KJW

If you constructed a structure out of bricks, how would it become a BH without matter-crushing forces? The structure would always have a radius larger than the Schwarzschild radius. No, the Schwarzschild radius of a black hole is proportional to the mass, and the mass of a ball of uniform density is proportional to the cube of the radius. Therefore: [math]r_s = \dfrac{2 G M}{c^2}[/math] [math]M = \dfrac{4 \pi \rho r^3}{3}[/math] [math]r_s = \dfrac{8 \pi G \rho r^3}{3 c^2}[/math] [math]r_s \geqslant r \implies \dfrac{8 \pi G \rho r^3}{3 c^2} \geqslant r[/math] [math]\implies r \geqslant \sqrt{\dfrac{3 c^2}{8 \pi G \rho}}[/math] That is, even for a material of low density, if the ball of material has a large enough radius, the Schwarzschild radius of the ball will exceed that radius. Whether or not the ball of bricks can support its own weight under its own gravity is irrelevant to the point I'm making. Even if the ball of bricks can support its own weight, the mere size of the ball if large enough is sufficient to produce a black hole. That is, it is not necessary for the ball of bricks to collapse under its own weight to produce a black hole.

There are two words that seem to me to be connected to the question you are asking: Ontology: The philosophical study of being. Epistemology: The philosophical study of knowledge.

As the size of the black hole increases, the total energy required to form the black hole increases, but because the energy density decreases, the mechanism that prevents the formation of the black holes (Schwinger Effect) is no longer significant. Therefore, with sufficient energy, even if that is the product of the total annihilation of a pair of matter and antimatter stars, black hole formation would not be prevented, at least not by the Schwinger Effect. I had not come across this in my reading. Actually, the event horizon expanding from the centre seems rather intuitive anyway. I didn't really give much thought to it until it occurred to me that before the event horizon has fully expanded to the surface, there is not enough mass inside the event horizon to form a black hole, and therefore the event horizon forms at the surface when all the matter crosses the Schwarzschild radius. However, this reasoning was based on an apparently incorrect understanding of the event horizon. Intriguingly, a corollary of an expanding event horizon is that if one encloses a black hole inside a spherical shell of mass, the black hole will expand. Yes, and a consequence of this is the shell theorem.

It's not clear to me that you have truly resolved the matter-antimatter asymmetry. You may have shown a cosmological balance with regards to isospin, but it is not clear that this is the same as a matter-antimatter balance. For example, there is still a cosmological imbalance between protons and antiprotons. You can't get around this imbalance by claiming a balance between protons and electrons because there are still antiprotons and positrons to consider. There appears to me to be a disregard of the fact that each type of particle has its own particular antiparticle, and that the balance between matter and antimatter is not simply a matter grouping particles into matter and antimatter categories. I've often seen it said that the asymmetric decay of neutral kaons and neutral antikaons might resolve the matter-antimatter asymmetry. But this doesn't seem possible to me because neutral kaons and neutral antikaons are both mesons that consist of a quark and an antiquark, and therefore are neither matter nor antimatter. This means that neither the neutral kaons nor the neutral antikaons are going to decay into an imbalance of matter or antimatter, even if the decay of neutral kaons and neutral antikaons is not symmetric.

What do you mean what about it? How does quantum mechanics fit in with your belief in determinism?

I guess you haven't watched the video. QED keeps the concentration of EM radiation from ever reaching an energy density allowing creation of a BH, due to the Schwinger Effect ( pair creation in a vacuum with sufficiently strong EM field ). It has nothing to do with conditions after BH creation. I did watch the video. It focused on small black holes for which the energy density is so high that the Schwinger Effect would prevent their formation. But to form a very large black hole, the energy density would be low enough for the Schwinger Effect to be negligible. The point I made about constructing a black hole from bricks is that one doesn't need matter-crushing forces to produce a black hole if the total mass is large enough. I will admit that I am somewhat of a black hole skeptic. Nevertheless, I recognise that simply accumulating sufficient mass into its Schwarzschild radius will create a black hole without the need of any forces that could possibly be thwarted. The invokation of the Schwinger Effect seemed to me to be not entirely unlike the "immovable object meets irresistible force" type argument that one might use against the formation of a black hole. However, I should remark that I fairly recently derived a result that contradicted a belief I had about black hole formation. I had believed that during the gravitational collapse of matter, the event horizon doesn't form from the centre and expand outwards, but rather forms at the surface of the matter at the instant the entire matter is inside the Schwarzschild radius. My error was due to a misapplication of Birkhoff's theorem and a failure to fully appreciate that the event horizon is due to the gravitational potential. The result I derived is that the event horizon starts to form at the centre of the matter before the surface of the matter is inside the Schwarzschild radius, and expands as the matter shrinks until it reaches the surface when the entire matter is inside the Schwarzschild radius.

