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The problem here is that you are using "for any" to mean the existential qualifier [math]\exists[/math] rather than the universal qualifier [math]\forall[/math]. There is a huge difference between [math]\exists a : \forall x\ af(x) + g(x) = 0[/math] and [math]\forall x \exists a: af(x) + g(x) = 0 [/math] The former means that either [math]g(x)[/math] is the zero function or that [math]f(x)[/math] and [math]g(x)[/math] are linearly independent. The latter is trivially true and has no meaning.

triclino, you can either continue to make silly, false analogies or you can try to understand. To use your bus analogy correctly, there are uncountably many buses heading out of town. Only one of them, bus number zero, goes to LA. That is not anywhere close to "all".

This is the root of your problem. It is what leads you to nonsense like this: You need to understand the difference between "for all" and "for any". Example: [math]e^x=1[/math] is true for one particular value of x. It most definitely is not true for all x. In comparison, [math]e^x>0[/math] is true for all real x, [math]e^x>1[/math] is true for all positive values of x.

You are using some nonstandard terminology. As far as I can tell, UMC means United Methodist Church. Note: 20037508.34 is pi times the radius of the Earth.

A laser is not perfectly columnated. At lunar distances, the laser beams used to target the retroreflectors spreads to about four miles wide. A tiny fraction of the light sent out by the laser hits the retroreflectors. The retroreflectors aren't perfect either, so an even tinier fraction of the emitted light returns to the receiver. A powerful, well-aimed laser is needed, but it is not an impossible task. Think for a bit about how large this conspiracy must be if the researchers at these observatories are lying about their lasers. Some advice: Learn to write more clearly. All that those ellipses (...) do is to make you look a bit incoherent. Those photos are not fake. It is you who wants to believe that they are. The claims regarding the lack of stars, the shadows, and the lighting have been fully rebutted. See http://news.nationalgeographic.com/news/2009/07/photogalleries/apollo-11-hoax-pictures/index.html, for example. These photos from the Lunar Reconnaissance Orbiter are not fake, either: I mentioned the size of the conspiracy a while ago. Too many people were watching the Apollo missions. The Russians would have called our bluff if there was any sign they were a fake. So would the uncounted number of ham radio operators who monitored the missions' progress. Too many people were involved in the project. The planning for the Apollo missions started in 1958 -- before NASA existed. The sheer magnitude of the effort precludes it from being a hoax. The only way two people can keep a secret is if one of them is dead. Too many people knew how to calculate whether the Saturn rockets and Apollo vehicles had the capability to go to the Moon and come back. Assuming it was a hoax, NASA would still have had to build the vehicles with the capability of sending people to the Moon and bringing them back alive. Why not just use that capability? If it were a hoax, why repeat the effort and risk the cover being blown? To claim that this was a hoax is incredibly insulting to the country and to the hundreds of thousands of people who worked on the project. Being incredibly insulting is of course the ultimate intent of most of the hoaxers.

I wrote my previous post, where I included a picture of Boltzmann's tombstone, before you wrote this. This equation pertains to a system in thermodynamic equilibrium. w (or [math]\Omega[/math] in modern terminology) is the number of micro states the system can be in that are consistent with a given set of macro observations of the system. Another view of [math]\Omega[/math] is that it represents how little one knows about a system. Consider the sum obtained by rolling a pair of dice versus the specific combinations that lead to a given sum. If you told me that you rolled either snake eyes or box cars I would know exactly what you rolled. Suppose, on the other hand, you told me you rolled a seven. What exactly did you roll? The picture is much less clear. There are six combinations that sum to seven.

