AD uses vector-Jacobian products.

𝟯. 𝗛𝗲𝘀𝘀𝗶𝗮𝗻 H_f
Takes a scalar function:

f : ℝⁿ → ℝ

Returns an n × n matrix of second-order partial derivatives.

It answers:

"How does the gradient itself change?"

That means the Hessian measures curvature.

When the second partial derivatives are continuous, the Hessian is symmetric.

At a critical point:
→ positive definite Hessian → strict local minimum
→ negative definite Hessian → strict local maximum
→ indefinite Hessian → saddle point

The clean mental model

Gradient = first derivatives of one output
→ tells you direction

Jacobian = first derivatives of many outputs
→ tells you sensitivity

Hessian = second derivatives of one output
→ tells you curvature

And the relationship between them is simple:
The Hessian is the Jacobian of the gradient.

For a scalar output, the Jacobian contains the same partial derivatives as the gradient, up to row/column convention.

Same idea:
measure change.

Different object:
direction, sensitivity, curvature.

Once this clicks, optimization stops looking like a pile of formulas.

It starts looking like a map of the problem.

[Tansu Yegen @TansuYegen]

So true☹️

📹 vetoshkin.evgeniy

[Paty Zavala @PatyZavalat]

🚫 NO AL DECRETO LORETO Península de Baja California en el Golfo de California 🚫

Las Ballenas nos necesitan! 🐳🐋💧🌊🇲🇽 Seamos su voz 🙌📢

#SalvaLoreto #BallenaAzul #LoretoSinCruceros #SantuarioDeBallenas #BahíaDeLoreto #BajaCaliforniaSur #México

[CopyRebeldia @CopyRebeldia]

La biología en PDF acaba de morir.

Un tío hizo una app donde exploras estructuras 3D como un videojuego.

UI: GPT Images 2. Código: Gemini 3.1 Pro.

Los libros de texto ya no sirven.

[Mathematica @mathemetica]

Shannon Entropy: Measuring Uncertainty in Information

H(X) = - ∑ P(xᵢ) log P(xᵢ)
This is the legendary formula by Claude Elwood Shannon (1916–2001); the father of Information Theory.
Entropy quantifies how much uncertainty (or average information) is contained in the outcome of a random variable X. The more unpredictable the outcomes, the higher the entropy.

From data compression and cryptography to AI and communications; this concept powers the digital world.

[φ @QuanticASI]

there are no atheists in quantum physics

[火ノ国人 @maekawa_4405]

ACRONYM® | J91U-WSKPが届きました！
前回買ったJ91U-WSDBからグレーになっただけで、なんでこんなに魅力的なんだ。
一着買ったら一着増える魔力。恐ろしい！

#KojimaProductions
#コジマプロダクション

[Shohan @Shohan_Tech_AI]

Scientific calculator functions and the uses...

[The Scientific Lens @LensScientific]

This is the longest equation in physics: the Standard Model Lagrangian.

It is the complete mathematical summary of the Standard Model, our best theory describing how all known elementary particles interact through the electromagnetic, weak, and strong forces.

[Mathematica @mathemetica]

One curve.

Three projections.f(t) = e^{-γ(t-t₀)²} ⋅ e^{iωt}

From the Re-t plane: damped cosine.
Im-t: damped sine.
Re-Im: perfect inward spiral.Same reality, different slices.

[Dyz∄r NiHil ψ @Dyzer_Nihil]

Recuérdalo siempre, y no dejes que nadie te contradiga.