Ask HN: What scientific phenomenon do you wish someone would explain better?
I've been studying viruses lately, and have found that the line between virus/exosome/self is much more blurry than I realized. But, given the niche interest in the subject, most articles are not written with an overview in mind.
What sorts of topics make you feel this way?
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[ 2.1 ms ] story [ 462 ms ] threadMyopia is definitely hereditary, especially the pathologic variants that can lead to retinal tears and the like.
That being said, there is the process of refractive development that occurs early on in life. The eye develops at a frighteningly fast pace, and you achieve near-adult globe size after about 18 months. The genes that drive this refractive development could be hereditary, if that is what you're trying to figure out.
Now, we can make a claim that this adaptation during infancy could eventually affect our genome, but I have not delved into the epigenetic literature to determine if that has been borne out or not.
One idea is known as the Copenhagen interpretation.
It basically says that the wave-like effects we associate with matter is merely a wave of probabilities. Or in terms of the double-slit experiment and in other words, light behaves like a particle, but the wave-like effects you see is just the result of probabilities where the particles end up. Dark areas are areas of low probability, and lighter areas high probability.
One might imagine the light particles streaming through the slit end up having slight variation in trajectory from one particle to another (for various reasons such as interference with other particles), which results in areas where most particles end up and others where few end up... representing a wave.
Why interpretations: There is an experiment you can do that is hard to explain: Either particles are able to somehow influence each other faster than light (non-local), or the particle somehow doesn't exist except when interacting with some other particle (non-real).
Try this video: https://www.youtube.com/watch?v=zcqZHYo7ONs the AHA moment in the video comes when you realize you can entangle the light and that adding a filter by one stream of light somehow causes the other stream of light to also be influenced.
1. Subatomic Matter is by default both mass and a wave, but when "observed" it becomes a particle as we know it i.e. with mass.
2. Atomic bonds are formed due to electrons (waves) being shared between adjacent atoms.
Hope I have some parts correct. Perhaps someone can shed some photons.
Here is a few books you can read on the subject. They do a pretty good job on describing what the issue is and what the interpretations mean:
Max Tegmark - Our Mathematical Universe
Sean M. Carroll - Something Deeply Hidden
Adam Becker - What is real?
Here are some things you can google if you want to just skim the subject: Wave–particle duality, The Measurement Problem, Quantum decoherence, Copenhagen interpretation, Bell's theorem, Superdeterminism, Many-worlds interpretation, Ghirardi–Rimini–Weber theory (GRW).
Last but not least, look at the Wolfram Physics Project. (https://wolframphysics.org). The take on quantum mechanics if you go along with the idea of hyper-graph is fascinating (to me)
And adding on to that: Will light inside a box redshift? If I weigh the box (i.e. weigh the light inside the box), then wait a bit for the light to redshift, then weigh the box again?
The underlying reason for this is that Noether's theorem tells us that every physical symmetry implies a conservation law for some physical quantity. Conservation of energy and momentum comes from the fact that the physical laws are the same throughout time and space. However, cosmological expansion violates that assumption, so there is no reason that energy and momentum should still be conserved. [2]
[1]: One side note here is that relativistically, energy and momentum are not really separate physical quantities, but instead two components of the same underlying physical quantity. Unfortuantely, this quantity does not really have a good name (despite Taylor & Wheeler's attempt to call it "momenergy"). It ends up being called the momentum 4-vector, but the temporal component of this 4-vector is energy.
[2]: This is only true globally. Locally, the laws are approximately the same from one moment to the next, so conservation of energy and momentum hold for small distances and short times.
I believe even folks at NASA have even said it helped cement their mathematical knowledge with a better intuitive understanding.
https://en.wikipedia.org/wiki/Protein#/media/File:Chaperonin...
I'm working on making a model of this chaperone complex relative to a folded protein to get a sense of how it might be interacting with the amino acid chain before it becomes globular
How immune system and medications work.
Why some plastics are recyclable and others are not.
The difference between thermoset and thermoplastic polymers has to do with irreversible chemical bonding during curing. With thermosets, you have chemical bonds between molecules preventing deformation, whereas with thermoplastics, you just have a viscous friction between molecules that varies with temperature. If you heat up a thermoplastic, that viscous friction goes away and the plastic can be remolded.
>The immune system: The most awesome thing ever.
It's actually two systems. One called the innate that we have in common with most forms of complex life, and the adaptive immune system, something we've only seen manifested in jawed vertebrates.
The innate immune system is a set of cellular signals/behaviors that are triggered by cells being exposed to damage and or stress.