That's appropriate. It was Boltzmann who (rather reluctantly) did much of the work on developing statistical physics. Boltzmann described entropy as disorder. His tombstone: Freeman Dyson similarly described entropy as a measure of disorder. I can't find his 1954 paper online, but the relevant parts are quoted in many undergraduate physics texts. For example, http://books.google.com/books?id=_ozWIrrBBxcC&pg=PA19&lpg=PA19 You are talking about physical entropy defined as [math]S=-k \sum p_i \ln p_i[/math], compared to information entropy defined as [math]S=-k\sum p_i \log_2 p_i[/math]. Wheeler's "it from bit". The problem with this view is that entropy/disorder does not translate all that well when viewed as information content, and even less so when viewed as complexity. See above. My opinion: Complexity is not the same thing as disorder, or lack thereof. Consider a perfect crystal versus a cloud of ideal gas in equilibrium. The first represents a minimal entropy (disorder) configuration; the latter, a maximal entropy configuration. Neither is not all that complex. Neither is anything close to chaotic. Complexity is more akin to the concept of chaos. Chaos lives on the borderline between the boringly predictable and utterly random.

Don't you think the primary reason is that the was raised at Obama's press conference, and that Obama acted stupidly by responding without full information?

Who started this weapon of mass destraction?

Three words: Disturbing the peace. Two more: Disorderly conduct. Freedom of speech is not an absolute right. You do not have a right to be a loud obnoxious asshole in public. (You can be an incredible jerk in private, so long as it truly is in private.) Laws on disorderly conduct and disturbing the peace are not unconstitutional. This guy got arrested for disorderly conduct. Just like this guy, or this guy. The latter case made it to the Massachusetts Supreme Court, which said Consequently, the Massachusetts disturbing the peace law which applies a normative standard and requires individual harm in fact is not unconstitutionally vague. ... In Cohen, particularly, the Supreme Court emphasized that speech, which in most cases will be constitutionally protected, can be forbidden if it is likely to provoke a violent reaction by the hearer, Cohen, supra at 20, or if it invades a "substantial claim to a recognizable privacy interest," id. at 21. The Court offers as an example of such a privacy interest, the interest in "being free from unwanted expression in the confines of one's own home." Id. at 22. Certainly loud and abusive speech, whatever the content, at 11:45 P.M. near dwelling places may invade the privacy rights of persons asleep in their homes. If they were in danger Gates would be in big, big trouble. Gates was acting exactly like drunkards who are picked up off the street, put in jail overnight, and released. These arrests are not specious, and neither was Gates.

Read the arrest report. The problems started when Gates didn't let it go. Gates went after the police, not the other way around. This is corroborated by multiple witnesses and a backup police report. Gate's counter claim sounds ludicrous in comparison and is uncorroborated.

Read the article again. After VASIMR completes some additional earthbound testing, its designers hope for it to be tested in orbit onboard the International Space Station. That article is a bit dated, written on May 15, 2003. Chang-Diaz' company has advanced to the point where those hopes to test the engine have come closer to reality. NASA and Ad Astra signed a Space Act Agreement for a flight test aboard the ISS. The agreement is structured in a series of “gates,” designed to allow the parties to assess the requirements on an incremental basis while proceeding to flight. Upon the successful achievement of the milestones set forth in the agreement, NASA and Ad Astra envision that VASIMR™ will be launched to the ISS where the rocket can be tested, for the first time, in its intended environment: the vacuum of outer space. The only plans NASA has now is to allow Ad Astra Rocket to test their engine. NASA obviously sees promise in the engine, but as far as "planning to attach a VASMIR thruster to the station to do orbital boosts"? Not yet. Not even close.

No, it's not. While NASA does spend some (not very much) money on advanced propulsion technologies, none of those are targeted for use on the space station. Short answer: They have to bring the fuel with them. Longer answer: The ideal rocket equation; read the linked article. If the fuel could be plucked out of space that would be a different story. Two ideas that do just that are Bussard ramjets and solar sails. (Google these terms.)

