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Just eight hours old, listen while its fresh!

I've only watched the short and entertaining animated video https://www.youtube.com/watch?v=83ZietQ1W_s

I've linked to the University of Aarhus webpage, which has lots more detail.

Interesting to see the numbers, about ten billion a year from tobacco, about four billion a year from wine, around four billion a year from spirits.

This video is about the market for expensive (say $10,000 to $100,000) paintings. Although made by an art lover, the account of how the market works is pretty brutal in its cynicism. His line is "Prices can fall, so just pay your money, hang it on your wall, and enjoy it." but maybe he is revealing that you $100,000 painting was never worth $10,000 to begin with

They lifted the siege of Vienna on the 12th of September 1683

A video about numerical methods for solving differential equations. With pretty animations. And a nice exposition of fourth order Runge-Kutta

The article contains a reminder of past events

>The United States and its allies have not operated in airspace as contested as Ukraine’s since the Vietnam War. Over the skies of Vietnam the US would lose over 10,000 fixed-wing aircraft and helicopters to Soviet-made air defenses employed by Vietnam’s armed forces.

while conceding that things have changed and "the rapid maneuver warfare utilized by US-led forces in Iraq" no longer works

My new favorite performance of this study. Very fast, but at the service of bringing out the tune in the left hand octaves. Plenty of variation in the dynamics and articulation. Beautiful handling of my favorite bit, bar 42 to 45, where the arpeggios down and up within a bar. Totally magical!

A classical cover of Gigi D'Agostino's anthem

This is a nicely paced 13 minute presentation. It goes slowly and carefully over root two, before going faster over root N, then finishing off rapidly with monic polynomials. Suitable for a variety of audiences. But I still cannot figure out how it avoids fancy properties of the integers. I expect that you must at least use Euclid's Algorithm, but it doesn't. Here is how I prove that square roots are either integers or irrationals:

Suppose that √n∈ℚ. Write √n = a/b where a and b are coprime. Then b√n=a. Also, by Bezout's Theorem there are r,s∈ℤ such that ra+sb=1. Multiply by √n

ra√n + sb√n = √n

We already have b√n=a so sb√n is an integer. Does anything similar happen for ra√n ? Yes, just multiply b√n=a by √n obtaining bn=a√n, from which we deduce that ra√n must also be an integer. Thus √n is the sum of two integers and itself an integer.

That is awkward; a rational square root always turns out to be a whole number! Since 1<2<4 we know that 1<√2<2. But there is no integer between one and two so √2 is irrational.

But the approach in the video has neither Euclid's Algorithm, nor Bezout's Theorem (which I view as a corrollary of Euclid's Algorithm).