Less Variance

Our definite article is endangered. Linguists have crunched the numbers, and over the course of the twentieth century, our use of the plummeted. If you treasure the the as I do, join the campaign to employ the the as often as the circumstances allow. (We started by putting it in the title of our magazine.)

Fix a prime number p, draw a p × p array and fill it with integers as above.

Can you always find p prime numbers in such an array with no two of them in the same row of color (a transversal of primes)? The images depict two possible solutions for p=11. This question remains unsolved today.

We talked about how integrals will tell you the “area under a curve.” Look up above where we drew a trapezoid to find the area under the curve. A little bit of the trapezoid is actually over the curve though, so it’s not exactly the right area. But it’s close. If we want to get closer, why not use two trapezoids? Then we can add their areas together. Great, how about four trapezoids? Even better, but still a little bit off because the top of a trapezoid is straight, and the curve is curved.

Affine transformations preserve parallel lines, and include rotations, scaling, shears and translations. Linear transformations can’t perform translations, but this can be achieved if we go to a higher dimension.

In this animation, a planar (2D) shape lying on the plane z = 1 is translated by means of a linear transformation in three dimensions: a shear along the z axis.

Rotations can be performed normally, also around the z axis. For rotations around any other axis parallel to the z axis, it’s just a matter of performing the appropriate translation that cancels out the translation of the axis performed by the rotation.

This way, all transformations are now linear in 3D, and can be represented by a single 3x3 affine transformation matrix that acts on two dimensions.

Statistics vs Machine-Learning

I stumbled across a game called Vax, which lets you simulate an epidemic breakout in a very graph-theoretic mindset. You only have two tools though: you can either vaccinate an infected vertex or you can cut out a vertex altogether. And that notion reminded me of this quote I had read sometime ago, in particular:

Take the web of interactions within a cell. If you knock out an important gene, you will significantly damage the cell’s growth rate. However, it is possible to repair this damage not by replacing the lost gene, which is a very challenging task, but by removing additional genes. The key lies in finding the specific changes that would bring a network from the undesirable state A to the preferred state B.

The curvature of curves.

x²

x³

sin(x)

exp(x)

Normal distribution (y=exp(-x²/2))

Ellipse

r=5/2+cos(3τθ)

x=(t-1)(t+1), y=t(t-1)(t+1)

Archimedes’ Spiral

Logarithmic spiral

If you want to try your own curve, try on Desmos graphing calculator!

Russian postmen fix an error caused by an ASCII-illiterate e-mail client in France. New aesthetic, anyone?

This letter was sent to a Russian student by her French friend, who manually wrote the address that she received by e-mail. Her e-mail client, unfortunately, was not set up correctly to display Cyrillic characters, so they were substituted with diacritic symbols from the Western charset (ISO-8859-1) The original message was in KOI8-R.

The address was deciphered by the postal employees and delivered successfully. Some of the correct characters (red) were written above the wrong ones (black).

Encoding problems are usually called Mojibake (from Japanese) but other languages refer to it as monkey’s code, letter salad, chaotic code and even little bushes. Read more at Wikipedia.