Fermats Library (@fermatslibrary )

Fermats Library

Bio A platform for illuminating academic papers. We publish an annotated paper every week. Our chrome extension for arXiv: fermatslibrary.com/librarian
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Account created 10-09-2015 03:54:59
ID 3511430425

Buffer : Universe


10²⁴


🌌 Milky Way


10¹⁸


10¹⁵


💫 Solar System


☀️ Sun

🌍 Earth
10⁶


🏔 Mountains

🐋 Whales
👫 Humans

🐞 Ladybug
10⁻³


10⁻⁶
🦠 Virus

🧬 DNA
⚛️ Atoms

10⁻¹²


10⁻¹⁵

⚡️ Electrons
⚫️ Quarks

Buffer : Proof that the inverse of a symmetric matrix is also symmetric

I=Iᵀ
AA⁻¹=(AA⁻¹)ᵀ → since AA⁻¹=I
AA⁻¹=(A⁻¹)ᵀAᵀ → since (AB)ᵀ=BᵀAᵀ
A⁻¹A=(A⁻¹)ᵀAᵀ → since AA⁻¹=A⁻¹A=I
A⁻¹A=(A⁻¹)ᵀA → since A=Aᵀ
A⁻¹A(A⁻¹)=(A⁻¹)ᵀA(A⁻¹)
A⁻¹I=(A⁻¹)ᵀI
A⁻¹=(A⁻¹)ᵀ ■

Buffer : Today is the 119 anniversary of the birth of Enrico Fermi. In 1945, Fermi calculated the power of the Trinity atomic bomb based on the distance traveled by pieces of paper he dropped during the blast.

Heres the 1-page report he wrote after the explosion: fermatslibrary.com/s/my-observati…

Twitter Web App : Clarification on the assumptions used:
We are assuming a small sag. In this case, the weight is uniformly distributed horizontally, the rope satisfies the 1D wave eq., the shape is a parabola.

For a big sag, the weight is proportional to the arc length and we get a catenary.

Buffer : Proving that the Maxwell eq. ∇.B=0 means magnetic monopoles can't exist.

• Using the Divergence Theorem ∭ᵥ∇.B dV = ∯B.dS
• If ∇.B=0, then ∯B.dS=0, which means all magnetic field lines that leave the surface eventually come back in - there are no monopoles.

Buffer : How should interest rates on goods in transit be computed when the goods travel at close to the speed of light? The time taken in transit will appear less to an observer traveling with the goods than to a stationary observer. - Paul Krugman

Great read: fermatslibrary.com/s/the-theory-o…

Buffer : So far no one has been able to prove if e + π is irrational or not, however theres one thing we know: either e+π or eπ has to be irrational!

Heres why: youtube.com/watch?v=ZxA-xD…

Buffer : Objects sliding with no friction under the effect of gravity along a Tautochrone curve will reach the bottom of that curve at the exact same time, independently of where they start. pic.twitter.com/JdDiLtJ6GV

Twitter Web App : This is a good example of an early use of the brute force approach to check mathematical conjectures.

The paper published with the result had just 2 sentences: fermatslibrary.com/s/counterexamp…

Buffer : In 1769, Euler conjectured that for all integers n>1, k>1, if the sum of n kth powers of positive integers is a kth power, then n≥k: aᵏ₁+aᵏ₂+...+aᵏᵢ=bᵏ⇒n≥k

In 1966, Lander and Parkin disproved the conjecture by finding a counterexample through direct computer search

Buffer : The physicist Arnold Sommerfeld was the first person to use the contour integral sign ∮ in his 1917 paper Drudes dispersion theory from the point of view of Bohrs model and the constitution of H₂, O₂ and N₂