Representing a complex line as a directed ellipse

Consider nonzero $v = v_r + iv_i \in \mathbb{C}^n$, It can be thought of as an ordered 2-tuple of vectors $(v_r, v_i)\in \mathbb{R}^n\times\mathbb{R}^n$.

The complex line generated by $v$ is
$$\{r[(\cos(\theta) v_r -\sin(\theta) v_i) + i(\sin(\theta) v_r +\cos(\theta) v_i)]:\\ r\ge 0, \theta\in[0, 2\pi]\}$$
So, essentially, it is a set of concentric ellipses. We consider one of them:
$$\{(\cos(\theta) v_r -\sin(\theta) v_i) + i(\sin(\theta) v_r +\cos(\theta) v_i): \theta\in[0, 2\pi]\}$$
As $\theta$ increases, both $(\cos(\theta) v_r -\sin(\theta) v_i)$ and $(\sin(\theta) v_r +\cos(\theta) v_i)$ rotate in the ellipse, always being conjugate to each other.

A very unexpected 333 that I found in math

While studying about dynamical systems theory for my ergodic theory course, I came across the extremely unexpected number 333. From Resonances and small divisors (Etienne Ghys, 2007), chapter 10 of Kolmogorov’s Heritage in Mathematics:

In 1962, Moser succeeded in accomplishing the feat of proving the theorem in the space of infinitely differentiable functions [Moser, J. On invariant curves of area-preserving mappings of an annulus (1962)]. 

In fact, Moser used functions which are 333 times differentiable and the topology of uniform convergence on these 333 derivatives... The mere fact that it is necessary to use as many derivatives shows the difficulty of the proof. Nowadays, it is known that the theorem is true with 4 derivatives and false with 3 [Sur les courbes invariantes par les difféomorphismes de l'anneau Herman, Michael R. Astérisque, no. 144 (1986)].

333 is half the number of the devil. Moser probably cut half of a deal with the devil.

Let's Watch: 10-on-10: The Chronicles of Evolution - Eörs Szathmáry

This post is supposed to be a prequel to my upcoming post on the major transitions in the evolution of life on earth. Eörs Szathmáry is a good lecturer and the lecture gives a very clean overview of the origin of self-replication, evolvability, and the relation of evolution with learning theory. I am reading about this, because I think AI might cause another major transition in evolution in a similar way.

This post will give annotations, references, and comments, to read along the lecture.

The "10-on-10" comes from Sydney Brenner's 10-on-10, of which this lecture is a part:

These seminars would begin $10^{10}$ years ago with the origin of the universe, then move on to the beginnings of biological life $10^9$ years ago, and so on, up to the development of modern human society in the present time, or $10^1$ years.

The Future of Human Evolution, The Modern Chicken Evolution

In The Future of Human Evolution (2004), Nick Bostrom argued that there are two possible ways for future evolution of humans to result in a bad end for humans, because what end up being evolutionarily fit is not "nice". Two possible bad outcomes:

Scenario I: The Mindless Outsourcers.  

Competitive uploads begin outsourcing increasing portions of their functionality: “Why do I need to know arithmetic when I can buy time on Arithmetic-Modules Inc. whenever I need to do my accounts?  Why do I need to be good with language when I can hire a professional language module to articulate my thoughts?  Why do I need to bother with making decisions about my personal life when there are certified executive-modules that can scan my goal structure and manage my assets so as best to fulfill my goals?”  Some uploads might prefer to retain most of their functionality and handle tasks themselves that could be more efficiently done by others.  They would be like hobbyists who enjoy growing their own vegetables or knitting their own cardigans; but they would be less efficient than some other uploads, and they would consequently be outcompeted over time.

We can thus imagine a technologically highly advanced society, containing many sorts of complex structures, some of which are much smarter and more intricate than anything that exists today, in which there would nevertheless be a complete absence of any type of being whose welfare has moral significance.  In a sense, this would be an uninhabited society.  All the kinds of being that we care even remotely about would have vanished.

Big Chicken and the Anthropocene.

Broiler chickens, in particular. The broiler chicken is the most popular breed of chicken produced by modern humans. If you go to a supermarket in any modern country, and see a chicken (dead, though living chickens are a thing too), it's probably broiler chicken.

Humans make lots of broiler chickens. 23 billion are alive at any instant, and each lives about 40 days (they grow extremely fast).

From ScienceDaily:

These chickens are an artificially evolved new 'morphospecies', the kind of thing palaeontologists recognise, that reflect a biosphere unrecognisable from its pre-human state and now dominated by human consumption and resource use.

"Since domestication there have been many strange and beautiful chicken breeds, but the broiler is perhaps the most extreme form of all. The body shape, bone chemistry and genetics of the modern meat chicken is unrecognisable from wild ancestors and anything we see in the archaeological record." 

It usually takes millions of years for evolution to occur, but here it has taken just decades to produce a new form of animal that has the potential to become a marker species of the Anthropocene -- and the enormous numbers of these chicken bones discarded worldwide means that we are producing a new kind of fossil for the future geological record.