These responses are generalized. Just about anything odd can invoke them, so they are typically the first line of defense. These include things like alteration of permeability of the local extracellular matrix (swelling), formation of impermeable tissue barriers to isolate damage (cysting, compartmentalization), and setting up signaling molecule gradients that attract phagocytic/cytotoxic cells to the terminate anomalous cellular activity/clean up the place (macrophage attraction), and alteration of metabolic activity to generate thermal stress (fever).
The issue with the general immune system though, is that it's non-specificity and versatility makes it a bit like a sledgehammer in the context of a complex organism. It can do as much or more damage as it can do good, and it isn't that good at only eradicating the exact thing causing the issue without excessive collateral damage.
Enter the adaptive immune system. The adaptive immune system is composed of various cell lines, and organ systems all specialized into dealing with specific facets of an immune response, and mediated through a set of special cellular surface receptors.
The facets of the adaptive immune response are: antigen recognition, coordination, moderation, and memory.
The major cell lines are T and B cells. T cells are further broken down into cytotoxic T, and helper T cells.
The adaptive immune system starts with naive lymphocytes. These cells rapidly multiply randomizing the ever loving crap out the region of the genome dedicated to the MHC receptor. By doing this, it'll cause the receptor to fold in ways that will allow it to bind with certain types of antigen (think of it as the antigen's key fitting the Mac's lock.) This proves of new receptor generation is mediated by the Thymus. The thymus tests every new variant to see whether there is any sensitivity to proteins that may expressed in other parts of the body. If it finds that to be the case, itinduces that particular cell to suicide to prevent the proliferation of immune cell lines with a high chance of being prone to autoimmunity. Those that survive are allowed to move out into the lymphatic and circulatory systems to patrol for their particular antigen. Upon meeting it, a few things happen. First off, the immune cell can help kick off or amplify a general immune response. Secondly, signaling proteins are released to attract more leukocytes to the area. Third, an antigen bearing cell will migrate toward the Thymus to recruit more immune cells. Once an antigen presenting helper T cell binds with a compatible B or Cytotoxic T cell line, that cell line undergoes massive replication without further modifying it's receptor, and the helper T cell does likewise.
B cells will create and secrete antibodies. Small snippets of protein that will bind to and foul up the workings of the antigen to which the are sensitive.
Cytotoxic T cells will patrol for and engulf antigen it encounters, either breaking it down with a burst of oxidative substances, or if the antigen is detected being presented on a cellular membrane protein, and a helper T cell is near by to enable the response, a cytotoxic T cell can induce cell death of an antigen presenting cell, but with much greater specificity and numbers than the mechanism used by macrophages. Once the cell death takes place, the cell will either clean up the remains, or attract macrophages to do so while it heads off for the next target.
The cytotoxic T cells are handicapped in their destructive potential by the need for a nearby Helper T cell. B cells just shotgun anti-antigens into the ...
Some fundamental concepts that might fly over someone without a biology degree might include the absolutely fundamental requirement of protein binding for any biological process (akin to a transistors function for a computing device).
I'm thinking that step one of communicating the entire immune systems complexity is probably omission, of anything that's not absolutely required for comprehension of the basic concept. In that regard I'm not convinced there's any need for bringing up the innate system first (there's a reason we discovered it quite recently, perhaps?). Other details can also be similarly "omitted" for simplicity perhaps. What are your thoughts?
There are so many odd facets of it with interesting implications; like did you know exposure to sex hormones actually contributes to the atrophy of the Thymus over time? This is posited to have some relation to the increased likelyhood of developing autoimmunity 0roblems as we get older.
Also, not all T cell lines undergo adverse selection in the Thymus. There is a smaller population of more autoimmune sensitive cells that develop and specialize in the extremities. It is theorized this is evolutionary selected for because there is a tradeoff between being able to develop to a wide variety of pathogens, and being free of auto immunity. So you keep a small group of possibly autoreactivr immunity cell lines just in case. This is theorized to explain the prevalence of autoimmunity issues in the extremities being relatively common.
The full 3d Coriolis force is more complicated than that (eg accounting for the Eötvös effect): The spinning disk example only gets you to the -2vω term (where v denotes radial velocity and ω angular velocity).
ZK proofs have a number of good explainers, mostly using graph colorings. Non-interactive versions, however, require quite a bit more than that explanation allows - and despite asking experts, I still haven't found a good, basic explanation.
https://en.wikipedia.org/wiki/One-electron_universe
https://www.youtube.com/watch?v=9dqtW9MslFk
https://www.youtube.com/watch?v=zcqZHYo7ONs
https://www.youtube.com/watch?v=f72whGQ31Wg
https://www.youtube.com/watch?v=qd-tKr0LJTM
The StackExchange sites have less coverage and answers tend to be more technical.
