KJW

It should be noted that expansion has units of inverse time, not speed, although it is usually expressed as speed per distance. For a flat (three-dimensional) universe described by the FLRW metric, the recession speed at a particular time is directly proportional to the distance, and therefore there will always be some distance beyond which is receding faster than c. However, it should be noted that when we look outward, we are looking at the past, so the observed recession speed is not necessarily proportional to the observed distance.

Thank you! There is another way, but I don't use it and am not exactly sure of what it is. Ok.

I use [ math] [ /math] (remove the space after "[")

Perhaps I'm narrowly focusing on the wrong area, but I don't see how this helps. Does it enable a solution without solving a non-trivial cubic equation? However, these are not solutions. Actually, they are solutions of the cubic equation. Bear in mind that numbers have three cube roots. Real numbers have one real and two conjugate complex cube roots. So there are three distinct values for [math]\sqrt[3]{7+\sqrt{50}}[/math] and [math]\sqrt[3]{7-\sqrt{50}}[/math]. There are also three distinct values for [math]ab=\sqrt[3]{-1}[/math], each leading to a distinct cubic equation. But it is not clear how many distinct values there are for [math]\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}[/math]. It should be noted that [math]a^3+b^3=14[/math] and [math]ab=-1[/math] determine a unique pair of values for [math]a^3[/math] and [math]b^3[/math], so the cubic equation in [math]u=a+b[/math] is not producing spurious values for [math]a[/math] and [math]b[/math]. Also, the above complex solutions demonstrate that there is indeed only one real solution for [math]u=a+b[/math].

[math]\text{Also:}\ \ \ \ \dfrac{u^3+3u-14}{u-2}=u^2+2u+7[/math] [math]u=-1\pm\sqrt{6}i[/math]

[math]\text{Let:}\ \ \ \ u=\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}[/math] [math]\text{Let:}\ \ \ \ a=\sqrt[3]{7+\sqrt{50}}[/math] [math]\text{Let:}\ \ \ \ b=\sqrt[3]{7-\sqrt{50}}[/math] [math](a+b)^3=a^3+3a^2b+3ab^2+b^3=a^3+b^3+3ab(a+b)[/math] [math](a+b)^3-3ab(a+b)-(a^3+b^3)=0[/math] [math]a^3+b^3=14[/math] [math]ab=-1[/math] [math]u=a+b[/math] [math]\text{Therefore:}\ \ \ \ u^3+3u-14=0[/math] [math]u=\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}=2[/math] [If the above LaTeX doesn't render, please refresh browser]

Yep, that's the only way I know. How about the one above it? I haven't figured that one out yet. It seems to me that 50 = 72 + 12 = (7 + i) (7 – i), but how that is used is not clear to me.

48 + 68 + 98 = (28 + 38)2 – (64)2 = (28 + 38 + 64) (28 + 38 – 64) = 8113 x 5521

One issue with discussions about metals is whether the metal is in elemental form, or as a compound. The title suggests the copper is elemental, but the article seems to suggest otherwise. However, it remains unclear to me. I'm not aware of any lifeform that biologically produces metallic elements, though I imagine copper would be fairly easy to produce, at least from an electrochemical standpoint.

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.