guyomes

See also the Wikipedia page on the unreasonable ineffectiveness of mathematics, notably in biology and economics: https://en.wikipedia.org/wiki/Unreasonable_ineffectiveness_o...

A generalisation of this idea is known as Taylor model in 1998 [1]. It might even have been known in 1984 as neighborhood arithmetic [2]. The generalisation works by taking a Taylor expansion of the function up to order n, and then by using a bound for the remainder using bounds on the partial derivatives of order n+1 [3].

[1]: https://www.bmtdynamics.org/cgi-bin/display.pl?name=rdaic

[2]: https://books.google.fr/books?id=2zDUCQAAQBAJ

[3]: https://en.wikipedia.org/wiki/Taylor%27s_theorem#Taylor's_th...

My anecdotal tests and several benchmarks suggest that Qwen2-VL-72b [0] is better than the tested models (even better than Claude 3.5 Sonnet), notably for OCR applications. It has been available since October 2024.

[0]: https://huggingface.co/Qwen/Qwen2-VL-72B-Instruct

For handwritten texts, the tool that works best for me is Qwen2.5-VL-72b [0]. It is also available online [1]. I'm surprised that it is not mentioned in the article since even the previous model (Qwen2-VL-72b) was better than the other VLMs I tried for OCR on handwritten texts.

[0]: https://huggingface.co/Qwen/Qwen2.5-VL-72B-Instruct

[1]: https://chat.qwenlm.ai

For the mantissa, the insight is probably that :

(1+e1)*(1+e2) = 1+e1+e2+(e1*e2)

If e1 and e2 are small, then e1*e2 is negligible.

I wonder if their curated set of 817 math problems is also useful as teaching material for training math students on a diverse set of problems.

> Aristarchus had come up with a heliocentric theory way back around 250 BC.

He notably did convincing calculations showing that the sun should be larger than Earth. However, this was not convincing enough for his opponents who argued notably that usually fire hardly remains in a fixed position. Also, if Earth was moving, it was hard to understand why there was no observed parallax on the stars at night. Actually, the first apparent proof that Earth was moving came after inventing instruments precise enough to observe parallax on far enough stars, in 1727, from James Bradley. It actually had different results from expectation, due to the finite speed of light [1].

Just after Aristarchus came also Seleucus of Seleucia [2]. He supposedly had a theoretical argument for heliocentrism, but it was lost.

[0]: https://en.wikipedia.org/wiki/On_the_Sizes_and_Distances_(Ar...

[1]: https://en.wikipedia.org/wiki/James_Bradley

[2]: https://en.wikipedia.org/wiki/Seleucus_of_Seleucia

Sambanova is not often mentioned either [0]. One of his co-founder is known as “father of the multi-core processor” [1].

[0]: https://sambanova.ai/

[1]: https://en.wikipedia.org/wiki/Kunle_Olukotun

> the utility will only let me put up a 4kW array due because they can't accept the extra energy and stay profitable.

Transmission congestion might be a more important issue than profitability: "Avoiding the congestion is essential for a competitive electricity market and is one of the toughest problems of its design." [1]

The course of Damien Ernst [2] gives an excellent overview of all the challenges related to decentralized electricity markets.

[1]: https://en.wikipedia.org/wiki/Transmission_congestion

[2]: https://damien-ernst.be/teaching/elec0018-1-energy-markets/

> I wouldn't want to breathe the stuff in powdered form.

This makes me think of wood dust being dangerous to inhale [1], despite wood being a perfectly safe material for furniture at home.

[1]: https://en.wikipedia.org/wiki/Health_impacts_of_sawdust

> I never heard anyone say that if a highly social person just tried harder they'd learn to like a quiet weekend at home

The full school institution actually do it. Teachers tell students to stop speaking to each others for most courses during the week. Moreover some teachers explicitly talk about the joy of quietly reading books, and directly ask students to do so.

I'm fine with the way school works, especially considering the limited resources they have. Nevertheless, teachers do not often encourage extroverted skills that are useful in group work, oral presentations, etc. Exams seem to favor people that are able to stay hours long alone thinking about abstract concepts.

