KipIngram

I think (a) looks right. It matches the value given in the Wikipedia particle in a box article, except they use k = n.pi/L. Your answer to (b) looks kind of fish to me; if I read it right you're saying that <x> = 2.pi^2 / L? That says that by making L large you can wind up expecting your particle to be as far over to the left edge as you wish. I thought <x> was the center of the box. Maybe I misunderstood what you were trying to write down. I didn't have time to look at ©.

+1 for "conscious, adult choice." All of that is very well said.

Thank you.

Hey, UT Austin! My alma mater! Matt Patterson: Goodenough has a solid reputation, so I'd be shocked if it's a "hoax" in the deliberate sense. Hopefully further work won't be disappointing. That article Strange linked directly addresses all of the areas where we need breakthroughs: energy density, lifetime cycle count, charging time, and SAFETY. If it works out it will change the game completely.

On the other hand, how could you stay focused on science when there was a baby to play with? Seriously, it would all work out, but as father of five I can attest to Bender's point that babies seem to plot against you containing everything they expel. Once when my 18-year-old was a newborn my wife and I went to a local sandwich shop / sports bar for dinner. There was a table full of college-age guys sitting next to us. My wife held the baby most of the time, but as we got up to leave she handed her to me so that she could gather her purse and stuff. Our daughter (still in the formula/milk fed stage) chose that very moment to throw up massively. She just coated the front of my shirt with it. The college guys looked like they were scared to death - I have a feeling each one was thinking "Christ, that's going to be me someday." Fun times.

Um, isn't x^2-5x+6 = (x-2)(x-3)?

I don't see that it's obvious at all that choice is an illusion. It might be, but I don't see solid proof of that. Regarding ideas just "coming to us," that's definitely a real phenomena; I've experienced it many times. Usually after I have spent some time consciously trying to solve a problem with no success; we clearly continue to work on those things unconsciously in some way. I see two different things to consider in discussions like this. First, free will. I feel fairly sure I have free will, but I'm not willing to stick a 100% stake in the ground on that claim. Second, there is self-awareness. I am willing to stick that stake in the ground; as far as I'm concerned the very fact that I recognize my own self-awareness indicates that it has to exist. If it didn't I wouldn't even be wondering about it - I'd just be robotically going through the motions of existence. The notion of self-awareness is more than enough to tie mainstream physical theory in knots. Even when we apply them to biological systems our most fundamental laws of physics are still theories of non-sentient stuff. Consciousness clearly doesn't enter into our theories at the bottom (fundamental) end, and I have a very hard time seeing how it's going to make its way in at the top end (emergence). Nothing anyone has shown me about emergence so far has seemed very convincing. Consciousness really seems to "bother" some people; such an effort is made to deny it.

"Adequately" means unambiguously. The amount of information required for the specification is the amount that lets you reconstruct the system as precisely as it is possible to do. Like Mordred alluded, some systems could have many, many degrees of freedom and yet still be "simple" if those degrees of freedom can all be specified with a small amount of information.

Yes, +1; public funds should not be spent to indoctrinate people in a singular belief. If there's any one "belief" I've tried to drill into my children (five daughters; I've been around the block a few times), it's to THINK FOR THEMSELVES. I don't care what they believe, as long as they arrive at those beliefs through their own path and not simply by "swallowing" someone else's belief system.

Yes, it's really quite amazing how we're just reversing the whole PC revolution. Computers were mainframes, with dumb terminals where the user was. Then we pushed functionality outward. Now we're pulling it back. I sort of wish they'd make up their minds. My own personal preference is to have dedicated compute power at my exclusive disposal.

In the OP he noted that each had speed wrt Earth of 0.5c; it would just be that an observer on earth would measure their speed wrt one another as c. But that's not the same as what was just said.

+1 for Feynman videos - I think I've seen that very one (I didn't play it, but I watched a flock of them recently and think I remember this bit). The man was nothing short of amazing in his ability to convey the scientific perspective. Joshua, I think the biggest piece of advice I'd share with you, and with anyone, is "Don't abdicate control over your own opinions. You have a mind. Train it, use it, and think for yourself in all things."

Off the cuff I would say that a system that requires more information to specify is more complex than one that requires less. So perhaps complexity <--> information content?

Responding to the OP, I think everyone is different and each finds fulfillment in his or her own way. I can't imagine myself without my scientifically inquisitive mind. Understanding as much as I possibly can is, quite simply, a craving that I've had my whole life. On the other hand, I don't think of myself as religious, per se, which to me implies a devotion to a particular organized religion. I am as curious about matters of the spirit as I am anything else - in particular I believe that science as it is situated today really tells us nothing about the nature of our self-awareness, whether or not we have free will, and so on. These questions are very important to me - I want to understand. That I have self-awareness seems directly clear to me, and I consider it possible that I have a non-corporeal spirit. But I don't regard that possibility as implying, necessarily, that an ultimate God-like spirit must also exist. It's certainly possible, but not proven conclusively for me. I view organized religion with distrust. The major religions have become "large organizations," and I feel that almost every large organization that exists / has existed winds up taking on an agenda of its own that's largely unrelated to its original stated goals. They usually become about power and control. So even if God exists, and even if he handed us His Word at some point in time, I feel that Word has been manipulated and revised over the years to reflect the political goals of the religious organizations. So I don't think there's a book I can read that can tell me how God wants me to behave. Instead, I just think and reflect on the world around me, and decide for myself what constitutes good behavior. So basically I think both of the things you mentioned are important. As human beings one of the biggest advantages we have over other life is the ability to think rationally and logically. I try to do that for questions about both science and ethical / moral behavior. I do not, however, think that one who chooses not to subscribe to an organized religion has somehow "failed" as a human being. I think there are many paths to being a good person.

