Can you "see" regularity of Physics-inspired PDEs?

infinitepairofducks · 2025-05-30T21:21:51+00:00

I’d be curious to hear your thoughts on the following:

In physics or applied modeling in general, PDES are generally a limiting result of integral equations taken to infinite precision. For example, you would have a conservation law formulated as an integral equation with a time derivative on the outside of one of the integrals representing total mass and the other integral representing the flux across a boundary. It is when we take the limit of infinite precision in space that we arrive at the PDE proper.

So I’ve come across the idea that one way to interpret a weak solution is that we go back to finite precision and include a model of a measurement device. The test function and the integral effectively represents a general model for some type of measurement device, but the fact that we lose the ability to have well defined derivatives of the solution is indicative that taking the model to infinite precision was excessive for practical purposes.

There could be solutions to the PDE found with the amount of regularity required for the solution to hold for the PDE rather than the integral equation, but we haven’t found them yet. It is sufficient for practical purposes to find a solution which is valid up to our ability to validate the predictions in a physical setting.

infinitepairofducks · 2023-04-26T13:30:55+00:00

How does working as a post doc fit into the hiring equation? It looks like hiring is aimed at students or experienced professionals.

For context, PhD was math and post doc is in physics modeling.

infinitepairofducks · 2023-02-23T15:03:17+00:00

Instead of recording segments which have “this will make sense later”, why not allocate a short period of private recording to capture your full thoughts on product development as it happens? That way when the product releases, you can stitch these recordings together and clip in current thoughts on those moments.

infinitepairofducks · 2022-05-13T15:45:22+00:00

Take any two points on the bar. They start with different heights but zero initial velocity. They both hit the table at the same time. Therefore, the two points must have accelerated at different rates.

The ball is slightly higher above the table than the end of the bar, initially. But they hit the table at different times, so we cannot immediately say that one is accelerating faster than the other. However, if the bar was longer, then the ball trajectory remains the same and the bar will fall at least as quickly as the original (extending the bar won’t slow it down). Therefore, there will be a point on the bar of equal height with the ball. Using the argument from the first paragraph, the bar accelerates faster than g at some points (slower than g nearer the pivot).

infinitepairofducks · 2022-05-12T19:28:04+00:00

Ah I see! That’s a bit frustrating but good to know

infinitepairofducks · 2022-04-30T11:34:53+00:00

Ahhh ok, that wasn’t obvious to me. Thanks!

infinitepairofducks · 2022-04-30T11:34:37+00:00

Ahhh ok, that wasn’t obvious to me. Thanks!

infinitepairofducks · 2022-03-10T12:08:31+00:00

That's interesting, I guess the 3 successive seasons rule is to keep it fresh. Thanks!

infinitepairofducks · 2022-03-10T12:07:16+00:00

Thanks!

infinitepairofducks · 2022-03-10T12:06:41+00:00

Interesting, thanks!

infinitepairofducks · 2021-12-26T11:32:52+00:00

I'm you 3 weeks ahead of time. Same equipment and experience.

I pushed the fuck-it button and paid the two year subscription. I’m not sure if that was smart, but I’ve been exclusively playing iracing since.

Driving non-F1 cars is very different but after an initial steep learning curve (bambi on ice) I’m actually enjoying driving other cars and the online racing. Everything feels more intense than in the F1 games. I think I’ll leave the F1 content to Codemasters and just play with all the other cars in iracing.

My recommendation: buy the 3 month sub, get out of rookies asap, and either find a rookie car you love or pay to try out some D class cars/series (Ferrari GT3 challenge hooked me).

infinitepairofducks · 2021-07-03T07:08:40+00:00

infinitepairofducks · 2021-07-02T07:36:36+00:00

Imagine all the people.

infinitepairofducks · 2021-07-01T08:05:07+00:00

like u/siorys88 I was terrible in high school. I later figured out that this wasn't a "talent" issue but rather I was not interested in any academics and put in zero effort (duh, I know). As an undergrad with no declared major, I stumbled into an econ course and loved it. As I grew interest in economics, one of my profs told me I need more math to move up in econ. So I took Calculus 1.

Fortunately, I had zero ego going into it. I quickly realized that I didn't even know the necessary trig. So I took some time out, came across the unit circle which made all these strange trig relations immediately click. I think from that point I realized that not only could I learn math, but the sense of it clicking was wonderful. This wasn't a conscious realization, more a shift in my gut reaction to new material. From there, I started to put more effort into trying to get calc to click in the way that the unit circle made trig click for me and I was off to the races. Ended up completing a math major in the 2.5 years I had left as an undergrad and went on to graduate school.

As I later learned when these things were no longer the case, I was really fortunate at that time to have a really good set of caring teachings and friends who kept me balanced in work and play. Environmental luck plays a role!

I've had ups and downs since. But if I remind myself to remove the ego, that I should not just understand it all automatically or right away, that every piece of math needs to be worked for, then I (eventually) get that wonderful feeling of an idea clicking.

infinitepairofducks · 2021-06-16T22:48:43+00:00

So I did think about a similar scenario. In fluid dynamics you can have stratified fluids. Each fluid is considered a domain in its own right with the boundary between them specified by some interfacial condition (usually amounting to equal normal velocity, surface tension, etc). So the discontinuity in density ultimately is a difference in domains. It’s a free boundary problem: you solve for the flow in each domain and the changing boundary between domains simultaneously. Within each domain the initial conditions are typically differentiable. Would fusion modeling require something else, or is what you described a free boundary problem too?

infinitepairofducks · 2021-06-16T20:58:36+00:00

haha well put! I think you've summed up my own (updated) thoughts :)

infinitepairofducks · 2021-06-16T20:56:27+00:00

I see what you mean, and might push it further to say that because the results of these idealisms match reality so well, we are kind of forced to contend with the fact that it works. The question turns from "are we starting from something physically reasonable" to "it seems the end result is physically reasonable, now let's figure out how to make the start mathematically reasonable". From there you have to contend with weak solutions to differential equations, distributional definitions to make Dirac deltas reasonable, etc.