University websites return reliable answers, but often neither short nor accessible.
Fermat's Last Theorem (book) by Simon Singh is the source to check out if you're interested in the details of how it eluded mathematicians and a general idea of how the problem was solved, without getting too technical. It's a great story well told.
But why that solves the problem? Because it connects two branches of mathematics (modular forms and elliptic equations) in a way that proves that equations of that form cannot exist (where the exponent is > 2)
Though there probably is an easier way of explaining it, it is strongly suspected that Fermat got the wrong idea there.
I also like that FLT follows easily from the Beal conjecture, which seems overlooked. Maybe its overlooked because its closely related to some other (harder to understand) conjectures.
@GP: I'd recommend to try to read an understand the proof for N=3. (And why that approach does not extend to bigger N.) It requires only undergraduate level math and it is much much much easier. It uses very different tools, so it will give you very little insight of the general proof, but it will give you some taste of the problems of the proof.
In some experiments the weird mathematical thing can be approximated as an almost classical particle. That approximation simplifies the calculation a lot, and sometime you can get some result intuitively. But it is never true, it is only a very good approximation.
In some experiments the weird mathematical thing can be approximated as an almost classical wave. That approximation simplifies the calculation a lot, and sometime you can get some result intuitively. But it is never true, it is only a very good approximation.
Try to read again everything you have read about the subject, but every time the text says "here light is a wave/particle" use a red marker to rewrite that sentence as "here light can be approximated as a wave/particle".
Basically: particle are the quanta of waves. So it's not really a duality in the end.
At the subatomic level, we observe that electrons have some extra angular momentum, beyond what we'd expect from their "orbits". We call that spin, because it's intrinsic, like the spinning of a macroscale object.
1: https://www.lesswrong.com/posts/AnHJX42C6r6deohTG/bell-s-the...
2: https://kantin.sabanciuniv.edu/sites/kantin.sabanciuniv.edu/...
But, we specifically have no way of proving that theory. So now we're back to the essence of the original question - if these things seem random why do we know that they're in fact deterministic without any hidden variables?
If you are properly amazed by it, rejecting MWI or any crazy-ish borderline-conspiracy theory seems suddenly a lot harder.
I feel the whole Yudkowsky's QM series in fact served to deliver that one post.
But the fact remains that it is impossible to prove and it is conveniently well equipped to handle this situation. I'd prefer an argument that presupposes the Copenhagen interpretation as that is when my intuition fails.
Theoreticians choose very different mindsets about the same equations, which (they say) somehow create them grounds to form various new hypotheses. As far as I know neither approach was very fruitful so far in terms of new science, so people try multitude of others.
What I've meant to say above, I have much trouble using Copenhagen to understand Bell's experiment. MWI fits the bill here for me.
Is that not like trying to get a better intuition for planetary movement by using an epicycle-based model? The fact that the interpretation is conveniently shaped in a way that a paradox isn't an issue is not a coincidental thing that should be overlooked in the spirit of fairness to alternative interpretations. Regardless, I think my post below is useful for answering your want.
>So now we're back to the essence of the original question - if these things seem random why do we know that they're in fact deterministic without any hidden variables?
The world is only deterministic under Many-Worlds, and it's deterministic in the sense of "each outcome happens (mostly) separately". It doesn't make any sense to try to make sense of the "deterministic" part separately from MWI. MWI is the only deterministic QM theory (unless you're going to consider "superdeterminism", but there's nothing concrete to that interpretation besides "what if there existed a way that we had QM+determinism but not MWI". There's no basis to it, besides a yearning from people that like the abstract idea of determinism and don't like the abstract idea of MWI).
EPR doesn't tell us that the world is deterministic. It tells us that local hidden variable interpretations (where experiments have a single outcome) of QM can't work, because it shows that a measurement on a particle can appear to you to affect the measurement made by someone else on a distant particle. The Copenhagen interpretation response to this is that the wave function collapse must be faster than light. Therefore, the Copenhagen interpretation is not a "local" theory. (The Copenhagen interpretation doesn't give us any answer for who we should expect to trigger this wave function collapse first when two measurements are taken simultaneously at a distance though.)
But IIUC, one of the remarkable things about MWI is that it would be a local hidden variable theory!
This is a very important property to have because the principle of locality is deeply ingrained in the way the Universe behaves. Note that (almost?) no other quantum interpretation is both realist and local at the same time.
Maybe you wonder, how is it possible that MWI can be considered a local hidden variable theory if Bell's theorem precisely shows that local hidden variable theories are not possible?
I think that it was Bell himself who said that the theorem is only valid if you assume that there is only one outcome every time you run the experiment, which is not the case in MWI.
