KJW

For me, 216 = 63 was the hard part of the problem without a calculator. Everything else is just applying logarithm identities.

Allow me to give the hint: 216 = 63 The hint for me was that this problem is going to have a simple solution and is therefore constrained so as to have a simple solution.

While this diagram is up, I'd like to mention something interesting about zero-point energy. If one considers a chemical bond to hydrogen and the corresponding chemical bond to deuterium, then because the mass of a deuterium atom is greater than the mass of a hydrogen atom, the zero-point energy of the deuterium atom is lower than the zero-point energy of the hydrogen atom. That is, in the diagram above, E0 for deuterium is lower than E0 for hydrogen. But the classical energy curve for the different hydrogen isotopes is the same, so that the top of the curve which corresponds to bond breakage will be same for both isotopes. However, because the zero-point energy for deuterium is lower than the zero-point energy for hydrogen, it takes more energy to break a bond to deuterium than to break a bond to hydrogen. Thus, chemical reactions for which breaking the bond to hydrogen or deuterium is the rate-determining step will be slower for deuterium than for hydrogen. This makes "heavy water" (deuterium oxide) somewhat toxic to most lifeforms.

The only such connection I see is, if the ASA for triangles depends on the parallel axiom. I don't know if it does. By "subtle", I was referring to a connection such as the use of the parallel axiom to formulate the case that leads to a contradiction. Such a connection may still be a logical implication of the parallel axiom even if the logic is less direct.

I was unable to find the direct connection between the parallel axiom and the case where α + β is equal to two right angles. However, I was willing to accept that the connection is more subtle (rather than non-existent). The assumption in bold is the assumption that the parallel axiom is valid for a+b equal to two right angles.

Without the validity of the parallel axiom for α + β less than two right angles, you could not make the assumption that the parallel axiom is valid for α + β equal to two right angles, and therefore would be unable to obtain the contradiction for α + β equal to two right angles.

Yeah, I understand the point you're making, which is why I said your question was a good question, and why my reply was almost but not quite an answer to it. However, you are using the validity of the parallel axiom for α + β less than two right angles to formulate the case where α + β is equal to two right angles that leads to the contradiction.

This is actually a good question. The best answer I can come up with is that the parallel axiom is being proven not to extend to the case where α + β is equal to two right angles (that for the parallel axiom, α + β is strictly less than two right angles). Note that this is a proof by contradiction (reductio ad absurdum) where the parallel axiom is assumed to be true for α + β equal to two right angles, then shown to contradict the uniqueness of a line through any two points.

In the US, the way it's currently going, back to leeches.

... for your calculator?

One thing I learnt fairly recently is that Δ9-THC found in cannabis is a partial agonist and therefore is self-limiting in its effects, whereas synthetic cannabinoids such as JWH-018 are full agonists and therefore are potentially more dangerous.

In general terms, the action of drugs/poisons is extremely complex. There really is no simple answer to the question of why drugs have side effects. An inkling to the complexity of drugs/poisons can be gathered by examining the Wikipedia article on the 5-HT receptor. This is a receptor that is activated by the neurotransmitter serotonin (5-hydroxytryptamine). The thing to note is that there are 14 known distinct 5-HT receptors in 7 families, and that these receptors are distributed throughout the body in locations as diverse as blood vessels, central nervous system, gastrointestinal tract, platelets, peripheral nervous system, and smooth muscle. Drugs/poisons that bind to 5-HT receptors can do so as agonists (full or partial) or as antagonists, and can bind to the different 5-HT receptors with different affinities, perhaps even acting as agonists on some receptors but antagonists on others. Thus, different drugs/poisons that that bind to 5-HT receptors can have quite different effects, and even different drugs/poisons that bind predominantly to a particular 5-HT receptor as an agonist may exhibit differences in effects.

Does that mean you don't subscribe to the commonly stated view that atoms are mostly empty space?

But what function did you use for a(t)?