I've said it before, but according to a quote I've seen, a neural network has the second-best solution to any problem.

What about quantum mechanics?

In the case of a sufficiently large black hole, the energy density is low enough for pair production to be insignificant. Whereas for an ordinary object, mass increases with R³, for a black hole, mass increases with R. Thus, very large black holes have low density and one could construct such a black hole with bricks if one had a sufficient number of them. This kills any "immovable object meets irresistible force" type arguments with regards to black holes.

It's worth noting that the Y-chromosome is a very small chromosome containing very little genetic information. It is my understanding that the Y-chromosome contains the trigger gene for producing male characteristics, but not the genes for actually producing male characteristics. I recall an episode of "House" in which a rather beautiful young woman had a Y-chromosome that contained a defective trigger gene, and therefore failed to develop any male characteristics, being physically female by default.

Are you sure ? @studiot is incorrect. Lightlike vectors are not orthogonal to timelike vectors. As null vectors, lightlike vectors are orthogonal to themselves. A vector that is orthogonal to a timelike vector is spacelike.

missed that one; simple: Eve impersonates Caesar during key distribution and gives Alice Eve's public key while claiming it belongs to Caesar. Alice then encrypts using Eve's key, for which Eve has the corresponding private key. How does Alice authenticate that the public key actually belongs to Caesar? In established public key cryptographic protocols this problem is well-known and solutions exist. @Trurl, it should be noted that this is a separate problem to the one public key cryptography intended to solve. Public key cryptography intended to solve the problem of an eavesdropper listening in on a private communication between parties, not a man-in-the-middle attacker who intercepts and possibly alters messages between parties who believe they are communicating with each other instead of communicating with the attacker. From reading https://en.wikipedia.org/wiki/Man-in-the-middle_attack, it appears to me that while there are approaches to solving the man-in-the-middle attacker problem, there does not appear to be solutions without their own vulnerabilities to attack. Also, solutions appear to be based on technology and protocols rather than mathematical theory, as is the case with public key cryptography.

Sorry, but I'm not understanding how this works. The purpose of Alice's public key is for anyone who wishes to send an encrypted message to Alice to use that key to encrypt their message before sending it to Alice. As a public key, it is not a secret and is available to everyone. But because it is Alice's public key that encrypted the message, only Alice's private key can decrypt the message. And as a private key, it is a secret known only to Alice. That is, Alice's private key is not known to the person who sent her the encrypted message, nor to anyone who intercepts this message. For Alice to reply with an encrypted message, she would have to use the public key of the person who sent her the message to encrypt her reply. This public key is available to her because it is a public key that is not a secret. The masterful brilliance of public key encryption is the existence of public keys that are not secret, and of course the underlying mathematics that support this existence. That is, it is the existence of keys that are not secret and can encrypt messages that allow secret communication to take place across public channels. Without effective public keys, secret communication would have to be across private channels vulnerable to interception. Note that it is the requirement of the sender to have the receiver's key that makes this key public.

Have you read the Wikipedia article about phytic acid? If so, what have you read that is contrary to what the Wikipedia article said? I don't have any independent knowledge of the topic, but what Wikipedia says rings "true" to me.

