The determinant of a polynomial mapping is a polynomial, which is the subject of the conjecture. To get the Jacobian determinant, all you need to do is compute partial derivatives of polynomials, and add, subtract, and multiply them together. All of these operations map polynomials to polynomials.
The crux of the assumption is that if a polynomial mapping is invertible everywhere (Jacobian nonzero everywhere), its Jacobian must be a constant. Why? Because the only polynomials which are zero nowhere are constants.
No. This is about polynomials. The assumption that the Jacobian is nowhere zero is what is doing so much of the work. This means the Jacobian must in fact be constant. But obviously there are many mappings whose Jacobians are not constant.
"At present, no language can rival, let alone surpass, Fortran when it comes to implementing long-lived, large-scale, massively-parallel scientific and engineering applications; not even C and C++."
Huh??? A tremendous amount of scientific computing happens in C, C++, and many other languages. I have worked with Fortran, worked with Fortran-heavy labs in academia, and this is just a nonsensical thing to say.
So far as we know, consciousness only takes place in biological organism. A computer isn't a biological organism, therefore the burden of proof for claiming that an LLM is extraordinarily high. To me, this makes it reasonable to dismiss the idea. If you want to demonstrate that LLMs are conscious, you better start working on your neuroscience and philosophy PhDs.
This is what the field of numerical analysis exists for. These details definitely have been treated, but this was done mainly early in the field's history; for example, by people like Wilkinson and Kahan...
This is just subdivision of Bernstein polynomials. The unifying way of looking at this is via the blossom of the Bezier curve (Bernstein polynomial). You can derive all manner of shifts, change of basis transformations, and so on using the blossom. Subdivision is just a shift of the domain of definition of the original polynomial.
The truck would come by once a week to pick up all of our dirty diapers and drop off a new set of clean diapers. The service provided a bucket with a cloth bag inside of it that had enough capacity for one kid for a week. Packaging up = closing the lid of the bucket and leaving it outside the door of our apartment. They would drop off a new one of these buckets full of clean diapers each week (visiting many other people at the same time). Beyond all the environmental benefits, living in the city this was actually super convenient.
With the service we used, they picked up and dropped off diapers once per week using a big truck. They also drove around to do the same for many other people at the same time. You can't call this diaper delivery "constant". And if the emissions from this are significant, you could always use an electric van.
How are you feeling: do you still think this is something only for performative environmentalists or "trad wives"? Neither of these descriptions apply in even the remotest sense to my wife or I.
What do you find confusing about the idea of a diaper service? It seems too unsanitary for you? The idea that it might cost slightly more than disposable diapers is intolerable to you?
Do you have any reason to believe washing diapers actually is unsanitary?
The energy saving is clear and gets better year after year as the grid continues to green. You can also line dry cloth diapers which saves a lot of energy.
I never had any trouble with cloth diapers. You can always have a blow out with any diaper, but I didn't have any issues beyond that. Of course, there's some technique, but it's easy enough to find a YouTube tutorial if you need it. Snappies and decent covers help.
Cloth shopping bags are a really bad comparison here.
Some things working in favor of cloth diapers here are general greening of the grid, mitigating issues with electricity consumption.
Beyond that, line drying diapers works very well and even preserves the life of the diapers.
Cloth diapers hold their value extremely well and can easily be bought/sold/given away on sites like OfferUp or groups like Buy Nothing.
ALso, "2 car trips per week": do you have no idea how this works? No diaper service in their right mind would send out cars to make bespoke trips to individuals. They're done using a big truck on a schedule to amortize the cost of pick up and drop off as much as possible.
This study has serious methodological issues. I would encourage you to carefully peruse it yourself before citing. This is far from the only issue, but the fact that someone who works at Procter and Gamble is on the study's advisory board is... fun.
Oops, talked to my wife and realized I got a detail wrong: we did not keep the diapers from the service, we gave them back. And the diapers we got from the service were almost surely used by plenty of other people before.
Someone else on here seemed confused about the logistics off the service. We lived in Brooklyn at the time. A big truck would drive around our neighborhood and pick up and drop off diapers from many people throughout the neighborhood at once. More amortization at work.
Also, one more thing: we're big on line drying. We will primarily line dry these diapers instead of drying them in a machine.
Oh, and one MORE thing: we got our current washer and dryer free from a friend who was about to throw them out when replacing them.
There are so many ways to mitigate and reduce environmental impact beyond the simple-minded apples-to-apples comparison many in this thread seem to be doing.
The crux of the assumption is that if a polynomial mapping is invertible everywhere (Jacobian nonzero everywhere), its Jacobian must be a constant. Why? Because the only polynomials which are zero nowhere are constants.