The experts from Belenios do not recommend to use remote e-voting for high-stake elections [1]. Some issues they mention are the risks that the users sells their credentials or that a malware on their computer leaks who they voted for.

[1]: https://www.belenios.org/faq.html

You can add to the list the colors of their wings, rendered by a complex nanostructure that is triply periodic [1].

[1]: https://math.ucr.edu/home/baez/butterfly.html

For a small survey of practical efficient methods for bignum arithmetic operations, the Algorithms section of the documentation of GMP [1] is excellent.

[1]: https://gmplib.org/manual/Algorithms

You're perfectly right. It is indeed perfectly logical then. It could be reformulated like this: if f(A) > f(not A) then f(not A) is not maximal.

f: a function indicating how complete the thoughts are.

A: writing about thoughts.

Taking the statement completely out of context, it states : if A implies B, then not A implies not B. This is a logical flaw.

The correct statement from a logical point of view is: if A implies B, then not B implies not A.

In this case, even if writing down your ideas makes them more precise, there might be other methods that make your ideas more precise. Again this is just the logical point of view, out of context.

The field evolves quickly. The Meta 3D Gen paper states that Rodin Gen-1 [1,2,3] has a clean topology. As a non professional, the wireframes from some examples indeed look nice.

[1]: https://github.com/CLAY-3D/OpenCLAY

[2]: https://hyperhuman.deemos.com/

[3]: https://huggingface.co/spaces/DEEMOSTECH/Rodin

For FFT with floating-point numbers, another paper from Arnold Schönhage in 1982 [1] already gives the bound in Psi(n l) operations, where n is the number of coefficients, and l is the desired precision (typically 53 for double precision). Psi(m) is the time to multiply two integers with m digits, which is known since 2021 to be O(m log m) [2]. So the current bound is O(nl log(nl)).

[1]: https://doi.org/10.1007/3-540-11607-9_1

[2]: https://www.texmacs.org/joris/nlogn/nlogn-abs.html

And it turns out that on the Klein bottle, the maximum is 6 colors. More generally, this number depends directly on the number of holes of the surface [1].

[1]: https://en.wikipedia.org/wiki/Heawood_conjecture

If you are ready to spend some precomputation time to compute a good approximation, you can use the Remez algorithm [1]. It is implemented in the Sollya library for machine precision [2,3]. It has notably been used to implement the Core Math library [4] to provide correct rounding for the math functions in the libc library.

[1]: https://en.wikipedia.org/wiki/Remez_algorithm

[2] : https://www.sollya.org/

[3]: https://www.sollya.org/sollya-weekly/sollya.php

[4]: https://core-math.gitlabpages.inria.fr/

Another approach is the one implemented in Estonia. A citizen there has the ability to query who has accessed their records [1]. Combined with proper laws, this feature lead to "some very public cases of government officials being caught accessing private data of Citizens - without any legitimate and authorized reason for such access."

Note that through this mechanism, the society is still a control society [2], even though citizen themselves have more to say in that control.

[1]: https://doi.org/10.1007/s12553-017-0195-1

[2]: https://en.wikipedia.org/wiki/Gilles_Deleuze#Values

You can fold a paper to get a torus [1]. With those foldings, the distances on the torus embedded in 3D are the same as the distances on the flat paper.

It is even theoretically possible to embed the flat paper as a torus in 3D with a C^1 surface, without polyhedral edges [2,3]. However, this surface has a fractal structure.

Finally, any torus surface embedded in 3D that is at least C^2 (with a continuous second derivative) will nessecarily stretch some distances [4].

[1]: https://www.imaginary.org/hands-on/diplotori-flat-polyhedral...

[2]: https://aperiodical.com/2012/05/torus/

[3]: https://www.pnas.org/doi/full/10.1073/pnas.1118478109

[4]: https://math.stackexchange.com/questions/2291382/c2-isometri...

> OpenAI is betting heavily on code as a way to improve LLM reasoning for AGI.

