This is extra bizarre to me, because for most purposes German law doesnt operate on a system of "legal precedent" the way countries which adopted the UK model do.
Am I missing something about Germany following a precedent system for patent/copyright or something, or is this even dumber than it sounds?
Sorry, I rushed through my comment and perhaps didn't make it clear.
They have a default judgement only. But they used it to demand US-based manufacturers recall European-bound inventory, destroy it and certify it destroyed.
Even though they know full well that inventory can legally be sold in the USA — which is part of the near-comical gaslighting walkback the FMIC CEO attempted the other day. They are already admitting it's not a USA thing.
As a legal theory, "this default judgement against an anonymous AliExpress seller is binding on literally everyone in the world" kinda reminds me of the Dune nft bros' "we bought a book about Dune and therefore now own the intellectual property rights to Dune."
Except this one is apparently coming from actual accredited lawyers? (Who knows, I'm not a lawyer, maybe it really does work that way and Fender is the first company to figure out how to exploit this)
Because the only way Trump Guitars can sell an LP-type guitar to US customers is that Gibson also lost a body-shape case like this (to Washburn, if I remember right?)
This might be the coldest summer for the rest of your life. European summers are only going to get hotter, and more humid. Meanwhile, we have a pressing need to decarbonize our home heating solutions both for energy sovreignty and climate reasons.
Modern air-to-air heat pumps (i.e. aircon) are a pretty good solution for that. I think we're just going to have to work our way around this as a society. While many Europeans live in historic buildings that will require a lot more care, most do not, and installing aircon at least for homes with elederly people and young children provably will reduce unnecessary deaths during our now yearly heatwaves.
But what about the coldest springs? It's wild, I was in Paris just this mid May and it was raining and hailing, berry sized icy projectiles from the sky. I had to pop into a Decathlon and get a waterproof jacket.
The rapid buildout of solar, and the rise of efficient air cons for electric home heating certainly makes it seem like a much less wasteful technology than before.
I still think though that people are severly underestimating the effectiveness of relative simple, low-tech options like awnings though.
I live in sweden and I am baffled by how uncommon awnings are. In my home country window shutters that also work as yawnings are quite common, especially in places without AC.
My partner was downright insulted when I suggested getting some blackout curtains to reflect sunlight back outside when the sun is hitting our living room. I eventually won that battle after the first summer.
> underestimating the effectiveness of relative simple, low-tech options like awnings though.
I mean, I'm currently in France, with 30°C (86°F) in my bedroom, do you really think that our shutters are opened ? At some point, sunlight is not even the issue. For the night to come, like the previous night, predictions are that we will have 35°C (95F) at midnight. I'm not even sure it was this hot when I was at Las Vegas eleven years ago. Las Vegas which is in the middle a of desert.
We are way past the awnings and we already have shutters and both of them are useless when the air itself is like a air dryer.
>For the night to come, like the previous night, predictions are that we will have 35°C (95F) at midnight. I'm not even sure it was this hot when I was at Las Vegas eleven years ago. Las Vegas which is in the middle a of desert.
Vegas's dry heat is much more tolerable than lower temperatures in more humid places (which is pretty much the rest of the civilized world).
Awnings are significantly better than shutters, but they at most mitigate the heat and the temperature range they make the ambient confortable is quite limited.
I guess there is a cultural difference on the word here and I should have be more precise.
In France, shutters aren't like most American ones, here we mostly use either plain wood with no gap for light (old houses) or for most of the recent houses, we use rolling shutters that let 0% of light (and therefore, 0% of radiative energy from the sun) get to the window and will make the room entirely black. Also, most modern rolling shutters are white by default so they are pretty reflective.
In the current situation, it's the air temperature that fuck us by not getting down at night and so our (concrete) buildings accumulate the heat even at night.
An awning protrudes out of the window so airflow can still pass and light still gets into the room (just not direct light).
Even if the shutters are wood and white when sun hits them they will radiate some heat into the indoors. But of course shutters are still better than letting the sun hit your floor directly.
I didn't say anything about shutters, I said awnings. It's a testament to how much our society has forgotten about building effective homes that people think shutters are just as effective as awnings.
That said, yes, if possible, you should be installing an AC unit as well (and the awning will help make the AC unit more efficient.
The idea is that it's an alternative way of talking about vectors, rotations, and geometry in general. I.e. a replacement for the vector notation you learned that makes it operate more like how complex numbers are used.
More or less agreed. I think though that one reason the geometric product is so tempting is that if you take matrix representations of all of these objects, then the geometric product is literally just straightforward matrix multiplication.
Because of that, it just becomes so tempting to try and phrase everything you can in terms of this geometric product. I'm very sympathetic to the temptation, and I even think the geometric product has some great uses (it shows up a lot in some physics I do), and using it makes writing rotations a treat, but I think it's still vastly overemphasized by GA people.
I still don't really know what my favoured notation for differential geometry is, I find myself switching around so much.
> From a mathematician's point of view, yes, you should write the Maxwell field equations, at least to see it once, that way because you're showing a very low-level symmetry that even the differential forms approach doesn't get all the way to. Differential forms is a standard approach for general relativity, e.g. MTW.
While it's neat to write them all as one equation, I disagree that it's an enlightening perspective to learn. While it seems like writing Maxwell's equations in one equation instead of two is a step forward with even more symmetry, what is actually going on is that you are obscuring the most important part of Maxwell's equations: the gauge structure. Without this, it actually becomes much more hidden just how geometric electromagnetism is.