This means that MWI is one of the few (the only?) interpretation we have that can explain how we observe Bell's theorem while still being a local, deterministic, realist, hidden variable theory.
If you take the Bell test experiment where Alice and Bob perform their measurements at approximately the same time but very far apart, I think you and I both agree that when Alice does a measurement and observes an outcome, she will have locally decohered from the world where she observes the other outcome.
But I don't see why the decoherence necessarily has to happen faster than the speed of the light.
It makes sense that even if Alice decoheres from the world where she observes the other outcome, the outcomes of Bob's measurement are still in a superposition with respect to each Alice (and vice-versa).
And that only when Alices' and Bobs' light cones intersect each other will the Alices decohere from the Bobs in such a way that the resulting worlds will observe the expected correlations (due to how they were entangled or maybe even due to the worlds interfering with each other when their light cones intersect, like what happens in general with the wave function).
I admit I'm not an expert in this area, but is this not possible?
Plus, I'd rather expect at least one professional (QED) physicist exists able to explain it and he isn't one. Mermin is, but the explanation is decidedly less clear.
BTW I came here to say Bell's inequality as well. For me it's as baffling as science could ever be.
https://www.physics.wisc.edu/undergrads/courses/spring2016/4...
AFAICS it was published in the American Journal of Physics in 1981 but it's addressed to the general reader. It requires no knowledge of quantum physics.
https://www.youtube.com/watch?v=zcqZHYo7ONs
https://www.youtube.com/watch?v=MzRCDLre1b4
Here's what I would have thought happens: After the first lens, you get polarized light, 90deg offset from the last lens, so no light passes. Then you introduce a 3rd lens in the middle, 45deg offset. This could alter the polarization (maybe it widens the band, or introduce some greater variance, shifts it who knows), and this is why now some light will pass through number 3. No need to create any light
https://www.amazon.com/Quantum-Non-Locality-Relativity-Metap...
If you don't want to read a whole book then I recommend this article:
https://kantin.sabanciuniv.edu/sites/kantin.sabanciuniv.edu/...
but the book will give you a much deeper understanding.
Then we choose some settings and press GO and record whatever number pops up. We do this many times so we each have a nice frequency chart. Now Bell proved that if you live in a local hidden variable universe, the correlations between these numbers is upper bounded, no matter how you choose settings on the boxes. Then, he also gave a prescription for choosing the settings, such that if you live a in quantum universe, the correlations between these numbers will be higher than the upper bound.
The rest is mathematics, which cannot really be simplified without leaving the reader unsatisfied.
I'm going to throw out an analogy that gets at what's observed and why it's surprising, but doesn't relate to the physics of spin, momentum, position or anything that's actually under observation in these experiments.
It's as if we have a pair of dice, and I throw my die and you throw your die many times. In a classical world, if I throw a three, it has no influence on what you throw; you're equally likely to throw 1-6. But in the quantum world it's as if when I throw a one, your die still has the expected uniform distribution, but when I happen to throw a three, you're a little bit more likely to throw a three. Your die is fair if I happen to roll a one, but it's weighted if I happen to throw a three.
Back in the real world, this is the strange behavior that is observed in experiment. Schroedinger's equation predicts the probabilities perfectly. But Bell shows that it's far from intuitive.
Imagine explorers on Mars find the ruins of an ancient alien civilization. In those ruins they find several small devices that have three buttons. Beside each button are two colored lights. red and blue. Above the buttons is a display. The linguistics team figured out enough alien writing to tell that the buttons are labeled with the alien's equivalent of A, B, and C, and that the display is a numerical display that goes from 0 to 38413 displayed in base 14 (which fits with other evidence found that the aliens have two hands with 7 fingers).
There is also some kind of docking station, which can hold two of the devices, and has a single button.
If two of the devices are placed in the docking station and its button is pressed, all the lights briefly flash on the devices, and the counter resets to 0. The lights stay on until the device is removed from the dock. Nothing happens if only one device is placed in the dock.
To try to figure out what these devices do, pairs are placed in the dock, reset, and then given to a couple people who go off and press the device buttons are record what happens.
Here is what those people observe.
1. If they press one of the buttons (A, B, or C), exactly one of the two lights next to that button comes on. When the button is released, the light goes out, and the counter goes up by 1, until it reaches 38413. After the next press/release, the counter goes blank and the device is unresponsive until reset again in the dock.
2. As far as anyone can tell, there is no pattern to which light lights. It acts as if pressing a button consults a perfect true unbiased uniformly distributed random bit generator to decide between red and blue.
