Renan Pighini

PhD student in Computational Mathematics at @icmc_usp. Researching Computational Fluid Mechanics, with a focus on Non-Newtonian Fluids.

[مريم | بيانات @mrym_data]

🙈

[Matheorems @matheorems]

The Tangent Plane Rides Along the Surface

At every point p of a smooth n-dimensional surface M there is a flat space of all the directions you can move in without leaving it: the tangent space T_pM.

If phi: U in R^n to M is a parameterization with phi(a) = p, the partial derivatives

d phi / d x_1 (a), ..., d phi / d x_n (a)

form a basis of T_pM, so T_pM is an n-dimensional vector space.

Here both surfaces are graphs, phi(x_1, x_2) = (x_1, x_2, f(x_1, x_2)), so the two basis vectors are exactly (1, 0, df/dx_1) and (0, 1, df/dx_2).

Top: the bowl f = (x_1^2 + x_2^2) / 3 lies above every one of its tangent planes.
Bottom: the cap f = 1 - (x_1^2 + x_2^2) / 3 lies below every one of its tangent planes, and at the top the plane is level.

The point a moves in the square U, p = phi(a) moves on the surface, and the plane spanned by the two partial derivatives tilts with it.

[Proswit @proswit]

How to calculate the Gaussian integral?

[(｀∀´)Ψ @0xcinzano]

uₜₜ = c²∇²u
uⁿ⁺¹ = 2uⁿ − uⁿ⁻¹ + c²Δt²∇²uⁿ

[Renan Pighini @matema_tia]

@CarlosMag30 Aí há um falso dilema. O fato de outras profissões não exigirem o mesmo exame não significa que ele seja injustificável para a medicina. São atribuições e responsabilidades distintas. A necessidade de avaliar a competência médica deve ser analisada por seus próprios méritos.

[matharium @matharium]

A 2×2 matrix A makes the plane flow by x′ = Ax, and each flow is one of nine kinds: points sink, escape, spiral, circle or settle on a line.

The kind is set by the eigenvalues of A, the roots of λ² − (tr A)λ + det A = 0. det A < 0 gives a saddle. Above det A = (tr A)²/4 they are complex: spirals, or a centre if tr A = 0. Below it, nodes; on it, star and degenerate node.

The stable spiral here has tr A = −1, det A ≈ 2.65, so λ = −0.5 ± 1.55i.

[Scholarship for PhD @ScholarshipfPhd]

My thesis is almost done. It has been almost done for longer than the research itself took.

[Mathematica @mathemetica]

Why can one measure be completely recovered from another using only a single density function?

The Radon–Nikodym theorem guarantees that whenever a measure is absolutely continuous with respect to a positive σ-finite measure, it can be written as the integral of an integrable function (the density) against the base measure. Absolute continuity simply means the second measure never places mass where the first has none.

This single idea underlies importance sampling in machine learning, change-of-measure methods in quantitative finance, and likelihood ratios throughout statistics and signal processing.

Mathematics turns the qualitative notion of “more concentrated here” into a concrete function that algorithms can compute and reweight.

[Math Files @Math_files]

Gradient, Divergence and Curl

[Miguel Nicolelis @MiguelNicolelis]

My home town making me proud to be a Paulistano and a Brazilian!
Verá que um filho teu ñ foge à luta!

[Miguel Nicolelis @MiguelNicolelis]

O mundo todo hoje descobriu que a maior democracia do planeta se chama:
BRASIL
Nem Europa, Nem EUA!
Aqui ex-presidente golpista vai em cana!
Aqui o povo na rua derrotou o algoritmo das BigTechs!
Aqui o Brasil enterrou o fascismo pelo voto popular!
Terra Adorada! BRASIL!

[Miguel Nicolelis @MiguelNicolelis]

Ouviram do Ipiranga as margens plácidas
De um povo heróico o brado retumbante
E o Sol da Liberdade, em raios fúlgidos
Brilhou no ceú da Pátria Neste Instante
Mas se ergues da justiça a clava forte
Verás que um filho teu ñ foge à luta
Dos filhos deste és mãe gentil
PÁTRIA AMADA
BR

[Physics In History @PhysInHistory]

We will always have STEM with us. Some things will drop out of the public eye and will go away, but there will always be science, engineering and technology. And there will always, always be mathematics. Everything is physics and math.

- Katherine Johnson