I don't consider the need to breath to be a barrier to freedom, unlike the need to have money to eat Do you really think you can get a meal at lower cost living in the jungle than I can living in suburbia? For example, for about half an hour of devotion to an employer, I can take the money they gave me, spend about ten minutes driving to a nearby Chinese restaurant, order a very nice takeaway meal, wait about ten minutes for that meal, spend another ten minutes driving home, eat the meal over a period of about half an hour (or maybe a bit longer to savor the meal), spend a few minutes washing up, and I am fed. So, for under two hours of my time, I had a meal containing lamb, rice, broccoli, carrot, onion, and a whole bunch of other things too numerous to mention. Furthermore, I could depend on getting that meal, or at least something similar. For two hours of time in the jungle, could you possibly and reliably obtain a meal of similar quality in terms of the variety of ingredients in the meal?

During a thunderstorm, or even just heavy rain, I'm usually indoors and can look out a window to see wallabies sheltering under trees. I'm perfectly dry inside a house while the wallabies have to put up with whatever limited protection a tree can give. I often wonder what the wallabies in the yard are thinking when they see me drive down my driveway in the car and then enter the house through a locked door with a key. I don't know if they are at all jealous of me, but it makes me feel a little bit superior to them. I'm certainly not jealous of them. A few days ago, I opened the door to my car when a small lizard ran from the gap between the door and the chassis to somewhere inside the car that I could not locate. As far as I could tell, that was a fatal mistake the lizard made. The wallabies can do pretty much anything they like, but my life is certainly richer¹ than theirs. And the lizard's free choice most likely ended its life. ¹ There's more to the enjoyment of life than freedom. There's no such thing as "pure freedom". Constraints take many forms, but they are always there. For example, if I get seriously ill, I can go to a hospital and my chances of survival are vastly improved. An animal in the wild would inevitably die from such an illness. I get to drink clean water from a tap or from a shop. An animal in the wild has to drink from sources that may be contaminated by whatever. A person may have to spend several hours a week in servitude of other people. But an animal in the wild also has to spend a substantial amount of time in search of food, water, and shelter, as well as avoid predators.

I can just enjoy remoteness while there is still some That seems a little abstract to me. For example, one can live in a location with a great view, but once the novelty wears off, what else is there? You can't live on a great view. Similarly, you can't live on remoteness. I feel that you are just substituting one lack of freedom for another. You have not convinced me that you would be any freer in a remote location without money than in society with money. However, I do agree that governments tend to impose excessive restrictions on people, and in particular that the restrictions tend to increase over time.

In my case it's mostly being able to build shelters without having to pay for the land Like make nice camps, make fire, sleep there... But if you're relying on remoteness to be able to illegally occupy land, how is that "freedom"?

I guess we all have things we want to do when living in society but are not free to do because it's illegal So, what illegal things do you want to do in the Amazonian rainforest that you can't do in the privacy of your own home?

I've never liked the 'absorption - delayed emission' explanation of refraction. Such an explanation would be expected to produce scattering as well as failing to account for the particular property of materials that produce higher refractive indices than other materials. I wrote the following about six months ago. It would appear that I thought I would need it again: This is a common misunderstanding of refraction. Refraction is easiest to understand in terms of classical electromagnetic waves. This can then be translated to the quantum picture provided the important aspects of the classical picture are maintained. When an electromagnetic wave passes through a medium, it exerts a force on the charges and charge dipoles of the medium. Depending on how easily the charges and charge dipoles of the medium can move in response to this force, the motion of the charges and charge dipoles of the medium creates an electromagnetic wave that combines with the original electromagnetic wave to produce a total electromagnetic wave that is delayed with respect to the original electromagnetic wave and therefore travels through the medium at a slower speed. Thus, the refractive index of the medium depends on how readily the charges and charge dipoles of the medium can respond to the passing electromagnetic wave. This depends on the frequency of the passing electromagnetic wave. Higher frequencies exert a greater force, but larger bulkier charges and charge dipoles respond more to lower frequencies. At visible frequencies, only electrons can significantly respond to the passing electromagnetic wave, and in this case, the refractive index depends on the polarisablity of the electron orbitals of the medium and increases with frequency due to the increasing energy of the photons. As for the question asked in the opening post about light passing through an absorbing medium, this can actually be somewhat complicated as one can get weird dispersion effects near absorption frequencies. These weird dispersion effects are sometimes reported as causing faster-than-c propagation of light through the medium. A proper understanding of the effects of dispersion requires an understanding of the notions of phase velocity and group velocity.