3. When they compare their results with those of the person who had the box that was their box's dock mate for reset, they find that if on each person's n'th press
-- if they both pressed A, or both pressed B, or both pressed C, they got the same color light.
-- if one of them pressed B, and the other pressed either A or C, they got the same color light 85.36% of the time.
-- if one of them pressed A and the other pressed C, they got the same color light 50% of the time.
4. These results do not depend on the timing between the two people's presses. Those correlations are the same if the people happen to make their n'th press at the same time, or at wildly different times. Even if one person goes through all their presses before the other even starts, their n'th presses exhibit the above correlations.
5. These results do not depend on the distance between the boxes. If a box pair is split up, with one person taking theirs back to Earth while the other remains on Mars, and the two then run through all their presses at nearly the same time, completing quickly enough that there can be no communication between the two boxes during the run due to speed of light limits, they still exhibit the correlations.
Challenge: try to figure out how such boxes could be built without using quantum entanglement. Assume the aliens have nearly unlimited storage technology, so you can include ridiculously large tables if you want, so you can even propose solutions that involve the dock preloading the responses for every possible sequence of presses (all 3^38414 of them). Anything goes as long as it produces the right correlations, and does not involve quantum entanglement.
To see it, imagine you have a struct with a data member for each local variable of your function, and replace your function with a member function that has no local variables, but uses "this" to get at what was local data.
Add one more data member, a number that is set differently right before each place the function returns.
Finally, insert some code at the start of the function that, according to the number, jumps to just after the last return statement executed.
Then, each time you call the function, what happens depends on what happened last time.
There are more details, but that is the gist.
You can write that yourself in C++98, with the body of the function inside a switch statement. Getting it past code review would be the real challenge.
Ultimately, though, you are right that you have to understand it all, once, even if you can't remember it all a month later. The explanations I find online are not good at presenting just the details you need when you need them, and building up to the full picture.
The measurements were finally shown to be effects of the immediate environment on the measurement apparatus.
That detectors used in labs may vary with time by >0.1%, unknown to their users, seems pretty important. How did everybody involved not know?
To make things more specific, those labs had uncertainty budget with something like 20 terms for the things they measured. Each of those terms had associated probability distribution etc. They had uncertainty budgets for all the methods they did etc., and some of those where probably dated, done by someone else, etc. etc. Who checks that? Is the check rigorous enough? Are some assumptions made that don't hold to scrutiny?
So it is actually very easy for error to creep in, I would say actually very likely.
I have trouble with this too. I think it's actually incorrect, or at least misleading. I think what it's _trying_ to say is that even if an entity can perform a complex task doesn't mean it can understand a complex task.
I think the more important result of this argument is that certain complex tasks can be "pre-baked" into rulesets _by an existing intelligence_. To me this just means that intelligent entities can sort of copy parts of their intelligence into other entities which are not intelligent i.e. computer programming.
I think with this argument they're trying to say "a series of sufficiently complex if statements isn't necessarily intelligent" by choosing something we know computers are good at - string manipulations and applying it to something we consider intelligence - language translation.
The argument holds that the computer is obviously not intelligent because it's just a function that takes a character and outputs another character.
But it needs to be a convincing translation, right? The computer would then be able to spit out not just accurate translations but also properly converted cultural idioms and new combinations of words where one didn't exist in the other language. That requires context of surrounding characters, memory of common language use, statistical analysis and creativity.
One implication that arises from this argument is actually about humans. How do we know that we aren't all just incredibly detailed rulesets ourselves without any actual understanding?
Well, first off - we technically can't prove it for anyone other than ourselves. More pragmatically, it's obvious that we, unlike the computer translator, can probe ourselves and be probed by others on whether or not we understand the subject. It's not like we're a bunch of Boltzmann's Brains that just happened into existence. We evolved intelligence in order to survive, not to "trick" other intelligent beings into thinking we're more intelligent than we are. There's no need for that. There's no one smarter around that we need to "trick".
Here is the chapter on Fourier transforms from my linear algebra book that goes into more details: https://minireference.com/static/excerpts/fourier_transforma...
As for the math, there really is no other way to convince yourself that sin(x) and sin(2x) are orthogonal with respect to the product int(f,g,[0,2pi]) other than to try it out https://live.sympy.org/?evaluate=integrate(%20sin(x)*sin(2*x... Try also with sin(3x) etc. and cos(n*x) etc.
I disagree with that. It's pretty easy to prove it in general by calculating \int_0^{2\pi} sin(mx)sin(nx) dx etc. for m ≠ n.
The "no other way..." was referring to me not having an intuitive explanation to offer about why an sin(x) and sin(2x) are orthogonal.
