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The Copenhagen interpretation describes what we can expect in terms of an experimental outcome. Whether that means anything about a larger "reality": The Copenhagen interpretation doesn't care. The CI takes an instrumentalist (aka non-realistic, as opposed to realistic or anti-realistic) point of view. Another way to say it: "Shut up and calculate". "Shut up and calculate." There are no hidden variables. The ket isn't quite real. The collapse of the wavefunction is what counts. "Does the moon exist if we don't look at it" is a nonsense question.

You would be correct were the computer programs that lurk behind many scientific papers as simple as your t-test example. They aren't. The problems here are software quality and software verification and validation. From years of experience with it, academic software is amongst the lowest quality software on the planet. It's bad stuff. Really. Really. Bad. Cyclomatic complexity in the triple digits, functions that are thousands of lines long, functions that access variables before they are assigned values, memory leaks galore, and the only comments are of three forms: "Well. Tell me something I don't already know!" (i = i + 1; // Increment i), "Well! Tell me why you did that instead of making a silly joke." ( i = i + 42; // Douglas Adams to the rescue!), or "Well!! I don't want to know that!" (// John Smith [1998]: Note to self: The following appears to violate the laws of physics). Even in high quality environments, simply replicating the effects of the code without replicating the code itself can be problematic. For a long time, one of the standard approaches to achieve robustness and correctness in software for safety critical systems was to have two different organizations independently develop the safety critical chunks of the software. This approach sometimes failed. Independent implementers sometimes made the same programming mistakes, sometimes made the same erroneous assumptions. Sometimes they did everything right and the code was still "wrong" because the requirements/algorithms were incorrect. Nowadays, that dual implementation scheme is being discarded in favor of independent verification and validation. Give the requirements, the code, the test procedures, and the test results to some independent organization and let them figure out whether the system is correct. One final point: There's a hidden flaw in your argument. You are implicitly assuming that the people who wrote the paper can describe what the software does. In many cases, good luck with that. Academic software is handed down from one grad student to another, then modified and extended, over and over again. Documentation is verbal. There might well be chunks of code written in Fortran IV that nobody understands and everyone is afraid to touch.

Entanglement and Bell's Theorem have been tested, many times over. There are some loopholes to some of the nasty implications that require you to throw out something rather important such as locality or counterfactual definiteness, but these loopholes are being closed one by one and are now rather slim. Schrödinger's hat did not say "its whatever answer you want". He was referring to various interpretations of quantum mechanics. All viable interpretations yield exactly the same answer to every experiment conducted so far, so in a sense it doesn't matter one bit which way you look at things. However, these interpretations color the way one looks at the universe. None of these interpretations is something with which we humans are quite comfortable. The Copenhagen interpretation is an offshoot of the 1920s Germany. It's chock full of bits of logical positivism. It is more than a bit autistic ("does the Moon exist when we don't look at it?") and is non-realistic. Many worlds is an offshoot of the 1960s America. Psychedelic.

Two things. One is that gravitational acceleration reaches its maximum at the core-mantle boundary, 10.68 m/s2. More important, however, is that force is not quite the right metric. Energy provides a much better perspective. There is a huge difference in energy between a protoplanet with a more or less uniform composition and a differentiated planet with dense material mostly segregated in the core.

Measure theory is such an important concept. Orthogonal polynomials, Fourier analysis, probability theory: All of a sudden these things make a whole lot more sense when looked at from the perspective of measure theory. Some mathematicians advocate for teaching Lebesgue integration from day one. BTW, Rudin is a very good place to start.

Hawking radiation will remain theoretical for a stellar blackhole or larger. For a solar mass non-rotating blackhole, Hawking radiation has a temperature of 62 nanokelvin, a peak frequency of 3.6 kHz, and a power of 9.0×10-29 watts sent over 4 pi steradians. Utterly undetectable.

Mind if I steal that as a sig?

Bonobos destroy your thesis.