Ok. There's no harm in admitting a lack of knowledge. Perhaps I was being harsh in expecting skills you genuinely do not have, especially given the festive spirit of the thread. I apologise. However, the questions I asked are legitimate: 1: Whenever you supply numbers to a formula, those numbers have units which should also be shown. Otherwise, how are we to know that the speed is in metres per second and not (say) miles per hour. Same applies to units of mass and energy. 2: It seems odd to specify a number as "5972200000000000000000000" instead of 5.9722 x 10^24 when you've already specified a number as 5.367545678×10^41. Also, the speed of light is 2.99792458 x 10^8 m/s when expressed in scientific notation. 3: This is about the accuracy of the numbers in the calculation. The accuracy of the final result is no greater than the least accurate number in the calculation. In your calculation, the least accurate number is the mass 5.9722 x 10^24 with only five significant figures. Therefore, the accuracy of the result (energy) is no greater than five significant figures. That is, 5.3675 x 10^41. The extra digits you supplied are just meaningless noise. It's obvious that you had just taken the number straight from a calculator.

For those interested, here is the metric I derived of a simple model of a ball of matter of mass [math]M[/math], radius [math]R[/math], and of uniform density: [math](ds)^2 = f(r^2) (c\ dt)^2 - \dfrac{(dr)^2}{f(r^2)} - r^2 ((d\theta)^{2} + \sin^2(\theta)\ (d\phi)^2)[/math] [math]\text{where:}[/math] [math]f(r^2) = 1 - \dfrac{2GM}{c^2} \bigg(\dfrac{1}{2 \sqrt{R^2}} \Big(3 - \dfrac{r^2}{R^2}\Big)\ H(R^2 - r^2) + \dfrac{1}{\sqrt{r^2}}\ (1 - H(R^2 - r^2))\bigg)[/math] [math]\text{and}\ \ H(x)\ \ \text{is the Heaviside step function:}[/math] [math]H(x) = {\begin{cases}1, & x \geqslant 0\\0, & x \lt 0\end{cases}}[/math]

For those interested, here is the metric I derived of a simple model of a ball of matter of uniform density: [math](ds)^2 = f(r^2) (c\ dt)^2 - \dfrac{(dr)^2}{f(r^2)} - r^2 ((d\theta)^{2} + (\sin(\theta)\ d\phi)^2)[/math] [math]\text{where:}[/math] [math]f(r^2) = 1 - \dfrac{2GM}{c^2} \bigg(\dfrac{1}{2 \sqrt{R^2}} \Big(3 - \dfrac{r^2}{R^2}\Big)\ H(R^2 - r^2) + \dfrac{1}{\sqrt{r^2}}\ (1 - H(R^2 - r^2))\bigg)[/math] [math]\text{and}\ \ H(x)\ \ \text{is the Heaviside step function:}[/math] [math]H(x) = {\begin{cases}1, & x \geqslant 0\\0, & x \lt 0\end{cases}}[/math]

1: Where are the units? 2: Why are there all those zeroes for the mass but the energy is given in scientific notation? 3: Why is the energy given to ten significant figures when the mass is only given to five?

I probably brought it up to illustrate how far-reaching the 2nd law of thermodynamics is. When it came to choosing between a violation of the 2nd law of thermodynamics, and a principle that I was unaware of but seemed necessary to prevent the violation of the 2nd law, I chose the principle that prevented the violation of the 2nd law.