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From the perspective of the observer on the embankment. You are completely missing the point of the thought experiment if you think this means the observer on the train also observed the lightning bolt strikes have occurred simultaneously. To the observer on the train, the lightning bolt strikes occur at different times. Simultaneity is relative.

Divides, in that context, is a synonym for "is a divisor of". 2 and 3 divide 24, but 5 does not. I was a bit hasty in my typing. I said that u+1 divides any polynomial of the form un+1. I should have qualified that with if n is odd. Fortunately, 3 is odd. In other words, there exists a quadratic q(x) such that (u+1)*q(x) = u3+1.

The implication here is that methane can only result from biological sources. That is not true. You can suspect that, but you cannot conclude that. There are known abiotic mechanisms for production of methane even on the Earth. Volcanos, for example, spew methane. Titan's atmosphere has a significant portion of methane. That does not necessarily mean Titan harbors life. That is intellectually dishonest. The vast majority of methane observed away from Earth is in the molecular clouds that form the interstellar medium. Are you truly claiming that this interstellar methane results from life? That all is a mighty big if, and it is demonstrably false. =========================== As an aside, Widdekind, your use of multiple fonts and colors makes your posts look very similar to posts made crackpots. You might want to rethink your posting style.

If you want to be taken seriously, write seriously. Use mathematics. Tell us how to test your concept. What does it predict? How do these predictions disagree with the prevailing science? If, on the other hand, you do not want to be taken seriously and would rather be viewed as a crackpot, write in an oversized font in impossible-to-read light green text on a light blue background. If you truly espouse to be viewed as a crackpot, using lots of pretty pictures without labels helps a lot.

The u-substitution u=e^x is a good place to start. Then du=e^x dx, so [math]\int\frac{e^x}{1+e^{3x}} dx\quad\to\quad\int \frac {du}{u^3+1}[/math] So this changes the problem from an exponential integral to a rational polynomial. Next step: Use the fact that u+1 divides any polynomial of the form un+1. You will get u+1 times a quadratic. Use partial fractions. You'll still have an ugly beast, but a completely solvable ugly beast.

What is your intent in this thread, navigator? Are you about to introduce some crackpot notion, or are you just confused? If it's the latter, re-read post #28. Also, google for conservation of energy.

Off-tangent, but Sheez! David Lynch?? Great films?? Those words do not belong together in the same sentence, the same paragraph, or even the same article. This is doubly true for the complete atrocity he made of Dune. =============================== Back on topic. Who are any of you to judge what salary someone "deserves"? If you think movie actors are paid too much, don't go to the movies. If you manage to convince enough people that your way of thinking is correct, and they do the same, actors will cease being paid their extravagant salaries. Part of the problem here as, that as bunch of science nerds, we live in a very egalitarian world. A fresh-out with a bachelors degree in a science or technology field can make $40,000 easily, $100,000 per year that with a masters degree in the right field / right geographic area. Sans a Nobel prize or being CEO of MegaTechnologies Corp, the odds of making a quarter million per year are rather slim. That is a very narrow salary range. Sports, acting, lawyering, and even doctoring do not suffer from (or benefit from, depending on your point of view) this egalitarianism.

Suppose each person is holding a clock and a meter stick. Let's look at what each person sees: Space twin sees platform twin's clock ticking at half of the rate as his own clock and sees platform twin's meter stick as being 1/2 meter long. Platform twin sees space twin's clock ticking at half of the rate as his own clock and sees space twin's meter stick as being 1/2 meter long.

In whose frame? You need to learn to construct space-time diagrams.

How does the spacecraft accelerate instantaneously along its entire length? The answer is: It can't. Doing so would require the compression or tension, depending on whether the thrust comes from the aft or forward end of the vehicle to propagate instantaneously. The best you can do is to place thrusters along the entire length of the vehicle. To bring this elongated vehicle to 0.8666c, the thrusters at the front of the vehicle will need to start firing years in advance of those in the back. The timing needs to be exact so that the wave of tensile force never occurs. Now comes the time for our adventurer to enter the back end of the vehicle. When the thrusters at the aft end of the vehicle finally ignite, the front end of the vehicle is well beyond has been Alpha Centauri. As far as the spacecraft is concerned, the distance between the Earth and Alpha Centauri has shrunk.

