Theorem of the Day (September 13, 2026) : Archimedes’ Equiareal Map Theorem
Source : Theorem of the Day / Robin Whitty
pdf : https://t.co/AJURNj3pFQ
notes : https://t.co/6xYBONdexA
#mathematics#maths#math#theorem
@anon597260576@AnyonCurve Every 6 months, people report it to Twitter, it is silenced into oblivion and then is posted again 6 months later. It’s called social media arbitrage
Placing regular polygons on complex plane so that some vertices or symmetry lines coincide with Re(s)=1/2\operatorname{Re}(s)=1/2\operatorname{Re}(s)=1/2 does not control the zeros of ζ(s)\zeta(s)\zeta(s). A visualization or reinterpretation! = a demonstration zeros lie there.
The Riemann Hypothesis is a precise statement about the non-trivial zeros of the Riemann zeta function ζ(s)\zeta(s)\zeta(s)
A proof requires showing that every such zero has real part exactly 1/21/21/2, using estimates on the growth of ζ\zeta\zeta.
#math
Three years ago, a user on "X" asked me if I could solve this problem. I found it interesting and solved it. I never asked him which book he got it from. I'm sharing the solution in case you're interested in analyzing it.
A(x) = ∑_{n≤x} a(n) is only a step function, yet it converts a discrete weighted sum into an ordinary integral. Abel’s identity states that whenever f is continuously differentiable,
∑_{n≤x} a(n) f(n) = A(x) f(x) − A(y) f(y) − ∫_y^x A(t) f′(t) dt.
The formula is summation by parts, the exact discrete counterpart of integration by parts, with the partial-sum function A playing the role of an antiderivative. Abel recorded the transformation in 1826; it has remained the standard device for moving from arithmetic functions to smooth estimates ever since.
Negation does not merely deny; it interchanges the two fundamental connectives.
¬(p ∧ q) ⇔ ¬p ∨ ¬q, and likewise ¬(p ∨ q) ⇔ ¬p ∧ ¬q.
The same pairing governs sets: (A ∩ B)′ = A′ ∪ B′. Augustus De Morgan gave the laws their modern form in 1847, after they had already been known, in words, to medieval logicians.
Paul Dirac’s 1928 equation put quantum mechanics and special relativity into one framework.
(iγ^μ∂_μ − m)ψ = 0
The γ matrices are 4 × 4 matrices, and their structure naturally accounts for the electron’s spin.
Back to basics: Algebraic Manipulation
by Abels Identity
Hint: Reciprocal 👌
Suggested solution on our YT channel
https://t.co/SHqAabKXO5
#sharingisthenewlearning
The number 7 is a tensor. So is every vector, every matrix, and the 4-way array that holds a color video.
Tensors simply keep adding axes. Rank 0: a scalar (no index). Rank 1: a vector. Rank 2: a matrix. Rank 3 and beyond:
A ∈ ℝ^{m × n × p}
A photo is height × width × color. A video adds time. A training batch adds one more. That is why vision models, climate codes and transformers all speak the same language.
Higher dimensions. Richer structure.
Three numbers - σ = 10, ρ = 28, β = 8/3 - turn a deceptively simple system into an inexhaustible source of surprise:
dx/dt = σ(y − x)
dy/dt = x(ρ − z) − y
dz/dt = xy − βz
The solution never repeats and never settles; it winds forever around two lobes of a bounded fractal set now called the Lorenz attractor.
A change of one part in a thousand in the starting point sends the future onto the opposite wing after only a few dozen time units.
Edward Lorenz extracted the equations in 1963 while stripping the physics of thermal convection down to its barest skeleton. What remained was the first widely recognized example of deterministic chaos.