For some intuition, consider music, especially on a violin. Fourier series applies to a periodic function (wave), and represents the whole wave as sine waves that fit the one period exactly. So, get sine waves at frequency 1, 2, ... that of the period. In music, these waves are called overtones.
Playing with a violin, the overtones are fully real and even important! E.g., get a tuning fork and tune the A string (second from the right as the violinist sees them) to 440 cycles per second (440 Hertz, 440 Hz). Then the D string, the next to the left, is supposed to have frequency 2/3rds that of the A string. So, bow the two strings together and listen for the pitch 880 Hz, that is, 3 times the desired frequency of the D string and twice that of the A string. So are listening to the second overtone of the D string and the first overtone of the A string; are hearing the third Fourier series term of the D string and the second Fourier series term of the A string. Adjust the tuning peg of the D string until don't hear beats. If the D string is at, say, 881 Hz, then will get 1 beat a second -- so this is an accurate method of tuning. Similarly for tuning the E string from the A string and the G string from the D string -- on a violin, the frequencies of adjacent strings are in the ratio of 3:2, that is, a perfect fifth. That's how violinists tune their violin -- which is needed often since violins are just wood and glue and less stable than, say, the cast iron frame of a piano.
For one more, hold a finger lightly against a string at 1/2 the length of the string and hear a note one octave, twice the frequency, higher. That's often done in the music, e.g., playing harmonics. And it's a good way to get the left hand where it belongs at the start of the famous Bach Preludio in E-major that starts on the E half way up the E string. Lightly touch one third of the way up the string and get three times the fundamental frequency, sometimes done in music to give a special tone color. Net, Fourier series, harmonics, and overtones are real everyday for violinists.
E.g., on a piano, hold down a key and then play and release the key one octave lower and notice that the strings of the key held down still vibrate. The key vibrating was stimulated by the first overtone of the key struck and released.
The Fourier integral applies to functions on the whole real line. Very careful math is in Rudin, Real and Complex Analysis.
Yes, Fourier series and integrals can be looked at as all about perpendicular projections of rank 1 as emphasized in Halmos, Finite Dimensional Vector Spaces, written in 1942 when Halmos was an assistant to John von Neumann at the Institute for Advanced Study. That Halmos book is a finite dimensional (linear algebra) introduction to Hilbert space apparently at least partly due to von Neumann. So, right, Fourier theory can be done in Hilbert space.
Fourier integrals and series are very close both intuitively and mathematically, one often an approximation to the other. E.g., if multiply in one (time, frequency) domain, then convolve in the other (frequency, time) domain. E.g., take a function on the whole real line, call it a box, that is 0 everywhere but 1 on, say, [-1,1]. Well the Fourier transform of the box is a wave, roughly a bell curve, that goes to zero quickly away from 0. A convolution is just a moving weighted average, usually a smoothing. Then given a function on the whole real line, regard that line as the time domain and multiply by the box. Now can regard the r...
A function is like a vector, but instead of having two or three dimensions you have a continuous number of them. Adding functions component-wise works just like adding vectors.
Just like regular vectors, you can choose to represent functions in a different basis. So you choose a family of other functions (call it a basis) that's big enough to represent any other you want. For a lot of reasons [1, 2], a very good choice is the set of complex exponentials g_w(x) = exp(2πiwx), for every real w. It's an infinite family, but that's what you need to deal with the diversity of functions that exist.
So you try to find the linear combination of exponentials that sum to your original function. You need a coefficient for each w, so call it c(w) for simplicity. After fixing the basis, the coefficients really have all the information to describe your function. They're an important object, and we call c(w) the Fourier transform.
How do you find the coefficients? Just project your original function onto a particular exp(2πiwx), that is, take the inner product. Usually the inner product is the sum of the products of coefficients. Since functions are continuously-valued, you use an integral instead of a sum. This is your formula for the Fourier transform.
I known there are technical conditions I am glossing over, but this is the intuition of it for me.
[1] There is an intuition for these exponentials. Complex exponentials are periodic functions, so you are decomposing a function in its constituent frequencies. You could also separate the exponential into a sin and cos, and will obtain other common formulas for the Fourier transform.
[2] Exponentials are like "eigenvectors" to the derivative operation (taking the derivative is just multiplying by a constant), so they're really useful in differential equations as well.
Weights on neural networks don't have to be independent functions.
Independence gives you a set of mathematical guarantees that insure you fully cover the space you're representing. For example that given a 2 dimensional space, X and Y are pointing in different directions. If they pointed in the same direction you could not fully decompose all vectors on the plane into two coefficients of X and Y.
If this helps, then it can also help with understanding other projections such as the Laplace transform (a dot project against the complex signal space).
