The Z3 was not a general purpose computer; it was a calculator that performed a predetermined sequence of operations that was written to its tape. It was remarkable for being all-binary in an era when differential gears and cams were very common in computing devices, and had some other advanced features. But the 1990s article that declared it Turing-complete is just silly. It would apply to every four-function calculator that supports rounding, and programming a computer like that is not just "impractical" - both the tape and execution time would grow exponentially in number of branches - but it is not the model that Turing proposed. The whole point of Turing's (theoretical) device is that a short program using the abilities of that device could perform unlimited computations; if you make the program length unlimited instead, that's a much less interesting model of computation.
The problem is that anything that gets into Wikipedia becomes ingrained in the Internet's collective mind, which then can't be changed.
The claim that the Z3 computer was Turing-complete is not true. There is a paper arguing for it, but a detailed reading of it shows that this is an extremely far-fetched and somewhat disingenuous stretch. (The disingenuous part is because any fixed-function calculator could then be claimed to be "Turing-complete", not just the Z3). The central point of the Church-Turing thesis is that a finite set of instructions, given an unlimited memory to work with, can perform any calculation we can imagine (where the "can imagine" part makes the thesis philosophical). The "finite set of instructions" is indispensable, however, since if the instructions are unlimited, you can simply encode any answer you want into them. The "Turing" mode of Z3, which was of course never used, involves a program which essentially scales in length with the total number of calculations it will perform - or even the exponential of that number, if there are many branches - which is not a good model of a Turing machine.
Of course, no computer is a true Turing machine, since the memory is always limited, but our computers are a useful physical approximation of a Turing machine because a small program can compute using a large memory. The Z3 is not that type of a device at all.
That's how this plane worked - the inflation pressure in flight was supplied by the motor. I don't think heating the air was desired, but some heat will inevitably end up in the air as it is compressed.
That doesn't say anything about Polonium being selectively taken up by the plant and used for growth. Just contamination on the sticky covering of the plants. Which doesn't make smoking tobacco good in any way, shape or form whatsoever, but it's not at all the same claim.
A quick search doesn't bring up any examples of radiophile organisms, in the sense of taking up radionuclides that are used for something. "Radiophile" bacteria exist, but that just means they are highly resistant to ionizing radiation. What are radioactive fertilizers? Potassium is radioactive to a small extent, but it doesn't seem like it's ever used for that property.
I am not the best person to ask, since it's not my field. I heard this from the neuroscientists that I worked with. My understanding is that there are spiking and non-spiking neurons in most nervous systems, including human, but most of the ones in ours are spiking. The earliest-evolved animals, such as nematodes, do not have spiking neurons, or myelin, or some of the ion channels in neuron membranes that more evolved neurons have. Their neurons still have axons and dendrites, but the signals propagate much more slowly and in different ways. I am not sure how well they are understood.
As I said, this is possibly out-of-date information. If there is someone here from the neuroscience field, they can probably make a better comment.
Ralph's main work was on neural impulses in the visual cortex, and on measurements of various potentials in the living brain. He published a memoir called "Another Day in the Monkey's Brain". I believe he had potential medical applications in mind, but I don't think anything that was close by. Unfortunately, he died of an illness in 2011.
I was lucky enough to do some programming work, very many years ago, in the 1990s, in the laboratory of Ralph Siegel (https://en.wikipedia.org/wiki/Ralph_Siegel_(scientist)), who among other things worked on this type of worm connectome models. He used the Hodgkin-Huxley equations to simulate neuron responses on the connectome. The Hodkin-Huxley model, as someone explained to me, is kind of like modeling a human leg as three rigid blocks connected by hinges - it's enough to be useful in many models, but of course it's not a full description. Also, it may not the right model for worm neurons, because worm neurons are non-spiking, and the HH equations describe neurons that produce trains of spikes; they exist in more complicated nervous systems. The HH equations are used in simulations because it's the mathematical model we have, and it seems that they're still used by the OpenWorm project. (I am not very sure about properties of worm neurons, I heard about this a long time ago and the information may be out of date).
I think it's great that this work is still going on, it may produce insights about functioning of nervous systems. But the difficulties are fierce, and we're making very slow and difficult progress in an immense unknown area.
I explained above what happens when the dimension grows - spheres and cones do indeed take up a smaller and smaller portion of their unit cube, eventually having negligible volume. This is important in the context of high-dimensional statistics and so on.
If you want to actually have infinite-dimensional volumes, you can't just assign finite values to them in a simple way, or you will have contradictions such as a certain volume being completely covered by a union of things which have 0 volume. In infinite dimensions, you instead have various measures like the Gaussian measure. Feynman's path
integrals are a kind of way to assign a value - called amplitude - to an infinite-dimensional manifold (a kind of "volume") of paths. But that takes us well to the side of the idea of the ratio between cube and inscribed figure volumes.
Actually, you have that exactly right, and it's a very important fact in mathematics and statistics. A unit sphere takes up a smaller and smaller part of a unit cube as the dimension grows (and a unit cone is similar). In other words, a unit circle fills up most of the unit square (~3.14 out of 4), a unit sphere fills a little over half of the unit cube (~4.2 out of 8), and as the dimension grows, the fraction becomes negligible.