That "fifth" force would come in mighty handy for as a propellant. We currently use chemical propulsion for most space vehicles.The best chemical propellant leaves the rocket at about 4.5 km/s, or about 1/67000 light speed. Even though the exhaust leaves the rocket at a tiny fraction of the speed of light, the only theoretical limit to a chemical rocket's speed is the speed of light itself. While there is no theoretical limit to speed in a non-relativistic universe, there is a huge practical limit. Suppose we want to use chemical rockets to get a small 1 kilogram payload to 1/10 the speed of light. Assume the rocket itself is made of some incredibly light material (for example, unobtanium) and ignore relativistic effects. The amount of fuel needed would be, via the rocket equation (see post 17) is the solution to [math]\frac{0.1 c}{4.5\,\text{km}/\text{s}} = \ln\left(1+\frac{M_f}{1\,\text{kg}}\right)[/math] or about 102900 kilograms of fuel, which exceeds the mass of the universe by a mere factor of 1052 or so. On the other hand our unobtanium rocket would require only 220 grams of this magical fuel that leaves the vehicle at half light speed to reach 1/10 light speed. Merged post follows: Consecutive posts mergedNow suppose we want to use get to 0.9 c. This magical fuel will come in mighty handy. First let's ignore relativistic effects and use the non-relativistic rocket equation: [math]M_f = \left(\exp\left(\frac{0.9 c}{0.5 c}\right)-1\right)\,M_r[/math] or about 5.05 kg of fuel. One cannot of course relativistic effects in achieving a velocity of 0.9 c. The relativistic equivalent of [math]\Delta v = v_e\ln\left(1+\frac{M_f}{M_r}\right)[/math] is [math]\tanh^{-1}\left(\frac{\Delta v}{c}\right) = \frac {v_e}{c}\,\ln\left(1+\frac{M_f}{M_r}\right)[/math] which yields 18 kilograms of fuel for our magical rocket.

Concur. Multiple physicists have derived special relativity by showing that causality coupled with unbounded velocities leads to a contradiction. One of the two has to go. Unfortunately, I don't have time to chase the papers down today. I have a big deadline tomorrow. I'll try to find some references later this week.

You can't have it both ways. If throw out c as an upper limit to velocity you inherently throw out the first postulate of special relativity. Some alternative derivations of special relativity start with by postulating a causal, flat universe. This postulate that the relative velocity of any two objects must have some finite upper limit and that this upper limit is the same to all observers. Reconciling this philosophical reasoning with Maxwell's equations says that c is that upper limit. While Galilean relativity appears to be correct at velocities that are small compared to c, it is not a universal law. If you want to pretend that it is a universal law (which is what I thought was the point of this thread), you need to throw out things like the speed of light being constant -- and you need to throw out causality.

The math is the non-relativistic rocket equation, which has no theoretical upper limit on velocity. The very best we can get out of chemical propulsion systems is about 4.5 kilometers/second exhaust velocity. Orbital velocity in low Earth orbit is about 6.7 kilometers/second. In other words, every time we put a vehicle in orbit we exceed the exhaust velocity. The exhaust velocity is not a limit to the velocity that can be achieved.

Rocketman: You are vastly underestimating how big, and how empty, space is and you are vastly overestimating the state of rocket technology. The US has launched five satellites that will eventually escape the solar system. The last one launched was New Horizons in January, 2006. It will reach Pluto in July, 2015. The other four (Pioneer 10 & 11, Voyager 1 & 2) were launched more than thirty years ago, and have left the planets far behind. Whether any of these previous four have left the solar system depends on what is meant by leaving the solar system. In the sense that leaving the solar system means transitioning from the heliosphere to interstellar space, one (Voyager 1) is in the boundary. The other three are still well within the heliosphere. If any of these probes were on a heading to the nearest star, it wouldn't arrive for another 75,000 years. Voyager 1 is the fastest of the five and it is moving at 1/20,000 the speed of light.

fminsearch is a local optimization method. Local optimization is a lot easier than global optimization. It remains the subject of a lot of research. There is no silver bullet. Do a google search. Look for optimization techniques.

