KJW

No, they are not.

You need to post something here that can be discussed without visiting an external website.

That's "un-American"! And I say "un-American" because I can't think of any other country that thinks like that. Yes, capitalism exists throughout the West, and conservative governments do push for privatisation, but this idea that any hint of socialism is thoroughly evil seems to me to be a uniquely American phenomenon.

I'll concede that point. On the other hand, you did say that "a clue should be the use of 'classical' in the name", indicating that it is obvious that the classical electron radius is not the actual size of the electron, and that it was not necessary to explicitly state this. However, I do think that the classical electron radius is in some sense an effective radius or size scale of a dressed electron. Where would one expect the size of an electron to appear in physics, anyway? It isn't as though the Bohr radius is used in physics that isn't about the size of atoms. The classical electron radius appears in non-relativistic Thomson scattering and the relativistic Klein–Nishina formula.

This may help: https://en.wikipedia.org/wiki/Periodic_table

I'm not an American, so I can't answer this question. However, I did have a rather strong interest in this US election, more than any previous US election (and even more than the recent state election at home). Based on the information I was receiving, it seemed to me that Trump was going to win. Although the political commentators were saying the polls were within the margin of error, the polls were nevertheless pointing to a Trump victory. And many of the "ordinary folk" that I saw interviewed were saying that they were voting for Trump. But I'm no expert and certainly wouldn't have locked in a prediction. Watching the election results come in, it did seem like a foregone conclusion from the start.

The flaw with this is that elections are not random and therefore applying probability is not valid. Some elections are easy to predict the outcome, others not so easy. If some methodology correctly predicts easy elections but gets the hard ones wrong, then that methodology is really quite useless, regardless of how many easy elections are correctly predicted.

The way I see it, different people have different ideas about how their country should be run, and therefore it is not possible to please everyone, regardless of the political system. That's not to say that all political systems are equal in terms of benefiting the population. But to think that one can solve the problems inherent in political systems simply by providing more democracy is misguided due to the fundamental differences within the population. There is such a thing as "tyranny of the majority".

What you said (to which I replied) was that size doesn't apply to fundamental things like an electron, whereas the classical electron radius is a size that does apply to the electron. That it was derived using classical electrodynamics is beside the point. It is not being suggested that the classical electron radius is the actual radius of the electron, but in some sense, it is an "effective" radius of the electron and a physical constant that appears in theories where the size of the electron is relevant. Another size associated with the electron is the Bohr radius. Even though this was derived using an obsolete model, it remains as a physical constant that appears in modern theories regarding the size of an atom. See above. I didn't actually say that the classical electron radius is the actual size of the electron. I referred to a Wikipedia article that describes the classical electron radius as "a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation".

This is reminiscent of the crystallisation of a racemic mixture, which can either crystalise as the racemate or as a mixture of enantiomeric crystals. This is especially relevant to this thread because the first resolution of a racemic mixture was of sodium ammonium tartrate by Louis Pasteur who manually separated the individual enantiomeric crystals into separate piles. It is said that he was quite fortunate to have found a racemic mixture that crystallises as a mixture of enantiomeric crystals because most racemic mixtures crystallise as the racemate.

https://en.wikipedia.org/wiki/Classical_electron_radius

Seems like a statement that there are conserved quantities in physics. Hardly groundbreaking.

I prefer to think of the critical point as where the liquid and gas become indistinguishable (the meniscus between them disappears). Interestingly, critical points can only occur between phases that have the same symmetry (e.g. liquid and gas).

Yes, that's the point I'm making. But the vapour pressure of a liquid at a given temperature does not depend on the local atmospheric pressure.

I should add that it is the higher kinetic energy water molecules in the liquid phase that escape to the gas phase. But once the higher kinetic energy water molecules in the liquid phase have overcome the potential energy barrier to escape the surface of the liquid, their kinetic energy is reduced to that of the average in the liquid. However, it is not clear to me that the kinetic energy of the water molecules in the liquid phase that escape to the gas phase corresponds to 100°C. 100°C is the boiling point of water at normal atmospheric pressure, whereas the phenomenon described above is independent of the atmospheric pressure.