While this analogy has helped me, I still have no clue why real valued signals result in an even FT.
edit: grammar
1. The magical orthogonal basis functions: complex sinusoids. Shifting of a time signal just multiplies the Fourier counterpart by a new phase (relative to its represented frequency). Thus transforming to the Fourier basis enables an alternate method of implementing a lot of linear operations (like convolution, i.e. filtering).
2. The magic of the fast implementation of the Discrete Fourier Transform (DFT) as the Fast Fourier Transform (FFT) makes the above alternate method faster. It can be most easily understood by a programmer as a clever reuse of intermediate results from inner loops. The FFT is O(N log N), a direct DFT transform would be O(N^2)
A mathy demonstration of this at https://sourceforge.net/projects/kissfft/
If you see any signal, it can be represented as a value at each time, x(0) = 1, x(1) = 2 .. x(100) = 5 etc. We can visualize them as you shouting 1 at time 0, 2 at time 1 and 5 at time 100. Alternatively we can also do the same with a larger number of persons.
Representation using dirac delta
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Lets say that you have 100 persons at your disposal. You ask first person to shout 1 at time 0, second person to shout 2 at time 1 and person to shout 5 at time 100. Other times they will be silent. So with these 100 people you can represent the signal X. We call each of these person as bases. Mathematically they are delta functions of time, ie they get activated only at their specified time. Other times they are silent, ie 0. The advantage of this representation is that you have fine control on the signal. If you want to modify value at time=5, you can just inform the 5th guy.
Introduction to bases
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Dirac delta is not the only bases. You can ask multiple guys to shout at multiple times. They can even tell negative numbers. All you have to ensure is that they add up to the value of X. The guys should be able tell any number that can come as a part of X. This we name the property "SPAN".
Instead of 100 guys, we can have 200 guys too, ie 2 guys for each time and they tell half of the original value. However, this is wasteful since you have to pay for extra guys with no use. Hence we say that the bases should be orthogonal, ie they should not have correlation with others in the group. So as we have uncorrelated and spanning guys, we can represent any signal using them.
Fourier transform
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In case of Fourier transform, each guy will shout according to a sinusoidal wave. Lets say sine wave. ie guy 1 at time 0 will tell the value of sine(f0 t). Second guy will shout value of sine(f1t) and so on. The f0, f1 etc are the frequencies for each guy. Now it comes out that these guys will be orthogonal to each other, and they can span all the signals. Thus we have Fourier transform. Hence instead of representing signal as value at each step, we can represent it as value at each frequency.
Why Fourier transform
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We have seen that as long as bases span and and are orthogonal, they can define a transformation. But why is Fourier transform so famous. This comes from the systems we use. The most common systems we use are LTI(Linear time invariant) systems. A property of the said system is that they work on sinusoidal waves. Ie if a sinusoidal wave of frequency f is passed through an LTI system, all it can do is to multiply with a scalar. Any other wave will have a more complex effect. Hence if we can represent signals as a sum of sinusoids, we can represent our system as just a amplifier at each frequency. This makes whole of system analysis into a set of linear equations which we are good at solving. So we love Fourier transform
The cool thing about this insight is that the converse is true. You can disaggregate any waveform into its additive harmonics. This means you can jam multiple signals into a single channel (eg a fibre optic cable) and then apply a fourier transform at the other end to "untangle" them.
To wit, the idea is that you cannot distinguish whether you are in an accelerated frame or in a gravitational field; alternatively stated, if you’re floating around in an elevator you don’t know whether you’re freefalling to your doom or in deep sideral space far from any gravitational source (though of course, since you’re in an elevator car and apparently freefalling... I think we’d all agree on what’s most likely, but I digress).
Anyway, what irks me that this is most definitely not true at the “thought experiment” level of theoretical thinking: if you had two baseballs with you in that freefalling lift, you could suspend them in front of you. If you were in deep space, they’d stay equidistant; if you were freefalling down a shaft, you’d see them move closer because of tidal effects dictated by the fact that they’re each falling towards the earth’s centre of gravity, and therefore at (very slightly) different angles.
Of course, they’d be moving slightly toward each other in both cases (because they attract gravitationally) but the tidal effect presents is additional and present in only one scenario, allowing one to (theoretically) distinguish, apparently violating the bedrock Equivalence Principle.
I never see this point raised anywhere and I find it quite distressing, because I’m sure there’s a very simple explanation and that General Relativity is sound under such trivial constructions, but I haven’t been able to find a decent explanation.
The first part of the argument is that for single point particles falling, the effect of gravity is the same for all particles. This suggests that we should model gravity as something intrinsic to spacetime itself, rather than as a field living on top of spacetime, which could couple to different particles with different strengths.