When you write Maxwell's equations as the pair `dF = 0`, `d*F = J`, the first of those two equations is exactly what tells you that this is a gauge theory, and thus may write `F = dA` where `A` is a vector potential. This vector potential then becomes the connection which defines a covariant derivative in a fibre bundle, and one then sees that charged particles follow geodesics now in spacetime, but in an enclosing fibre bundle. This is foundationally important to modern physics, and IMO obscured by writing Maxwell's equations as `∇F = J`
____
n.b. I'm not a particularly big fan of differential forms either, I think it leaves a lot to be desired, and it's super awkward to constantly have to pull out Hodge Duals every time you want to do something that involves the metric, but I'm also unconvinced that geometric algebra is the answer here.
What interests a mathematician isn't 100% the same as what interests the physicist. All I'm saying is there is some math there that's interesting and people should see it once for the math.
And then there are us engineers. I don't care much either way whether Maxwell's equations are ∇F = J or some other form, as long as it makes the problem easier to solve.
If I were in the GA Marketing Committee I'd publish a paper with suitably hand-picked worked examples where the vector approach is long and tedious, and GA version is short and sweet.
I like this idea but I get the sinking feeling GA proponents don't really solve problems with GA. Like how Haskell advocates don't write programs and modular synth enthusiasts don't write music.
Application of the Method of Moments to solve full wave formulation of Maxwell's equations. To derive the EFIE using maxwell's equations is a massive pain. With geometric algebra, all you need is ∇F = J and the MoM becomes a mechanical process.
I guess I'd say my point though is that the gauge structure is the mathematically interesting part of Maxwell's equations. (i.e. the fact that `F` is a closed differential form).
Without it, I think it'd be of significantly less mathematical interest because it'd lose almost all of its geometric properties.
There isn't just ONE interesting facet of this. There isn't just ONE mathematical formalism of a lot of these things. GA is just one of those approaches and you should see it just once, just like you should see the group structure and all of that as well. For most applications, the standard vector calculus approach is fine. But the math underlying all of this is full of richness and no one approach is the skeleton key.
Same with programming languages. Some people are like RUST RUST RUST and some are like C C C! I'm like, you guys only use one language?
I've found differential forms to be more useful than GA, but that might just be that I was brought up in the MTW tradition and don't quite get GA.
Whenever I look at GA, I try to figure out where the metric comes in, and I just don't see it.
For context, way back when I did astro theory and wanted to do things like figure out things like the magnetic field structure in the curved spacetime near highly-magnetized rotating conducting spheres, and then do some basic plasma physics in that environment.
The differential geometry approach at least gives the structure to think about that, then you can go down to the index-style notation to actually get the differential equations you need to solve. The GA approach, I'm not even sure how to frame the problem.
> pull out Hodge Duals every time you want to do something that involves the metric, but I'm also unconvinced that geometric algebra is the answer here.
I don't know, I recently tried to work out how the metric on vectors/1-forms induces a metric on higher-degree forms, and if the geometric product magically gives this for free I'd say it's a win (same for the Hodge star).
That comes from exterior algebra on its own, it's the k'th exterior power of the metric. Best not to conflate that with GA (unless I'm misunderstanding what you're talking about).
IIRC there's a fairly natural positive definite quadratic form on GA (used as the canonical norm) that takes the scalar part of the geometric product of a multi-vector and its reverse.
On the other hand, there's the k-th exterior power of the metric where one asks that wedge/interior products be adjoint in order to extend the metric to higher-degree forms.
I was under the impression that these metrics are the same, but maybe I'm completely wrong? Assuming I'm not, then the GA approach seems more natural to me.
You can see how the dot product (which uses the metric internally) is being applied "in-to-out" : the adjacent terms are dotted, at which point they become commuting scalars; then the next terms, etc. Which, frankly, is dumb. This is why the GA version of a scalar product has the "reverse" operation involved... because the GP is doing this in-to-out thing, the scalar product has to undo it by defining (abc) . (xyz) = (abc) (xyz)^~ = (abc) (zyx) = (a.x) (b.y) (c.z), with ^~ meaning reverse.
Whereas the standard exterior algebra inner product is always left-to-right, giving
(abc).(xyz) = (a.x) (b.y) (c.z)
IMO the GA version is a mess because it's conflating two concepts. When the GP works, it is composing operators, so AB = A ∘ B. But the inner product, at its core, is more like division---it wants to have (a).(a) = 1, since its job is to say say "how many copies of (a) are there in (a)?" To make this work for multivectors (ab).(ab), it needs to be left-to-right. GA does in-to-out to copy quaternions with their i^2 = -1, but that's not necessary -- i^2 = -1 follows from the fact that for a rotation, R ∘ R = -I, so it is composing two rotations, not measuring one in terms of the other. Really i^2 = -1 should not be interpreted as a dot product at all. This is very clear when a metric is involved: R_xy ∘ R_xy = -I is a degree-two tensor which transforms with two factors of the metric, whereas (xy).(xy) = 1 is a degree-zero tensor, a coordinate-invariant scalar. They are just different operations, which happen to overlap in simple cases.
agreed, when you start needing the the hodge star, diff form loose quite a lot of their interest.
i'd add it's quite nice in string theories for RR fields and coupling to D-branes, where writing 10 anti-symmetrized indices quickly gets annoying.. and topological field theories..
No, that's not my own argument — it's just how I understood the article's claim. I only know GA as something used for specific purposes. I was just sharing my thoughts on what the article was saying.
It's a very fun framework when you're learning it. It constantly feels like you're learning something extremely profound and useful, but I've also found that feeling to be a bit of a mirage.
Despite trying many times to make greater use of it, I've found that it often just makes a lot of actual physics work less clear, and with very little practical benefit.
There's times where it affords quite pretty notation, but often you have to actually unpeel all that notation before you actually do something with it. And what's the point of nice notation if none of your colleagues can even read it? The only time I ever really found that GA was actually a benefit to me was performing rotations.
https://news.ycombinator.com/item?id=48672732
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