Wrong. Quantum mechanics allows non-local phenomena. This is one of the things that bugged Einstein about quantum mechanics. His is the first initial in the EPR paradox, named for Albert Einstein, Boris Podolsky and Nathan Rosen. Einstein should have known better. He came up with several of the paradoxes in relativity, and the answer to those questions is "yep. That's what happens." The answer to the EPR paradox is similarly "yep. That's what happens." And that is exactly why TEW is wrong. A valid theory of quantum mechanics cannot be both local and deterministic. Aspect experiments, QHW experiments, and so on, and so on. Entanglement does exist, and it is "spooky action at a distance." That's a half-truth. That TEW cannot explain the Innsbruck experiment (better known as the QHZ experiment, for Daniel Greenberger, Michael Horne, and Anton Zeilinger) is absolutely true. This experiment falsifies TEW. That quantum mechanics cannot explain these results is an out-and-out lie. The results were consistent with the predictions of quantum mechanics to within 30 standard deviations. Baloney. Bell's Theorem is a mathematical theorem, not a physical theory. You have to prove that the mathematics is wrong (which it isn't), or that TEW can pass through one of the loopholes in Bell's Theorem (which it can't). The QHZ experiment, along with Aspect's experiment before it, let alone the many experiments since, are a death blow to TEW. TEW is nonsense.

Your parentheses don't match in 1/((Sqrt(x))*(atan(x)), but given your u-substitution, I gather you are asking about [math]\int \frac 1{\sqrt x\,\arctan x}\,dx[/math] Your integrand is a transcendental function. There is no analytic solution in the elementary functions to this integral.

From what I've seen, neither males nor females are rational beings for the most part. Humans are rationalizing beings. Rationalizations and bits of truthiness from a female might differ from their male equivalents, but they're still equivalents. Not rational, and not factual.

No, there isn't. That's a wikipedia article on a fringe subject. Those kinds of subjects tend to attract lots of cranks with lots of free time on their hands. Do not trust them. Wikipedia's quality has gone downhill as its popularity has increased. The main source of information from this article comes from Umberto Bartocci, an anti relativity and anti Einstein crank. There are lots of those cranks out there, and many of them have lots of free time on their hands. The premise of this group of nuts is "Relativity is wrong, and even if it isn't, Einstein didn't even come up with it." And that's leaving the ugly antisemitic bits out of their arguments. Given any two physicists in Europe in the early 1900s and it is almost a certainty that a three degrees of separation connection can be found between them. Just because the connection existed does not mean that it has any meaning. It is pure happenstance that Olinto's brother's coworker's nephew worked in the same office as did Einstein.

If Einstein did see that badly written paper published in an obscure little journal (a very dubious if), the path was from Olinto De Pretto to his brother to one of that brother's co-workers to one of that coworker's nephews to Einstein. That's three dubious degrees of separation. The world of physics was fairly small in the early 1900s. Any two physicists in Europe in 1905 were most likely separated by three degrees or less. Just because a degree of separation connection did exist does not mean that it was used.

That's crap put out by crackpots who for one reason or another want to knock Einstein down a notch or ten.

There is no proof of the parallel postulate in Euclidean geometry. It's a postulate. Regarding flaws in the proofs, the flaws start from the very onset with proposition 1 in Euclid's book #1. There are many others. One is his proof of the side-angle-side proposition. Many books make SAS an axiom because of this.

Yes, yes, yes, and students should also be forced to walk ten miles to school like they did in the old days, trudging uphill both ways, oftentimes encountering deep snow drifts in blazingly hot 120 degree temperatures (Fahrenheit). This notion that nobody has come up with a better way of teaching geometry in the 2300 years that have passed since Euclid is downright ridiculous. The notion that Euclid's Elements is the bible with regard to geometry is even more ridiculous. These notions hearken back to the pre-scientific notion that the best way to learn is to read from the great works, replete with monks chanting as background music. From the perspective of history of science and mathematics, Euclid's Elements and Newton's Principia are incredibly important works. From the perspective of how best to teach science and mathematics, there are better ways to teach geometry and introductory physics. Newton's Principia book is wall o' text page upon page of geometric reasoning, largely devoid of algebra and calculus. In fact, nobody teaches physics via Newton's Principia. So why this urge to teach geometry from the 2300 year old Euclid's Elements? Some of the proofs are flawed, others are rather roundabout, and the order of teaching does not match up with how kids think and progress.