Matlab has an optimization toolbox. Some of the capabilities are also implemented in the plain old vanilla version of matlab. fminsearch might do the trick if you don't have the optimization toolbox. Optimization toolbox: http://www.mathworks.com/products/optimization/description6.html fminsearch: http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/fminsearch.html

Bill Gates is tilting at windmills is a better description. http://www.usatoday.com/weather/research/2009-07-15-gates-hurricanes_N.htm "By cooling a region in the path of a hurricane (over 60 square miles), models suggest we could knock a half-a-category in wind speed out," says Philip Kithil of Atmocean in Santa Fe, an ocean-pumping firm mentioned in Gates' applications. "All the models indicate the path of the storm would be unaffected." I suspect that if they cooled a mere 60 square miles the magnitude of the storm wouldn't be affected much, either. 60 square miles is a circle 8.74 miles across. Large hurricanes (and this is the subject at hand) pass over small islands without diminishing in size a bit. How much larger an area would they need to hit? Large hurricanes (and this is the subject at hand) are several hundred miles in diameter, a hundred thousand square miles in area. Hurricanes do diminish somewhat when they cross over Cuba. Let's say they need to cool a 100x100 mile section of ocean: 10,000 square miles, a mere 1/10 of the size of a large hurricane. Cooling just the top 10 feet of the ocean won't be very effective, and the cooling won't last very long. To be effective the cooling needs to be much more pervasive -- say 100 feet deep. Assume a temperature difference of 50 degrees between the surface and the deep waters and assume the goal is to cool the surface by 5 degrees. You need to exchange 1/10 of the total volume to be cooled with deep water. Multiplying these factors, 1/10*100 feet*10,000 square miles is about 20 cubic miles, or 80 cubic kilometers, that needs to be exchanged with deep water. The volume of Lake Mead: 35.2 cubic kilometers. Let's say I'm off by a factor of two. This plan requires transferring one Lake Mead's worth of cold water from 500 feet deep or more with the same volume of hot surface water. More from the article: The patents envision sail-maneuvered barges, with conduits 500 feet long, pumping warm water down to the depths and bringing cold water up. So, Gates wants to bring up one Lake Mead's equivalent volume of water from the depths and pump the same volume of surface water down to the depths -- with sailboats! Now that is tilting at windmills.

Obviously he would go about solving by posting it as a question on this forum. Hint to externet: What google search terms might help you solve this problem?

It is worse than an opinion. It is a logical fallacy. We observe methane to be ubiquitous across the cosmos because we see signs of methane in interstellar gas clouds. This interstellar methane has nothing to do with life. It has everything to do with non-biological chemistry in interstellar gas clouds. We have seen signs of methane on one exoplanet, HD 189733 b. HD 189733 b is a hot Jupiter. We also see signs of methane, and a whole lot of methane, in the atmosphere of Jupiter, Saturn, Uranus, and Neptune. The vast majority of methane in the universe has absolutely nothing to do with life. You are invoking a non sequitur in claiming that life might be ubiquitous because methane is ubiquitous.

Please don't ask such open-ended questions. We are not going to write a book for you and we are not going to do your homework for you. What have you done, what do you know, and where specifically are you having problems?

I doubt the simulation require this. Rather it is your formulation of the problem is what requires this. There are often ways around such problems. Scaling and normalization techniques solve many such problems in a neutral way. When these don't work, people sometimes resort to heuristics to get rid of the problematic areas. An example: spherical harmonics. A naive spherical harmonics expansion of some function will often lead to extremely small coefficients. The application of these coefficients leads to some extremely large factors. By use of appropriate normalization factors, the problem of small coefficients and large factors pretty much disappears. Where it doesn't, heuristics such as not bothering to compute the 100, 100 term (or any large term) for large radii are employed. (The computation of the contribution of the n,m term with large n and m and large radii will result in underflow. These tiny terms might change the sum in the hundredth decimal point. Why bother?)