Bignose

This page, http://mathworld.wolfram.com/Combination.html, shows you how to compute the combination. (Note that the tree used a superscript, then a subscript and Mathword uses both subscripts, but they are the same thing... read "n choose k" in this case "500 choose 54"). I am hoping that you can compute the powers yourself using a calculator or computer.

If you are into some academic rigor, check out almost anything by Clifford Truesdell. Handbuch der Physik III/1 (that's book 1 in volume #3, oh and don't worry, that volume is in English) is just about everything you could ever want. The article is "The Classical Field Theories" by Truesdell and Toupin. Volume III of Handbuch der Physik is edited by Flugge. Truesdell's The Non-Linear Field Theories of Mechanics is more modern (the third edition was published posthumously from his notes in earlier editions), and his Introduction to the Mechanics of Fluids is innocent enough sounding, but is definately at least a graduate level text. But that Truesdell and Toupin reference will probably be more than enough.

I repeat myself: look up the binomial distribution: http://mathworld.wolfram.com/BinomialDistribution.html The example you gave is equivalent to 54 successes in 500 trials, where a success has a 1 in 8 chance. The binomial distribution lets you compute exactly that.

Look up the binomial distribution

Lift L = C * rho * 0.5 * v^2 * A rho=density of fluid, v = velocity A = surface area of the lifting surface C = lift coefficient Now, sure, you can direct the engine in any direction, look at a rocket, or the VTOL Harrier jets. But, there are other options, like lift, and if you have a big surface area, A, which the fixed wings do, the choice is to put the engine's enegry into increasing v. That way you get a big combination of v^2 * A, rather than just using the engine to power the entire craft. Given the same amount fo fuel, a Harrier will fly a lot farther using horizontal thrust and the wings for lift, than lifting itself by turning the jet wash. The pilots pretty much only use the adjustable wash for short take offs and landing and occasionally for super-fancy dogfighting. Oh, and airshows.

Bernoulli's efffect (this difference in pressures) is often not sufficient to produce lift to lift the entire plane... most wings also have significant flow turning. The fancy fluid mechanics word for this flow turning is circulation. Look at wikipedia's entry on lift, for instance. Both the complete pressure calculation and the circulation calculation should be the same, but when the flow is redirected downwards, the pressure calculations become much more difficult, which is why the circulation calculations are usually easier. The difficult thing is that the pressure acts normal to the wing, so that integral becomes very difficult for fancy shaped wings. Whereas the circulation can be measured on any closed loop drawn around the wing, and typically a large circle is picked for ease.

Wikibooks has a great deal of math textbooks online: http://en.wikibooks.org/wiki/Wikibooks:Mathematics_bookshelf

1991=996 + 995

these problems almost always work out very nicely via a change of variables. The definition of erf should give a very large clue as to what change should be made.

You should read several of the articles in the journal you want to put it in, that will let you know what level of sophistication and subject area that particular journal is publishing. Be sure to thoroughly check the existing literature to make sure that your work is fully original, or at the very least other articles of similar topics that may need to be cited. One of the databases I use to search journal articles is Web of Science. There are other ones, check them thoroughly.

A great test of basic statistical knowledge (adapted from radio-doc Dr. Dean Edell): A recent (made-up) study shows that you can reduce your chances of colon cancer by one-third if you eat 30 carrots a day. If the general populations' chances of getting colon cancer are only 0.02 %, if you ate 30 carrots a day (which is a heck of a lot of carrots), what are your new chances of getting colon cancer? If a different study showed that drinking 1 soda a day doubled your chances of colon cancer, how many soda drinkers would it take to statisitcally expect at least 1 colon cancer sufferer, i.e. on average, 1 in x soda drinkers will have cancer. Are the odds better or worse for the soda drinker that eats 30 carrots, or the person who does neither? Dr. Edell uses examples like this all the time to 1) show that studies often give conflicting advice... just look at the history of studies on coffee. and 2) shows that typically, the percentages are so small in the first place, that even if something is bad for you in one way, so long as it is in moderation, there is really no harm. For example, if drinking coffee doubles your chances of brain cancer... well, the chances of brain cancer are probably 1 in 10's if not 100's of thousands. 'Doubling' the chances sure sounds scary when heard on the evening news, but going from 1 in 200,000 to 1 in 100,000 doesn't look so bad. Looks even better if you write 0.0005% to 0.001%. That coffee that some people absolutely love is probably among the least important things to worry about. Same thing if eating something you don't like, like carrots, or fish... sure, it may reduce your chances a little, but what were the chances in the first place? Life is too short to fill it doing things you may hate just for the tiniest of chances of living longer so you have more time to do more of the things you may hate. This is not to say don't eat healthy, etc., but at the same time don't obsess.

I am relatively sure that some of the bonds are double and some are single. Look at the wikipedia article: http://en.wikipedia.org/wiki/Buckyball "The 6:6 ring bonds (between two hexagons) can be considered "double bonds" and are shorter than the 6:5 bonds (between a hexagon and a pentagon). " Carbons can be really tricky, benzene's carbons really aren't 1-sided and 2-sided... the bond energies sort of get all smeared out so that both sides of the beneze carbons look like 1.5's. I suspect that the drawings of C60 and so on are not drawn with the single and double bonds for conveneince. If every carbon only had 3 bonds, it would have quite a charge (-60 right?), and I think would be exceptionally unstable with that large of charge

Chemicals like ethers, with -O-O-'s in the middle of their structures, or chemicals with even more -O-'s (like -O-O-O- or more) will degrade and generate its own oxygen which can then be be used for combustion. I know of at least one horror story of some poor student heating up a beaker full of old ether that exploded on him. Maybe not exactly what you wanted, but a different answer to the 1st question.