No, the gas is still at 20°C. For example, if one placed a flask containing water at 20°C under a high vacuum, sealing the flask after some of the liquid water had boiled away and all of the air had been removed, then after the temperature of the contents of the flask has been allowed to return to 20°C, both the liquid water in the flask, and the gaseous water above it will be at 20°C. The gaseous water will exert a pressure of 2.3388 kPa (0.0231 atm)¹. And since all the air in the flask has been removed, the entire contents of the flask is water (liquid and gas), and the total pressure inside the flask will be 2.3388 kPa (0.0231 atm). Furthermore, if the flask had not been evacuated, then the partial pressure of the gaseous water molecules (the pressure the water molecules themselves exert within the water-air mixture) would also be 2.3388 kPa (0.0231 atm). That is, the presence of air in the flask does not affect the pressure of the gaseous water molecules in equilibrium with the liquid water at a given temperature. ¹ Source: https://en.wikipedia.org/wiki/Vapour_pressure_of_water

Perhaps I misinterpreted your original question concerning the field (0, 1). Anyway, I was focusing on the notion of Bring radicals because I thought they were interesting in terms of solving general quintic (and perhaps higher) equations.

[math]x^5 + x + 1[/math] In general, the root of [math]x^5 + x + a[/math] is called a Bring radical of a, denoted BR(a). Bring radicals extend the notion of radicals with regards to the Abel-Ruffini theorem that states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. By extending the notion of radicals, quintic equations that have no solution in ordinary radicals have solutions in Bring radicals. Note that asymptotically, [math]\text{BR}(a) \sim -a^{1/5}[/math] for large a.

[math]\big(\sqrt{7} + \sqrt{5}\big)\big(\sqrt{7} - \sqrt{5}\big) = 2 \\ \big(\sqrt{7} - \sqrt{5}\big) = \dfrac{2}{\big(\sqrt{7} + \sqrt{5}\big)} \\ \sqrt{5} = \dfrac{1}{2}\Big(\big(\sqrt{7} + \sqrt{5}\big) - \big(\sqrt{7} - \sqrt{5}\big)\Big) \\ = \dfrac{1}{2}\Big(\big(\sqrt{7} + \sqrt{5}\big) - \dfrac{2}{\big(\sqrt{7} + \sqrt{5}\big)}\Big) \\ = \dfrac{\big(\sqrt{7} + \sqrt{5}\big)}{2} - \dfrac{1}{\big(\sqrt{7} + \sqrt{5}\big)}[/math]

I thought the section https://en.wikipedia.org/wiki/Resource_curse#Examples_in_biology_and_ecology to be interesting. Whereas one might think that the resource curse is an absurdity of human nature (or of the capitalist system), the appearance of something similar in natural systems seems to indicate that the cause is more fundamental.

I don't think so. By the time Trump and his ship of fools are done with America, the current strong growing economy will be a distant memory.

At the time of my initial reply, I had not proved that the map from G to {1, –1} is a homomorphism. And instead of focusing on that proof, I digressed to a discussion about the alternating group. It was later that I devised a proof of the homomorphism based on Cayley tables. However, I subsequently discovered a direct proof that any subgroup of order half that of the group is normal in that group. The proof is that the left and right cosets of the subgroup must be equal, a defining property of normal subgroups.

This is drawing a longbow. Choosing examples where cognition is important to survival isn't sufficient to show that cognitive abilities are essential in natural selection. What about examples where cognition plays no role in survival? For example, what cognitive abilities do trees have? And yet trees are also evolutionarily successful. The various ways that organisms can become evolutionarily successful is unlimited. For example, animals such as cattle, sheep, pigs, and chicken have become evolutionarily very successful simply by being good food for humans. I doubt that is a choice those animals would have made cognitively.

Hmmm. I did not realise THAT was what I had proven. From what I said above, proving that a map from G to {1, –1} is a homomorphism completes the proof that a simple group cannot have subgroups of order half that of the group.

This actually seems like a bad idea to me.