Try using that technique to find a non-zero root of [math]x_{n+1}=\tan(x_n)[/math]. It won't work, period. It is unstable. It doesn't even work for finding the trivial solution. "Throwing more computer power at it" is not what everyone else does. At least not in the world of numerical computation. It is nice to have ever more computing power to throw at a problem because this makes formerly intractable problems become tractable. Use bad techniques and even formerly tractable problems suddenly become intractable.

There are two issues here, Vay. You asked for the "y-intercept of y= x+cosx", but then you asked for how to solve x+cos(x)=0. Issue #1 is that this is not the y intercept. That's the x intercept, not the y intercept. The y intercept is trivial: just solve for y with x=0. Issue #2 is the solution to x+cos(x)=0. This is a transcendental equation. There is no solution in the elementary functions. There are a number of ways to solve this numerically. One simple technique is to use a fixed point iteration scheme. Start with some initial guess, say x1=0. Now use this for your next guess via x=-cos(x). This yields x2=-cos(x1)=-cos(0)=-1. The next step yields x3=-cos(x2)=-cos(-1)=-0.54. After several of these iterations you will find that the sequence converges on a particular value. Fixed point iteration doesn't always work, but it does work in this case. It's not very fast, either, but it is simple. Ewmon's scheme is quite a bit faster in this case, but his scheme also fails in some cases, and it can be agonizingly slow in others. Try solving x=1.002*sin(x) using Ewmon's scheme, for example. His scheme is slow, slow, slow for this problem -- even if you start with a good guess such as 0.1.

You still haven't given us anything that we can use to help you. What is the homework question, word for word?

WHat's the question?

The idea is very well founded in a mathematical sense. This is an extremely simple probability problem. What exactly is your objection here? Are you truly contesting that switching doubles your chances of winning?

The probability is 50/50 if you flip a coin to decide whether or not you will switch. If your strategy is always stick with your original choice the probability is 1/3 (two to one against). This reverses to two to one in favor of winning the car (probability of winning the car is 2/3) if your strategy is to always switch doors.

One simple possibility: The OP phrased this oldie but goodie incorrectly. Google "wallet paradox" and you'll get tons of hits. Use Google scholar and you'll still get quite a few hits. Here's just one such result: http://academic.scranton.edu/faculty/carrollm1/wallet.pdf In this paper, Carroll, Jones, and Rykken analyze the wallet game from the perspective of Nash equilibrium. They find that there is none. While you're googling "wallet paradox", you might also want to do the same for "necktie paradox" and "two envelope paradox".

Close. In the winner-take-all formulation of the wallet paradox, it is the man with the lesser amount of cash in his wallet (not the greater amount) who wins everything. Each man will think "I have X dollars in my wallet, so that is what I stand to lose. If I win, I'll keep that original amount and I will win more than that. Therefore it is to my advantage to play." In the swap formulation of the problem, each man will think "I have X dollars in my wallet, so that is the most I stand to lose. That other guy might have ten times that, or a hundred times that. so my potential winnings are unbounded. Therefore it is to my advantage to play." It really doesn't matter which version of the problem you choose to analyze. Phrase the problem correctly and it will appear to each man that playing the game is advantageous to him. That is the paradox: How can each person see a positive outcome when this is clearly a zero sum game? The answer is that a probability distribution cannot be both unbounded and uniform. The arguments for why it is advantageous to play the game implicitly assume such an unbounded, uniform distribution. The supposed paradox is a simple matter of garbage in, garbage out.