I'd guess that it winds up heating the mirrors?

Well, please don't take my posts as being necessarily pro-Trump. I'm fairly shocked by some of the stuff he pulled, and recently I saw a story that claimed he more or less said that our constitutional checks and balances are "bad for the country." Later I decided I was unsure whether the quotes had been placed in proper context; I think may news outlets absolutely distort things. So I retracted my initial reaction to that one and put it in the "watchfully cautious" category instead. What I am trying to convey in my posts is support for sane, rational process for government goings-on. We have such processes and they provide mechanisms for dealing with problems. While I don't like every single thing about all of our processes, and don't imagine there is any such process that everyone thinks is perfect, I still support sticking to them in the name of order. It's a bit like disagreeing with the outcome of an election to start with. I can wish a candidate hadn't been elected while still recognizing him or her as the duly elected person - I can dislike an individual while still loving the fact that we choose our leaders via a democratic process. The bottom line is that the country's still going to be here and more or less in one piece when Trump's administration is over. When Obama was elected you got the same sort of howling and consternation from the right, and yet here we are. I disagreed with many, many of Obama's policies, but I accepted him as my duly elected President.

I imagine Mar's natural gravity would be enough. It's less than Earth's, but it ought to be enough. I think the notion of terraforming Mars would be great, if it will last. Is Mar's atmosphere poor because it just never had an appropriate atmosphere for humans, or is it because it doesn't have enough gravity to hold on to one? If the former then trying to change it might make sense, if we could manage the tech in an acceptably cost-effective way, but if it would just leak away again then it probably doesn't make sense.

Well, I think that by definition a perfect mirror would not absorb any energy. It would just reflect the photon and leave it otherwise unchanged.

That would be pretty rad, wouldn't it? Early in the video it had a frame about something patented in 1823.

Thanks, swansont. I'm a bit ashamed I didn't know that, but now I do. So ok, then, that means it's on Congress, and we get to judge them based on how well they do their jobs and make a new decision about them at a future time.

No. I suspect the whole story isn't being portrayed there. That is just not how the world works. As I said, if that worked we'd have many, many devices in regular use based on the process. Especially if it was patented in 1823.

You're taking the position that I'm wrong? Let's say the recent election had been turned the other way (Hillary won, but lost the popular vote). Do you think the same people would have complained? Issues like this should be considered and opinions formed based on the processes themselves, as abstract entities, not in the heat of the moment when people are upset. The Founders had specific reasons for structuring the things the way they did, and parties didn't even exist then so we know that those reasons had nothing to do with favoring one 21st century party over another. To be specific, the whole process of designing the Constitution was based on finding a structure that was equally acceptable to both populous and lightly populated colonies. The small colonies refused to support a 100% population driven structure, and the heavily populated colonies refused to support a 100% equal weighting amongst all states. So the system was put together to offer advantages to both. Equal representation in the Senate is one of those decisions that favored small states; proportional representation in the House was one that favored the heavily populated states. The Electoral Colony contains the same balanced mixture of influences. I don't know what to say, except that it was considered then to be a reasonable way to provide balanced treatment to states of different population natures. Of course, none of the Founders ever dreamed the federal government would wind up with the powers that it has; in their view of the world the state governments would continue to have the most significant impact on the lives of their citizens.

My point is simply that there are rules for all of this. I don't know what they all are, myself; I wouldn't presume that becoming President makes it so that the only law enforcement that can be carried out against you is Congress-driven impeachment. But I don't know. If the public is unhappy with the Senators and representatives that they elected (because they won't impeach Trump or for any other reason), then the public is free to vote them out the next go round. Our process simply does not allow for en masse recall of elected officials from office. You're certainly free to promote the idea of changing the rules so that it does, but I for one like the relative stability that our system has compared to some others, so you won't have my support. Just remember that any change you campaign for and get implemented will then cut in both directions, both when you want them to and when you don't.

An absorbent sail acquires the momentum of the photons that impinge on it. A reflective sail reverses the direction of the photons that impinge on it. So the photons undergo a momentum change equal to twice the momentum they were carrying prior to impact. The sail acquires that full amount of momentum.

Time isn't created by one of the four forces. Time is simply a coordinate label in the manifold in which physical processes are presumed to occur. Relativity posits that there is no such thing as "absolute time"; rather, every observer has a "local coordinate system" that applies to the portion of the manifold that's in their immediate vicinity. Each of those local systems will have a time axis and three spatial axes, but in general you and I will disagree as to whether various events are or are not simultaneous.