The 21 cm line was chosen specifically because it’s the brightest line in the radio regime in our galaxy. Any civilization in the Milky Way capable of developing electromagnetic sensors would see that emission. There is a game theoretic component to this, as these other civilizations would also know that we could also see that line, and thus understand its importance
Do you know how to program? It’s super easy to write a very simple rejection-based program to estimate the volume of a hypersphere (you can do it in <10 lines with numpy). Try it yourself for dimensionality 50 and see how long it takes before the estimate rises above precisely 0
Sorry, yes, you did miss the point. Specifically for the case of estimating volume, you just need enough points interior to the hypersphere to get good sampling statistics (which would become a problem in and of itself, by the way, at sufficient dimensionality). But estimating volume (or pi) is normally just used as a pedagogical example of rejection sampling, it’s not (usually) what people actually use the technique for. Typically its used to draw samples from one distribution, X, that is notionally hard to draw from, by instead drawing from Y. The sphere case is just an example (its actually easy to draw from a hypersphere without rejection sampling) - but if you did try to use rejection sampling specifically, it would take you a long time and lots of compute resources to generate those samples (in high dimensions)
Because if you are trying to draw samples from a hypersphere via rejection sampling from samples of the hypercube, your sampling efficiency will go down exponentially with dimensionality. A smaller and smaller percentage of your samples will go unrejected as dimensionality grows
I have yet to see differential entropy used successfully (beyond its explicitly constructed-for purpose for calculating channel capacity). Similar to your thought experiment is the issue that the differential entropy value depends on your choice of unit system. Fundamentally the issue is that you cant stick a quantity with units into a transcendental function and get meaningful results out
> If I say "well I suppose you'd at least need to do A to qualify" and then later say "huh I guess A wasn't sufficient, looks like you'll also need B" that is not shifting the goalposts.
I dunno, that seems like a pretty good distillation of what moving the goalposts is.
> I’ll know it when I see it, and I haven’t seen it. QED
While pithily put, thats not a compelling argument. You feel that LLMs are not intelligent. I feel that they may be intelligent. Without a decent definition of what intelligence is, the entire argument is silly.
Peoples’ memories are so short. Ten years ago the “well accepted definition of intelligence” was whether something could pass the Turing test. Now that goalpost has been completely blown out of the water and people are scrabbling to come up with a new one that precludes LLMs.
A useful definition of intelligence needs to be measurable, based on inputs/outputs, not internal state. Otherwise you run the risk of dictating how you think intelligence should manifest, rather than what it actually is. The former is a prescription, only the latter is a true definition.
Yes, we have several techniques for detecting exoplanets, and at least two of them have that bias. However, what I said still holds - our best estimates are that most stars with planets have (at least some) planets in orbits closer than is Earth’s, and therefore asserting that “planets with sufficient energy density at their surface to support complex life are rare” is unwarranted
> Smaller stars last longer, but they also emit less energy and so there is less likely to be higher life forms just because there isn't enough energy to support life.
This isn’t really true. The energy density at a planet’s surface is a function of both the star’s luminosity and the planet’s distance. Most star’s (at least in the Milky Way) are small, and most stars have planets in orbits closer than is Earth’s.
The so called “Goldilocks zone” is expected to have at least one planet in it for a very large number of stars (again, at least in our galaxy).
Norton’s dome is a valid paradox, in the sense that the math really does admit two valid equations of motion. The link you provided doesn’t dispute that fact (other than commenters pointing out that you need a proportionality constant to make the units work out).
My favorite intuitive explanation for the presence of the paradox is well summarized by the Wikipedia article on the dome: “To see that all these equations of motion are physically possible solutions, it's helpful to use the time reversibility of Newtonian mechanics. It is possible to roll a ball up the dome in such a way that it reaches the apex in finite time and with zero energy, and stops there. By time-reversal, it is a valid solution for the ball to rest at the top for a while and then roll down in any one direction.”
For point source astrometry, there are a few ways to beat the diffraction limit. For instance, some observers will purposely defocus their optics to spread the target photons over a larger number of pixels, which with sufficient SNR lets you gain a better lower bound (from a Cramer Rao perspective) on precision. I think Gaia actually does something similar to this but “blurs” through time, rather than across space, by purposely not perfectly tracking stars so that they drift at sub-pixel rates across the FPA.
While I agree with your general assessment, I think your conclusion is a bit off. You’re assuming 1kw/m^2, which is only true with the sun directly overhead. A real-world solar setup gets hit with several factors of cosine (related to roof pitch, time of day, day of year, and latitude) that conspire to reduce the total output.
For example, my 50 sq m set up, at -29 deg latitude, generated your estimated 30 kwh/day output. I have panels with ~20% efficiency, suggesting that at 60% efficiency, the average household would only get to around half their energy needs with 10 sq m.
Yes, solar has the potential to drastically reduce energy costs, but even with free energy storage, individual households aren’t likely to achieve self sustainability.
That reminds me of my favorite algorithm, which can find the majority element in a list with any number of distinct entries while using O(N) time and O(1) space (provided a majority element exists). I sometimes pose deriving this algorithm as a puzzle for people, no one has ever solved it (nor could I).
Boltzmann had a whole lot of math backing up his assertions. Among the many issues with what you posited, I highly doubt you have bothered to actually check to see if the force necessary to keep electrons from colliding with the nucleus would also be commensurate with what’s needed to counteract gravity at cosmological scales.
Your theory is analogous to saying that we don’t need to invoke gravity to explain why the earth doesn’t spiral into the sun, because it could just be explained with solar photon pressure.
> It also keeps the electrons flowing around the nucleus, but not into it.
Asa cautionary note to other readers, this statement is not consistent with any theory of which I am aware, and it’s certainly not a concept accepted by mainstream physics
I worked and did research atobservatory for my PhD - you are exactly correct. It was well known to the scientists and onsite engineers that a collapse was imminent, for years. Ironically, prior to the collapse the agency responsible for funding performed a study to see would need to do to stop supplying money to the observatory, and found that due to environmental impact, it would actually be more expensive to stop funding. After that study we all suspected that the agency was just waiting for decay or a natural disaster to do the dirty work for them
The column mass (ie, the spatial density multiplied by the length of the observer’s line of sight through the material) is highest at the limbs when looking at a gaseous shell. So shells like this look like rings from all angles. A related effect called limb darkening is the edges of the sun always look less bright, independent of viewing angle
The parent comment said nothing about using deep learning. Convolution is not the same as using a CNN. I interpreted their comment as meaning they used a 2D convolution (presumably a 2D cross correlation, actually) to find regions of overlap
This isnt a correction to your post, but a clarification for other readers: correlation implies dependence, but dependence does not imply correlation. Conversely, two variables share non-zero mutual information if and only if they are dependent.
Another issue is that likelihoods associated with continuous distributions very often have units. You can’t meaningfully assign “magnitude” to a quantity with units. You can always change your unit system to make the likelihood of, say, a particular height in a population of humans to be arbitrarily large.
From a physics perspective, complex frequency results in “evanescent waves” - ie, waves that decay rather than oscillate (technically a fully complex frequency of the form a+ib will both oscillate and decay)