Here are a few interesting links I browsed recently, listed in no particular order:

**“Mathematicians Tame Turbulence in Flattened Fluids”** [^].

The operative word here, of course, is: “flattened.” But even then, it’s an interesting read. Another thing: though the essay is pop-sci, the author gives the Navier-Stokes equations, complete with fairly OK explanatory remarks about each term in the equation.

(But I don’t understand why every pop-sci write-up gives the NS equations only in the Lagrangian form, never Eulerian.)

**“A Twisted Path to Equation-Free Prediction”** [^]. …

“Empirical dynamic modeling.” Hmmm….

**“Machine Learning’s `Amazing’ Ability to Predict Chaos”** [^].

Click-bait: They use data science ideas to *predict* chaos!

8 Lyapunov times is impressive. But ignore the other, usual kind of hype: “…*the computer tunes its own formulas* in response to data until the formulas replicate the system’s dynamics. ” [*italics* added.]

**“Your Simple (Yes, Simple) Guide to Quantum Entanglement”** [^].

Click-bait: “Entanglement is often regarded as a uniquely quantum-mechanical phenomenon, but it is not. In fact, it is enlightening, though somewhat unconventional, to consider a simple non-quantum (or “classical”) version of entanglement first. This enables us to pry the subtlety of entanglement itself apart from the general oddity of quantum theory.”

Don’t dismiss the description in the essay as being too simplistic; the author is Frank Wilczek.

**“A theoretical physics FAQ”** [^].

Click-bait: Check your answers with those given by an expert! … Do spend some time here…

**Tensor product versus Cartesian product.**

If you are engineer and if you get interested in quantum entanglement, beware of the easily confusing terms: The tensor product and the Cartesian product.

The tensor product, you might think, is like the Cartesian product. But it is not. See mathematicians’ explanations. Essentially, the basis sets (and the operations) are different. [^] [^].

But what the mathematicians don’t do is to take some simple but non-trivial examples, and actually work everything out in detail. Instead, they just jump from this definition to that definition. For example, see: “How to conquer tensorphobia” [^] and “Tensorphobia and the outer product”[^]. Read any of these last two articles. Any one is sufficient to give you tensorphobia even if you never had it!

You will never run into a mathematician who explains the difference between the two concepts by first directly giving you a vague feel: by directly giving you a good worked out example in the context of finite sets (including enumeration of all the set elements) that illustrates the key difference, i.e. the addition vs. the multiplication of the unit vectors (aka members of basis sets).

A third-class epistemology when it comes to explaining, mathematicians typically have.

**A Song I Like:**

(Marathi) “he gard niLe megha…”

Singers: Shailendra Singh, Anuradha Paudwal

Music: Rushiraj

Lyrics: Muralidhar Gode

[As usual, a little streamlining may occur later on.]