Public key encryption is a one-to-one function. It must be. Otherwise, it wouldn't be possible to decrypt the encrypted message. For example, the RSA (Rivest–Shamir–Adleman) cryptosystem works because prime factorisation is unique (up to order) but extremely difficult (for very large numbers). Starting from the private key it is easy to obtain the public key, but obtaining the private key from the public key is extremely difficult even though it is in principle possible to do so (by brute force). It seems to me that you don't have a proper handle on how public key encryption works in general. Consider a two-way conversation between Alice and Bob. Alice uses Bob's public key to encrypt all of her messages to him. Bob uses his private key to decrypt all of Alice's encrypted messages. Similarly, Bob uses Alice's public key to encrypt all of his messages to her. And Alice uses her private key to decrypt all of Bob's encrypted messages. So, both Alice and Bob must each have their own public key and private key pair. Alice must somehow have access to Bob's public key, and Bob must somehow have access to Alice's public key, but neither Alice nor Bob share their private keys. Also, there is a function that inputs the plaintext message and the public key and outputs the corresponding encrypted message, as well as a function that inputs the encrypted message and the private key and outputs the corresponding decrypted message. These functions are shared between Alice and Bob so that in principle Alice and Bob do not need to use the same cryptosystem (Alice and Bob can each send their own encryption function along with their public key to the other person). The encryption and corresponding decryption function do not need to be the same function. However, one can curry the encryption function with the public key to produce another function that simply inputs the plaintext message, one can curry the decryption function with the private key to produce another function that simply inputs the encrypted message, and that these two single-input curried functions are the inverse of each other. So, for a given cryptosystem, the curried encryption and decryption functions are each members of a one-parameter family of single-input functions. Each curried decryption function is the inverse of a corresponding curried encryption function. The parameter of the curried encryption functions is the public key, and the parameter of the corresponding curried decryption functions is the corresponding private key. Furthermore, the public key and corresponding private key must be related to each other such that the public key can be easily obtained from the private key, but the private key cannot be easily obtained from the public key, requiring that the private key be created first and the public key be obtained from that. From the above, I think it should be clear that creating a cryptosystem is far from straightforward. Nevertheless, there are cryptosystems other than RSA, and these may have the advantage over RSA by not being vulnerable to cracking by quantum computers.

Though clicking on the above yields Sorry! That page does not exist. Error code: 2S136/1Z Could this be because of the changeover from the old forum to the current forum?

Thank you. When I first saw it, I thought S referred to entropy, but now I'm almost certain it refers to action.

The OP asked for the simplest equation. Strictly speaking, an equation is an expression to be solved rather than a formula, identity, or definition. Thus, the simplest equation is: ax + b = 0 to be solved for x.

It should be noted with regards to ciguatera that it's not the fish itself that is poisonous, but rather the toxins that the fish have ingested. Thus, it is not a matter of identifying fish species, although some species are more likely than others to contain ciguatoxins. According to Wikipedia: Maitotoxin (MTX) is an extremely potent toxin produced by Gambierdiscus toxicus, a dinoflagellate species. Maitotoxin is so potent that it has been demonstrated that an intraperitoneal injection of 130 ng/kg was lethal in mice. Maitotoxin was named from the ciguateric fish Ctenochaetus striatus—called "maito" in Tahiti—from which maitotoxin was isolated for the first time. It was later shown that maitotoxin is actually produced by the dinoflagellate Gambierdiscus toxicus. The chemical structure of maitotoxin is believed to be:

Tina Turner was in "Mad Max Beyond Thunderdome".

Not all Amanita mushrooms are hallucinogenic. Amanita phalloides (death cap) contain toxins that inhibit RNA polymerase II, and is the most poisonous of all known mushrooms. It is estimated that as little as half a mushroom contains enough toxin to kill an adult human. Based on the chemical structure of some of these toxins, I suspect that the mushrooms would not taste bitter (unlike those containing alkaloids), so a taste test would not be advisable if death cap or similar mushrooms are a possibility.

Clever. I’m sure the Clay Mathematics Institute would like to know. 😂 My understanding is that this may have been tried but not yet arrived upon a successful solution (and the word Yet is doing a lot of heavy lifting there) And if after spending US$1 million worth of tokens, the problem has not been solved, should one continue? It seems to me that this sort of thing could be something new on which to gamble.

LOL

Has anyone given it The Riemann Hypothesis to solve? It would be interesting to see if it can be solved for less than US$1 million worth of tokens.

I don't know about "one has to account for the self-interaction of gravity", but general relativity has a somewhat peculiar form of energy conservation that is the zero covariant divergence of the energy-momentum tensor field, a form that is true in all coordinate systems of an arbitrarily curved spacetime. However, due to the restriction of covariance and the nature of the energy-momentum tensor field, unlike a conserved vector field, conservation of the energy-momentum tensor field as it is normally understood only works locally. That is, there is no integral form of the conservation of the energy-momentum tensor field that would provide a global notion of the conservation of energy-momentum. By contrast, for a conserved vector field, the generalised Stokes theorem provides the integral form from the differential form that allows the global notion of conservation.