The second part of the argument, which is what you point out, is that gravity can have nontrivial tidal effects. (This had better be true, because if all gravitational effects were just equivalent to a trivial uniform acceleration, then it would be so boring that we wouldn't need a theory of gravity at all!) This suggests that whatever property of spacetime we use to model gravity, it should reduce in the Newtonian limit to something that looks like a tidal effect, i.e. a gradient of the Newtonian gravitational field. That leads directly to the idea of describing gravity as the curvature of spacetime.
So both parts of the argument give important information (both historically and pedagogically). Both parts are typically presented in good courses, but only the first half makes it to the popular explanations, probably out of simplification.
Can you please explain to me how you went from"looks like a tidal effect in the Newtonian limit" to "a gradient of the Newtonian Graviational field"?
Tidal forces occur much more due to the difference in the direction of gravity than due to the difference in magnitude.
This point isn't raised anywhere because it's mostly a pedantic point that has nothing to do with the thought experiment. You shouldn't try and decompose thought experiments literally, otherwise you'll get caught up in unimportant details like this. Just assume the elevator is close enough to the earth such that the field lines are effectively parallel, or better yet, just pretend the elevator is in an infinite plate field.
The real principle of relativity is a bit more subtle (sometimes called the strong principle): that the effects of gravity can be explained entirely at the level of local geometry, without any need for non-local interaction from the distant body that is generating the gravitational field. To describe the geometry of non-uniform fields, we need more sophisticated mathematical machinery than what is implied by the elevator car thought experiment, but nonetheless, the elevator example is a useful launching point for that type of inquiry.
Clearly it will fail given a big enough lift to experiment in, since a big enough lift would essentially include whatever object is creating that gravitational pull (or enough to conclude its existence from other phenomena). However these effects are nonlocal, you need two different points of reference for them to work (like your two baseballs). In fact most Tidal forces are almost by definition nonlocal.
The precise definition involves describing curved spacetime and geodesics, but that one is really hard to visualize as a thought experiment. The thought experiment does offer insight though, as it is possible to imagine that, absent significant local variations in gravity, you cannot distinguish between free-fall and a (classical) inertial frame of reference without gravity. This insight provides the missing link that allows you to combine gravity with the laws of special relativity and therefore electromechanics, including the way light bends around heavy objects, which provided one of the first confirmations of this theory.
If you think it's sneaky to "implicitly" assume they're in the same direction, I would point out that this is no different from assuming they have the same magnitude. It would be kinda dumb to say "well this 1m/s^2 acceleration can't possibly be equivalent to gravity because gravity is 9.8m/s^2, so the statement is obviously wrong and they're trying to trick me!!"... same thing for direction.
I had to apologize and say that the explanation was over simplified and really it would work, say, only for some creatures living exactly on the floor of the elevator.
One of the two, at a challenging high school, made Valedictorian (surprise to her parents who didn't know she had long been first in her class) then in college PBK, got her law degree at Harvard, started at Cravath-Swain, went for an MD, and now is practicing medicine. Bright niece.
Turn "clock + baseball" into a light of increasing brightness.
That's what nature sees.
It's easier to speed up your clock so you get an extra second, move, and slow down your clock than just move.
Einstein did not differ between a mass's gravity and "wanting to move in a straight line unless acted on by another force."
Before Einstein, Planck proved an object needs energy before glowing the next color, from red to orange for instance. Einstein came along and said you have a start, you have an end, you have a particle.
The photoelectric effect requires photons of a minimal wavelength to produce a voltage in material. The photon sees the material all at once. No matter how many photons of a lesser energy you fire, no electricity until you hit that wavelength.
Not differing between gravity and inertia allows Einstein to talk in terms of photons instead of clocks changing.
Today our questions involve reality and locality. Do we transfer heat into everything at once? Or does heat go one way, and we can't trace it without smudging the trail.
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So my view of gravity is "something already exists". We got to account for it. If we want to generate something from nothing, we have to convince everything out there.
What stumps me:
Can we convince everything out there something exists? If so doesn't it just pop into existence if everything agrees something should exist there? What happens when we do? Does whatever pop into existence give us an accurate picture of everything out there?
If so, does everything that's out there just part of the material here? Galaxies precipitate in existence if we leave the Hubble on too long. They are only real when humans from Earth touch down on them.
The force that would be exerted from acceleration versus gravity is different. The force you we think of as gravity comes from a center point that changes with your position while acceleration comes from a uniform direction without regard to your position.
I don't know if this was it, but an explanation nonetheless https://medium.com/@omaraflak/automatic-differentiation-4d26...