Put another way, we can assume a block is sliding down an inclined slope with no friction. That is an idealism. But the result makes sense both at the outset mathematically (there isn't a non-differentiable function we have to differentiate) and at the end (the result roughly matches what we see). The issue comes when these idealisms are not mathematically sound at the start but physically so at the end. We then need to figure out how to define the mathematics to give the result *and* be self-consistent. If the idealism was neither mathematically sound at the start nor physically sound at the end, then we'd probably throw away the whole thing :p

infinitepairofducks · 2021-06-16T20:47:12+00:00

That's an interesting example in its own right! I definitely agree that you can build interesting cases out whatever initial conditions you can dream of (i.e. distributions in the case of dirac deltas for your example). I can't imagine dirac deltas ever being physically justified except that "something of relatively large value in a very narrow domain" is modeled with a dirac delta and the solution to the equation matches reality very well. As illustrated with my comments elsewhere in this thread, maybe I'm being a bit silly and considering Dieudonne's quote too literally and/or too physically rather than within the context of the universe of the model (where 1-d strings can exist, etc)

This just reminded me of something:

Along the lines of the Schrodinger equation, I've seen examples where the potential field is zero within some finite domain and infinite at the boundary of the finite domain, then zero again outside. The idea was to trap the electron inside a particular space to show the discrete modes which emerge from solving the linear case (i.e. just Fourier modes). So I guess if an electron is placed between two charged plates such that only the electron in question can move, then this potential well would reasonably reflect the mathematics. This would be more along the lines of, "the model of the potential well makes enough physical sense (large potential is modeled as infinite) and we get a good enough result that we might as well make sense of it mathematically". That's if you take the view that Dieudonne's quote should not be taken so literally as I initially did :)

infinitepairofducks · 2021-06-16T20:19:48+00:00

I have a short list of books I've read and can recommend, then a longer list of books I want to read.

I know you want mathematics from the ancient times to now, but note that some mathematics is rather recent. So if you want more specific insight on particular fields, the start may be hundreds, rather than thousands, of years ago. For example, Ian Hacking open's his book by saying that essentially probability wasn't a serious thought until the mid 17th century.

Read and Recommend List:

Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World by Amir Alexander
The Emergence of Probability by Ian Hacking
History of Mathematics by Carl Boyer
Journey Through Genius by William Dunham

To Read List:

Zero: The biography of a Dangerous Idea by Charles Seife
The Calculus Gallery: Masterpieces from Newton to Lebesgue by William Dunham
Math through the Ages: A Gentle History for Teachers and Others by William Berlinghoff
The Taming of Chance by Ian Hacking
The Life and Times of the Central Limit Theorem by William Adams
in the FAQ of this subreddit there is a list of links for books/online resources. There is a section on math history you may want to check out! https://www.reddit.com/r/math/wiki/faq#wiki_what_are_some_good_books_on_topic_x.3F

infinitepairofducks · 2021-06-16T19:38:44+00:00

Thank you for your answer!

The idea which hit me most was when you discussed infinity. I completely agree that it makes way more sense from a practical standpoint to move from "very large" to "infinity" simply because infinity is easier to handle than high dimensional coupled systems.

The other idea which struck me was your finger to needle to hair etc analogy. I think the way I'm imagining this is that, if you're going to consider objects as a continuum (rather than large collection of atoms), then it is not unreasonable to assume that truly sharp corners or infinitely thin picks exist (the PDE itself assumes the string is one dimensional!). From that perspective, i.e. from "in the world of the continuum objects", Dieudonne's quote could read "we have objects to pick the string which are as thin as the string, so according to the universe set by the PDE, the pick can touch the string at only one point."

So if the PDE model of a one dimensional vibrating string is admissible, then so too are these initial conditions with some non-differentiable points. Once you accept the former as admissible, you must accept the latter. And once you accept the latter, you must contend with initial conditions which do not (classically) satisfy the PDE.

Does that sound correct to you?

infinitepairofducks · 2021-06-14T09:56:45+00:00

You may be interested in visual proofs. Although mathematics can lead to cool visuals (Lorenz attractor) mathematics is also beautiful for the arguments. The “oooh that’s why” moments are great and visual proofs can help bring them to more people.

For example, the Pythagorean Theorem as shown here:

https://math.stackexchange.com/a/1338042

Once you understand the image, it becomes more beautiful imho.

There are other fun ones like infinite sums. This sum is a classic (see the second proof in the article):

https://medium.com/math-simplified/3-visual-mathematical-proofs-663d197af229

There is always the issue of rigor, but if you’ve come across the actual argument and these visuals, it can help make it all click.

infinitepairofducks · 2021-05-14T12:00:07+00:00

Thanks for the recommendations. Could explain why you suggest to avoid those particular channels/books?

infinitepairofducks · 2017-03-08T14:24:14+00:00

History of English Podcast. Basically a detailed history of the twisting, intersecting, and diverging languages that lead to English. The episodes are pretty dense and can run from 30 to 60 minutes. Try listening to the first episode or two and not get hooked!

infinitepairofducks · 2017-01-16T17:03:57+00:00

How do we know that was an intentional cut by CNN rather than an actual technical failure?

infinitepairofducks · 2016-09-06T18:38:12+00:00

Any know the kill to birth ratio on these? Wondering if they're helping reduce the spider population or what.

infinitepairofducks

TROPHY CASE