We've already given one example of how to define the Dirac delta distribution in terms of a function: think of it as a limit of a series of functions. The normal probability distribution forms one such series. A couple more are a series of square or triangular pulses. Consider a family of square pulses given by some parameter a: f(x;a)=2/a if -a<x<a and zero elsewhere. Now take the limit as a→0. Tada! The delta distribution. A series of triangular pulses is similar. Imagine a family of unit area isosceles triangles characterized by a single parameter a, with base length 2a and height of 1/a. Place the base on the triangles on the x axis and the apices on the y axis. Now form a family of functions based on these triangles. Here, f(x;a)=(1-|x/a|)/a if -a<x<a and f(x;a)=0 elsewhere. Re impulses: Look at the integrals of these families of functions: [math]F(x;a) = \int_{-\infty}^x f(\xi;a)d\xi[/math] For the square pulses, F(x;a) is zero for x≤-a and one for x≥a. For -a<x<a, F(x;a) forms a linear ramp from 0 to 1. In the limit a→0, F(x) becomes a step function: An impulse response.

Not necessarily. Think Komodo dragon. Dwarf elephants are an example of island dwarfism. Komodo dragons are an example of the flip side of island dwarfism, island gigantism. Merged post follows: Consecutive posts mergedEven better than Komodo dragons: http://en.wikipedia.org/wiki/Flores_Giant_Rat.

My personal take: We are essentially alone. Not all alone; there are 100+ billion galaxies in the universe, after all. However, if our nearest neighbor is over 1000 galaxies away, or even 10 galaxies away, how does that differ in any significant way from being utterly alone in the universe? The key argument against us being alone (or even essentially alone) is the mediocrity principle. My arguments against the mediocrity principle are three-fold: What justifies that principle? It isn't scientific at all. IMHO, the mediocrity principle is yet another example of why the divorce between science and philosophy at the dawn of the scientific age was a very good thing. What makes us special? Maybe nothing. What makes the winner of the $100 million lottery jackpot special? Nothing. He just got lucky. What makes us special? A lot of things. Multiply a lot of small probabilities together and you get an exceedingly small probability. We got lucky and hit the jackpot. The Moon is one of those things that potentially make us special. "... even if Earth had no moon the loose rotational axis would move so slowly that it would hardly pose a threat to early man" is an exceedingly fallacious argument, not one becoming a scientist. Early man would never have arisen if the conjectures about the Moon's role in stabilizing the Earth's rotation are correct. The Earth would instead have suffered the same fate as Venus: A very slow retrograde rotation. Venus didn't suddenly switch from rapid prograde rotation to slow retrograde rotation one million years ago. The transition occurred slowly and gradually over the last 4.5 billion years, with most of the damage done early on in Venus' history.

No. It means it runs through all j (i.e., from 0 to n) but skips over the j=i term.

What you just did is very, very bad in three ways. First off, Janus was talking about muons traveling at nearly the speed of light, not airplanes moving at nearly the speed of sound. You moved the goalposts to suit your argument. In this case of muons produced in the upper atmosphere, gravitational time dilation is negligible. Secondly, the Hafele-Keating experiment had planes flying in two directions. While the westbound plane's clock did run faster over the course of the experiment, the eastbound plane's clock ran slower. Selectively quoting only the part of some experiment that supports your evidence and ignoring the remainder is false attribution. Thirdly, general relativity alone does not explain even the behavior of the westbound plane's clock in the Hafele-Keating experiment. Both general and special relativity are needed to explain the behavior of both clocks. Use of fallacies is strongly discouraged at this site. Merged post follows: Consecutive posts mergedThread moved to pseudoscience.

Of course you did. You specifically said information has been sent faster than light. That would disprove special relativity. Information never has been sent faster than light. Every claim that it has has turned out to be either a mistake on the part of the researchers making the claim, or more often misinterpretation/misrepresentation by someone writing a lay article about the topic. There's a lot of crud out there on the internet. Take care that you don't step in it.

Quantum entanglement does not violate the speed of light. Information cannot be sent faster than light. That doesn't disprove SR. No, it hasn't. First off, you cannot decrease your rest mass. Secondly, massless (e.g. photons) travel at the speed of light.