Imagine that you have something which depends on many variables (hundreds), and you're trying to predict its behavior based on your previous experience. There is a high chance that the next combination of variable values that you see will be in one of the corners of the many-dimensional cube, because that's where the volume is (the central part of the cube has negligible volume, as we said above). This means that every measurement is in effect an outlier along several dimensions, making predictions very difficult. This is part of the "curse of dimensionality" in statistics. I have seen some people with excellent understanding of mathematics trip themselves up in this area.
"Workers of the Soviet Union! Let us consolidate ourselves even more around the Communist Party and the Soviet Government, mobilize our forces and our creative energy for the great work of building a Communist society in our country! Long live the unbreakable union between Party, Government, and People!"
The closest thing to a universal approach would be a Kalman filter. It's usually where you start when you have noisy measurements coming in, and you need to maintain state such as value and derivative.
Since the original question was about computing the velocity of a car, and since I work in the automotive field, let's take a real example: you want to know the approximate position, acceleration, and velocity (linear and angular) of your car. Your inputs are driven wheel speed (noisy, affected by wheelspin), non-driven wheel speed (noisy), accelerometer output (inaccurate, only present for some axes), GPS position (updated occasionally, has errors), and steering angle (pretty accurate, can be put into a chassis dynamics model). Almost certainly, you would use a Kalman filter to estimate the state of the car. Naive approaches such as subtracting two wheel speed values to obtain acceleration will not work well.
My point is that we should remember that numerical algorithms are a developed field with a lot of knowledge, and we should take advantage of the proven approaches. Sometimes, programmers who are not specifically from the physics or numerical fields, and who need to perform some computation, reach for a very simple approach such as the rectangle-rule integrals, and get bad results.
This calculation amplifies any noise present in the values of the function, often to the point of the output being unusable. There are many methods that can be used to approximate derivatives, depending on the problem. Just as we shouldn't try to invent our cryptographic methods from scratch, we should take advantage of the extensive knowledge already in use for numeric methods.
I've seen naive numeric methods cause everything from jerky motion in video games to incorrect navigation data for cars.
Even if you're limited to uniform sampling, something as simple as the trapezoid rule will give you quadratic convergence instead of linear for the naive Sum(f(t_i) * delta). In other words, error proportional to 1/n^2, instead of 1/n, where n is the number of samples, which is going to be a huge difference. There are many methods depending on the constraints of your problem - your ability to choose sampling intervals, knowledge of the bounds of your function or its derivatives, etc. The PSA is to study these things, instead of just writing the first thing that seems familiar from a long-ago calculus class.
Can't pass by without making a public service announcement about the average speed example: don't compute integrals using the calculus definition of Sum(f(x_i) * delta). Look up quadrature or numerical integration methods instead.
Although it is true that you can approximate the average speed by taking an average of instantaneous speed measurements, that's usually a very bad way to do it in any real world situation. Numerical difference values are always noisier than the underlying quantity, sometimes to the point of being unusable, so of course if you can just read off the quantity you want directly (total difference over time), you should do that. But even if you can't, you should use a proper integration method instead of the calculus definition.
I have seen the Sum(f(x_i) * delta) calculation in a lot of real-world code. It has bad convergence properties, bad errors when the function has large derivatives, and bad performance when the data has noise. Some of the code I've seen produces garbage results, or has thousands of function evaluations when you need, like, four. "Quadrature? I think I heard that before, but I don't remember what it means."
In summary, please don't compute derivatives as (f(x_i+1)-f(x_i))/delta, or compute integrals as Sum(f(x_i) * delta), and especially, please don't do the first immediately followed by the second. Which also happens. Look up numerical methods instead.
Wouldn't work, since 5 is not a member of the set of indescribable numbers. I am still not sure that the original example is good, since serious difficulties with the axiom of choice have to do with infinite collections of sets, especially uncountably infinite, not with choosing a member of a single difficult-to-describe set. But you can't handle these non-describable numbers very easily.
I don't know who applied it to LLMs, but it is/was the standard term used for an image processing model producing a detailed signal not justified by its inputs. For example, "face hallucination" means that the model produces a detailed-looking face when given very noisy data, but of course the face will not actually be the original face. In fact, the original image may have had no face at all. Hallucination can be either desired (as a kind of generative technique) or very harmful - imagine using image enhancement to identify a criminal in a noisy image, and getting a detailed face looking like someone in your training set - but not the right person's.
Any image enhancement technique, deep learning-based or not, can result in hallucination - you're producing information which was not in your input, which you're able to do because you have priors. But this can always result in incorrect information.
I don't know the details about the showing of that movie, but America's mass car ownership certainly came up in the comparisons of life in the USSR and in America. In one discussion with my classmates (in Moscow, USSR, ca. mid-1980s) I mentioned that in the suburb of New York in which my relatives lived, most of the high school students in the upper grades drove their cars to school and parked near it. "Come on", someone said, "I am with you when you tell us not to believe our propaganda, but you shouldn't believe theirs, either. Who is going to give a car to some kid, unless he is the son of a senator or something?"
Oh, and some years later, a friend of mine in America was asked by a family in the USSR (which would not be the USSR much longer) to bring some package or item to their relatives, who lived in some other suburb of NYC. My friend asked for the address. "Don't remember right now, but you'll be able to find them: they have two cars!"
The problem is that anything that gets into Wikipedia becomes ingrained in the Internet's collective mind, which then can't be changed.