In grad school I managed to take advantage of the pacing effect in an educational setting. I was teaching linear algebra. What I did was make the homework incremental - 1/3 of homework on today's material, 1/3 on the previous week, and 1/3 anything in the course. Those thirds were in increasing order of difficulty.
I also started every class with a question/answer period. The rules were simple, the questioning will last at least 10 minutes, and you don't want me to ask the questions. :-) Anything that had come up in the questions that seemed to be a point of confusion was sure to be added to the next homework set.
I won't go into what else I did with that class, but the end result is worth thinking about. First note that I gave a ridiculously hard final. Other grad students who saw it thought that the class would bomb. Secondly they aced the test. What do I mean by aced? Well I had a bonus question which fellow grad students thought nobody would get. 70% of the class got that question, and a good fraction were over 100% on the test. So they must have studied hard, right? Nope. I ran into some students several months later. They told me that they tried to study for the final and stopped after a few minutes because it was useless, they knew everything. And several months later they still knew much of the material cold!
The thing is that none of what I did was very radical. The principles have been known for a century. Psychologists have been trying to get people to listen for that long. I learned about it in the 80s from a university course I watched on TV. (British Columbia had a TV channel devoted to lectures for correspondence courses.)
Yet, despite how dramatic the effects are, nobody listens and nobody takes advantage of it.
Since the time of the cannon shot is unknown, only the time when it can be heard at a known location, you can't even pinpoint its unknown origin in space to a circle. Instead, the range of possibilities forms a cone in spacetime (exactly like the lightcone of electromagnetic waves, except with sound and also treating space as two-dimensional for simplicity).
The intersection of two such cones is a parabola. The intersection of the third cone with the plane containing said parabola gives another parabola that can intersect the first one in zero, one, two or an infinite number of points. In the zero-point case, you could still find the location where the two parabolas are closest.
(Or you could forget about geometry, define an objective function and do gradient descent to it.)
The question that the referenced paper (1) is trying to answer is "do the 3D incompressible Euler equations develop a finite time singularity from smooth initial data of finite energy?" This is an important question in the theory of nonlinear partial differential equations, but is probably not as relevant to real fluid flow as a lay reader might imagine.
The incompressible Euler equations model a very strange and unphysical kind of fluid. Incompressibility means that the speed of wave propagation in such a fluid is infinite, which means that normal causality is not respected. Effects in such a fluid happen simultaneously with their causes.
For example, if you apply a force to one end of a pipe full of Euler fluid, the fluid instantly starts coming out of the other end of the pipe, with no time taken for this effect to propagate from one end of the pipe to the other. You could use a long pipe full of Euler fluid as a superluminal communication device!
Intuitively, it seems reasonable that in such an unphysical fluid, it would be possible to form a singularity even from smooth initial conditions. The difficulty, of course, is proving that intuition, which is what the paper is trying to do.
1) https://arxiv.org/pdf/2210.07191.pdf "Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data", Jiajie Chen and Thomas Y. Hou.
There's also bats who have much more flight control than birds. Since they have a hand in their wing which is covered with skin and muscle, they have a large amount of additional control over birds. Here's a great video of a study.
Since we're talking about the Shannon limit, why does a 56 Kbps modem not violate it?
The common wisdom in the early '90s was that the fastest possible modem for a dial-up connection would be ~35 Kbps. I even remember reading this "fact" in a communications theory book from that era. Analog modems could never be faster. (It's assumed that US and Canadian analog telephones have a 3 kHz channel.)
Then why does a 56 Kbps modem not violate the Shannon limit for data transmission on a 3 kHz channel? How is it possible to break a fundamental principle of information theory?
This question has bugged me a lot. Based on some research on web, I tried to put together an intuitive answer to what happened. Here's my answer, but I welcome insight or corrections from people knowledgeable in telephony or information theory.
At the time the book was written--when people believed that dial-up modems could never be faster than 35 Kbps--the telephone network was perceived as being analog end to end (though lots of it was already becoming digital), like this:
A critical assumption in calculating the Shannon limit was the noise floor of the network, which was taken to be 35 dB, a figure based on an all-analog network. (A higher dB number is better for transmission quality.) The Shannon limit would be 35 Kbps based on a 3 kHz bandwidth and that particular signal to noise ratio of 35 dB. ( bps = BW log2 (1 + P/N) = 3000 log2 (1 + 10^(35/10)) = 34881 bps )
However, by the mid-'90s, most of the telco network became digital. Only the customer loop--the connection between the user and the telephone company--remained analog. So now, with respect to the Shannon limit, only the noise floor of the customer loop matters, since the rest is digital:
It turns out that it is possible to achieve a much better S/N ratio when only the customer loop is analog. The Shannon limit would be based on a 3 kHz bandwidth (the same bandwidth as before) and but a much higher noise floor of 98 dB for the customer loop. The Shannon limit would now be ~97 Kbps. ( bps = BW log2 (1 + P/N) = 3000 log2 (1 + 10^(98/10)) = 97664 bps )
The Shannon limit still exists, but other factors related to the network would limit the data rate before this bigger Shannon limit takes effect.
So the key thing that the experts and the communications theory book got wrong was to neglect the hidden assumption about S/N ratio. The bandwidth matters, but the S/N ratio matters too.
I teach a graduate course in optimization methods for machine learning and engineering [1,2]. Julia is just perfect for teaching numerical algorithms.
First, it removes the typical numpy syntax boilerplate. Due to its conciseness, Julia has mostly replaced showing pseudo-code on my slides. It can be just as concise / readable; and on top the students immeditaly get the "real thing" they can plug into Jupyter notebooks for the exercises.
Second, you get C-like speed. And that counts for numerical algorithms.
Third, the type system and method dispatch of Julia is very powerful for scientific programming. It allows for composition of ideas in ways I couldn't imagine before seeing it in action. For example, in the optimization course, we develop a mimimalistic implementation of Automatic Differentiation on a single slide. And that can be applied to virtually all Julia functions and combined with code from preexisting Julia libraries.
James Burke, and Richard Rhodes as well, are the reason I began to see history as something important and useful, a way of understanding both "how science happens" as well as "why our society is the way it is".
This knowledge, in the way it is presented, leaves the viewer or reader with a sense of agency, that what they do can either intentionally or not cause outcomes with great consequence.
It leaves one feeling one can do important things.
Sorry for me spamming this but this is extraordinary times:
For people who are curious what living in China feels like now (video uploaded Mar 14th), from a Japanese director living in Nanjing, China: https://www.youtube.com/watch?v=YfsdJGj3-jM
Things are getting back to normal, with tons of precautions.
I also started every class with a question/answer period. The rules were simple, the questioning will last at least 10 minutes, and you don't want me to ask the questions. :-) Anything that had come up in the questions that seemed to be a point of confusion was sure to be added to the next homework set.
I won't go into what else I did with that class, but the end result is worth thinking about. First note that I gave a ridiculously hard final. Other grad students who saw it thought that the class would bomb. Secondly they aced the test. What do I mean by aced? Well I had a bonus question which fellow grad students thought nobody would get. 70% of the class got that question, and a good fraction were over 100% on the test. So they must have studied hard, right? Nope. I ran into some students several months later. They told me that they tried to study for the final and stopped after a few minutes because it was useless, they knew everything. And several months later they still knew much of the material cold!
The thing is that none of what I did was very radical. The principles have been known for a century. Psychologists have been trying to get people to listen for that long. I learned about it in the 80s from a university course I watched on TV. (British Columbia had a TV channel devoted to lectures for correspondence courses.)
Yet, despite how dramatic the effects are, nobody listens and nobody takes advantage of it.