It depends on what you mean by "proved". If you mean it in the absolute, mathematical sense of proof, general relativity has not been proven to be correct. You need to understand how science works. Science operates much the same way as does law. Short of a time machine that can look back into the past and view the actual commission of a crime after-the-fact, there is no way to conclusively prove that a person charged with the crime truly is guilty. Law operates on the premise of a preponderance of evidence. Science works much the same way. That said, the evidence for special relativity is overwhelming, much more overwhelming than the evidence for general relativity. Just to name a few, particle colliders, observations of muons formed in the upper atmosphere, and last but not least, quantum electrodynamics. QED is one of the most accurately verified of all theories in physics. Einstein didn't mention any of those experiments in his paper. He was a theoretician, after all. The motivation for Einstein's postulate was Maxwell's electrodynamics theory (aka Maxwell's equations). These equations describe the propagation of electromagnetic waves through space. The speed of the radiation is independent of the source and observer; it is constant. The Michelson-Morley experiment was designed to prove Maxwell's theory wrong. It didn't work. It was a failed experiment. There is a lot of crud out there on internet forums. Some of that crud is simply misinterpretation, some is deliberate misrepresentation. Exactly wrong. The Michelson-Morley experiment was designed to show the speed of light is not constant. It is one of the most famous failed experiment in all science. The Michelson-Morley experiment was the first of several attempts to assess the constancy (or lack thereof) of the speed of light. Of course it's accuracy isn't all that good. It was the first. The results have been reproduced many, many times over since then, with ever increasing accuracy. While there is no way to unequivocally prove that the speed of light is constant, there is very little wiggle room. The constancy of the speed of light ranks right up there as one of the most accurately verified concepts in physics. Once again, reality doesn't care what you believe.

No. A point is a point. Attaching an attribute to a point does not necessarily mean you are using, implicitly or explicitly, delta functions. Delta functions arise when you try, implicitly or explicitly, to take a gradient across the point. There is no need to use delta distributions in modeling point masses because Newton's law of gravitation strictly speaking only pertains to point masses.

Distant objects are treated as point masses, but that does not necessitate a Dirac delta formulation. There is no need to do so, and aerospace engineers simply don't. If this thread is still alive next week, I'll ask my coworkers who were trained as aerospace engineers if they even know what the Dirac delta function is, let alone use it. I know the answer to the latter: They don't. I suspect the answer to the former question is the similar.

Janus is right, you're wrong. You are once again mixing frames and once again ignoring relativity of simultaneity. Your long spacecraft is very similar to the pole-barn paradox. See, for example, http://math.ucr.edu/home/baez/physics/Relativity/SR/barn_pole.html.

The first thing to realize regarding the delta function is that it isn't a function. Looking at the delta function (OK, I'm a physicist at heart) as a limit of some series of functions is a very good approach to visualize what is going on. Bignose (post #2) and Severian (post #9) talked about it terms of a normal distribution. That's a very nice way of looking at it, but of course not the way. Some of the reasons this is particularly useful: It's already normalized for you, the normal distribution is analytic everywhere, and it generalizes to multiple dimensions. That said, Nobody does this. For one thing, that is in essence a time-varying mass distribution function the size of the solar system! For another thing, why bother? The errors in timing; vehicle position, velocity, and attitude; thruster alignment and thrust variance; sensor measurements; ... completely overwhelm the contribution of Pluto, or Uranus, or Jupiter for that matter. (Unless of course the probe is close to Pluto or Uranus or Jupiter.) Trajectory planning uses fairly simple (often extremely simple) gravity models. They do rely on precision models of the relevant planetary bodies to have been pre-computed (which they are; we have ephemeris models that predict the locations and orientations of the solar system bodies well into the future). For yet another thing, point masses models are absolutely the wrong thing to use for a vehicle in low Earth, Moon, or Mars orbit. The Earth has a large equatorial bulge, and the Moon and Mars are *lumpy*.

Are they? The last question in particular is awful. It is ambiguous: Absolute or relative humidity? The selected answer only looks at half of the picture, the vapor. None of the answers look at the whole picture (the frost and the gas), and none of the answers is anywhere close to correct if the question is asking about relative humidity. Multiple guess questions. Yuck.

Which two are you missing? You have one answer marked with an asterisk for each question. Are these your asterisks? We do not do your homework for you here at Science Forums. Doing that would not help you learn. Instead, we help you to do your own homework. That means we need more input from you besides just restating your homework problems.

Yes and no. The no answer first: There is no way to sense gravity, period. From the perspective of Newtonian mechanics, a gravity sensor would require a gravity shield. We don't know how to build a gravity shield, and neither do our bodies. From the perspective of general relativity, the answer is even easier: Gravity isn't a real force. It is instead a fictional force, like centrifugal force and coriolis force. Fictitious forces can't be measured because they aren't real. The yes answer: We can build devices that measure the net non-gravitational acceleration of an object, and so can our bodies. Mechanical versions of these devices are called accelerometers. Our inner ears have otoliths -- the biological equivalents of an accelerometer.