And researchers from Google Deepmind, University of Wisconsin-Madison and Laboratoire de l’Informatique du Parallélisme, University of Lyon, actually publish some of their results in that direction [1,2].

[1]: https://deepmind.google/discover/blog/funsearch-making-new-d...

[2]: https://www.nature.com/articles/s41586-023-06924-6

In academy for example, including the publicly funded institute mentioned in the article (European Molecular Biology Laboratory’s European Bioinformatics Institute). Also, there is a blog page on the deep mind website where they cite "millions of researcher" around the world using alphafold [1]. Some of those institutions might be happy to develop the new alphafold.

On the other hand, even in academy, scientific articles are not always the best examples of open and reproducible research [2,3]. And in computer science, it is common for articles to be more focused on algorithms than on implementations.

[1]: https://deepmind.google/impact/meet-the-scientists-using-alp...

[2]: https://en.wikipedia.org/wiki/Reproducibility

[3]: https://en.wikipedia.org/wiki/Replication_crisis

Even though it might help to start with a mechanism at a different physical scale, changing the scale matters indeed.

The mass being different, this leads to different forces required to get the same acceleration or deceleration. That follows directly from the formula: force = mass * acceleration.

Moreover, the resonance frequency [1] of an object is different for different scales. For example, piano strings will vibrate at different speeds, depending on their sizes. That would be the same for the robot cables.

Also, when changing the scale of an object, the materials often change. The cables and the motors would probably be built differently, with different physical properties.

In general, to design a new mechanism, it is common practice to start with software simulations, 3D prints, and small scale prototypes. However, the properties of the new mechanism are only validated when you have built a real size prototype.

[1]: https://en.wikipedia.org/wiki/Mechanical_resonance

All-in with the Foldit paper [1,2].

[1]: https://en.wikipedia.org/wiki/Foldit

[2]: https://www.nature.com/articles/nature09304

> I've heard before that the people doing this 'tween' frames make less than minimum wage on contract.

An employee of an animation company describes in a comic book his experience on working with people drawing the in-between frames. They were paid literally with rice bags [1].

[1]: https://en.wikipedia.org/wiki/Pyongyang:_A_Journey_in_North_...

> I can remember the fine details of complete songs seemingly forever

If you didn't buy the song, does it count as an illegal recording? /s

This was the main subject of a science fiction short story, unfortunately I cannot remember its title or author.

That would probably be true if all the coefficients were complex numbers. The biais here comes from the fact that all the coefficients are real numbers.

Expressing proofs as formal logic is a great tool to make them more rigorous. Formal proofs are easier to check with a computer, provided that all the cases have been fleshed out, and that it has been written in a proof-assistant language. However those proofs are tedious to write, and a human reader has a limited capacity for handling high number of cases. Omitting cases and writing out informally why those cases can be omitted is crucial to help the reader to follow the proof. That is a step where errors can arise.

An example of wrong proof due to sketching quickly some cases is the proof that all triangles are isosceles [1].

Here is also an example of an apparently obvious result with a non-trivial proof, were all the cases are written out formally in the proof-assistant language Coq [2]. It is the proof that if we compute the multiplications and the square roots with floating-point arithmetic in base 2, then sqrt(a*a) is actually |a|. This was assumed without proof in some previous papers, and it is easy to understand how the authors may have missed it. Note that this result is not true in base 10.

Finally, sometimes non-formal proof can actually be more convincing than formal proof. For example, it was proven formally in 1957 that it is possible to turn a sphere inside out continuously, without cutting it, tearing it or creating any crease [3]. With more work, this result was later proven with a video. The video proof is arguably easier to follow and more convincing for a human. The formal proof has not yet been formally checked by a proof assistant, although work in this direction is on the way [4].

[1]: https://www.themathdoctors.org/false-proofs-geometry/

[2]: https://inria.hal.science/hal-01148409v1/document

[3]: https://en.wikipedia.org/wiki/Sphere_eversion

[4]: https://leanprover-community.github.io/sphere-eversion