geometric optics from the wave equation!
turbulence modeling with RANS and LES et al!
homogenization!

The second idea here is a natural tie in to a whole lot of amazingly cool applied math — starting from a baseline fine-scale model, approximating its solution in a nonlinear way that abstracts away the fine-scale variations so we can focus on the larger scale behaviour

aww thanks 😊

(and for a rigid body now the natural time sampling frequency comes from its angular velocity!), or it could mean modeling the time evolution of the slowly decaying _amplitude_ of the 440Hz vibration (like replacing x(t) with A(t)sin(2 π 440 t) and trying to find a nice ODE for A(t) alone)

In a case like this there really are only two choices to do it “right”: either take the *physically* required small time steps (according to Nyquist at least), or change your model.

Changing the model in this case could mean e.g. abstracting out all elastic behaviour and making it a rigid body

or adjust physical parameters etc. The results you are producing are outputs from a discrete system which is _not_ approximating the continuous physical system you thought you were modeling.

And even if you’re happy-ish with the results — probably rapid damping of an oscillation around 30Hz — as somehow being visually representative of the “averaged” behaviour maybe, the results will change unphysically and unintuitively if you adjust the time step (eg occasionally do two substeps),

If you want to simulate a 440Hz tuning fork but use time steps of 1/30 s, sorry: you just can’t do that in any useful or meaningful way.

If you use Backards Euler with those big steps anyways, you’ll get some kind of solution if the implicit equations are solvable, but it obviously won’t be 440Hz

But what if you the user have a hard computational budget and want to take even bigger time steps than that? That in itself does not make the problem “stiff”, and a stiff integrator may fail in bad ways — probably remaining stable, but giving aliased garbage in some sense.

Stiffly accurate time integration schemes, such as members of the BDF family, allow you to get an accurate solution only using time steps that resolve the solution

(in signal processing terms, at a sampling frequency high enough to avoid aliasing)

A stiff problem is in reality one where the time step needed to accurately resolve the solution (this is the actual definition of “want”) is a lot larger than the time scale of how fast perturbations from the solution _decay_

(if the perturbations don’t decay, they’re part of a nearby solution!)

The general stiffness issue is that some time integrators are only conditionally stable, with a time step restriction that is smaller than the time steps you want to take

— except the “want” here is a loaded term. It’s *not* just the user’s preference, eg. to fit some computational budget.

One of the basics of numerical analysis which took me far too long to internalize, and for which there is still a lot of misunderstanding out there in graphics, is the notion of stiffness in time integration, and associated ideas of time step selection, stability, and modeling.

…we have far too much bad visibility, bigots projecting their awful fantasies on us to rationalize hatred and violence and legal warfare, setting up trans people as a visible target but replacing our reality with concocted libel

On this trans day of visibility, again I am thinking how _good_ visibility is helpful for people who need some kind of role model to be able to figure out that they themselves are trans — it would have helped me realize before my late 40’s!! — but…

How Writing Changes Mathematical Thought | Quanta Magazine David E. Dunning explores how mathematical notation is a social, world-building technology.

David Dunning studied both English and math as an undergraduate. “They felt related to me,” he said. “They were both about the worlds and structures that can be summoned by writing.”

i have anxiety over turning 50 later this year only for the question of how to celebrate it (i do have a dream, which would be perfectly feasible if it weren’t for my pesky personality disorder ruining everything as usual)

I always loved Kathleen by him

…so my next step as a pragmatic applied mathematician is generating data from a lot of numerical solutions to try and guess the Taylor coefficient (as a function of problem parameters) or at least something close enough with the right asymptotic behaviour.

It’s honestly delightful 😊

The context is solving an ODE which is itself an approximation to what I actually want to work on, which alas is an unsolvable nonlinear BVP. My asymptotic analysis took me pretty far but not quite all the way to an accurate approximation near the boundary I care most about,

A plot of a curve f(x) where x goes from 0 to a little over 30, and f(x) drops from 0 to -10 or so, in a slightly convex way.

Currently playing "guess the function" (given lots of numerically determined approximate data points)

Disagree with the very concept of the debate but it's less that we care what people think and more that we're punished if we don't. Men are generally rewarded for sociopathy.

Oh dear. Now that I think about, it's the same for me with fireworks and garbage trucks...

Sigh. A reminder that just because a trans woman is online and even posts the occasional (sfw) selfie it does not mean she is a sexual object; porn-focussed accounts who engage will be blocked.

But I do my best to avoid people and places where I might not be safe, and then remind myself nobody else really cares how I present, and the good people already have made it clear I’m accepted and welcomed. That helps get me through the dysphoric days

Aw thank you — but I cannot stress enough how much I curate my selfies, throwing out 5+ for each one I keep, and some days I simply cannot find a single one that I like no matter how I pose and mess with lighting etc.

I still have days I lack confidence in public, cringe at my reflection, cry…

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shout out to babeswig.com for making human hair pieces that are halfway affordable!!

Selfie in a green fleece with loose hair: dark brown from a topper, slightly more auburn from my scalp, but mixed together and difficult to tell apart.

I have been growing out what hair I do have for 1.5 years now, and that lets me go with just a 9”x9” topper instead of a full wig, and it is working really well for me (looks good, convenient, and mostly secure enough just with clips). I think I’m mostly at peace with this now but it’s been rough.

I tried minoxidil foam a decent length of time (washing hands really carefully thinking about our dog) but did not find it did all that much, and eventually gave up on it 😕

Labour continues to troll the trans community - and to prioritise pleasing the right wing press. We know Hilary Cass was selected for her transphobic views: Kemi Badenoch told us exactly that.

So why is she running trans policy for Labour?