Well, firstly, these two statements are completely incompatible. "Air goes through at the same rate" would mean (at least to me) that there is the same volumetric flow rate, which would be the same average axial velocity. Axial velocity is a key phrase there, since there would be a swirl velocity as well, but in terms of fluid through the pipe it should be the same. Next, air through a tube at any significant speed is probably turbulent. At a still wall, the turbulent kinetic energy (k) goes to 0, but when the wall is moving k at the wall will not be zero. I am not 100% sure, but I think that this would mean an increase in k across the entire pipe. Even a static mixer would add a theta velocity component, which would lead to a higher k. A higher k, would mean a higher turbulent viscosity, and a higher turbulent viscosity (total apparent viscosity of the flow would be the sum of the intrinsic and the turbulent viscosities) means that it would take more energy to push the fluid through. So, no, the swirl would not make the flow easier. (Not to mention, as above, the extra energy needed to swirl the pipe) I can think of two reasons, however, to do this. 1) increasing the turbulence does increase the mixing in the pipe, good if you want the air to be a more uniform temperature if you are heating the walls or something. Or if you are mixing methane or something with the air. or 2) the swirl will keep the flow a little more coherent, in that the vortex will persist for some distance after leaving the pipe. It should persist longer than if the pipe just sprays in a jet straight out. A swirled outflow is probably easier to control that just a straight jet out. I think that this is done for flames, for example.

Engineers, in just about the most general terms, use equations as tools to help solve problems. The extra tool in a chemical engineers' belt would be the ability to also use chemistry to solve problems, this will always be necessary. Now, that said, traditional chemical engineering is disappearing. One of the real indicators of this is the large of amount of departments that are no longer "chemical engineering" but now "chemical and biological engineering." AIChE is even considering a name change. Not only biological interests, but nano technology is another huge area chemical engineers are in. The traditional core, or studying distillation and reactor design, is beginning to be supplanted by bio-engineering or nano-engineering. It used to be that the overwhelming majority of ChE undergrads went to a refinery or bulk chemical company, like Dow. Nowadays, graduates go to just as many pharmaceutical and agricultural plants as chemical companies. I guess, in terms of if you look at all the areas ChE researchers are in, all of those will become increasingly important in upcoming years. On the other hand, what chemical engineering was, and to a degree is, may never be what chemical engineering will be in the future. I can see the possibility that chemical engineers will eventually fracture into the different disciplines, not unlike how chemical engineering split from chemistry departments several years ago.

In most draining situations, where the height of the water is much above the drain, the velocity in the tube is proportional to the square root of the height above the drain. But, this is just a special case of the Bernoulli's equation, and if you use Bernoulli's equation you can include the friction factors from the contraction into the tube, the friction in the siphon itself (total friction depends on the total length of the tube). The inviscid Bernoulli's equation can be devired by integrating the Navier-Stokes differential equations along a streamline, but energy and mass balances are the typical way it is presented.

See http://mathworld.wolfram.com/MatrixInverse.html for a pretty long discussion. There are easy formulas for 2x2 and 3x3 matrices, after that it gets complicated. See especially the links in the above URL on Gauss-Jordan elimination, Gaussian elimination, and LU decomposition.

Maybe I am missing something here, but volume is a length cubed, so it is derived from the length measure. That is, the m is defined, and the cubic meter is derived from the definition of the meter.

What we feel as wind is the momentum of the air molecules striking us or any object. Notice you don't feel a wind-force when you and the air is still. If there is no air, there is nothing to carry the momentum, which is what we call wind.

This is a fairly interesting claim. Would anyone happen to know how many viruses 'die' within 15 minutes outside their host anyway? For that matter, I am pretty sure that whether a virus is alive in the first place is still an open question / being debated. If it is not alive in the first place, I'm not so sure it can ever be dead, either.

I suspect you could develop an entire working system of physics based on impulse instead of work. You would have to redefine everything to work in the new system of definitions. Nevertheless, the current definitions we have no seem to work pretty darn well at describing what is seen in nature. And in that respect, even if you redefine the mathematics, nature will take no notice whatsoever. It will take a certain amount of gas to change the velocity a certain amount no matter if you call that quantity impulse, or work, or energy, or ether, or breakfast cereal, or sloths, or footballs. Nature takes no notice of what equations we use to describe nature. And the judgement we use on how good an equation is, is how well does that equation describe nature -- certainly not the other way around like you are trying to do.

I suppose then it's all going to depend on what your definition of complete understanding is. In you example, if the media the object were travelling in were turbulent, by today's knowledge it would be impossible to know what the exact velocity is at every point. In fact, if the flow is complicated, it may be very difficult to even get just the time-average flow. If you want to be super-accurate, you can use a computational fluid dynamics technique called direct numerical simulation, but expect it to take several months to maybe even years. Is this complete understanding? I would rather use correlations and have an answer that is pretty darn close in a matter of minutes than months. It is the same why with the vapor-liquid equilibrium problem above. We know the complete problem -- the chemical potential of each phase must be equal. And there have been several models proposed for many, many situations. These work spectacularly well for the situations they are designed for. The software used by engineers to simulate entire chemical plants use these descriptions all the time. But the complete problem -- simulating each molecule to see how it moves around, what its chance of leaving the liquid to become a vapor, etc? That is not perfectly understood. But, like I said above, there are several models that are coming pretty darn close. So, again, it all comes down to your definition of complete understanding.

Yes, as anyone above who has taken a calculus-based physics class has said, this is fairly easy to prove mathematically. I would suspect that it would be very difficult to do experimentally however -- from a few posts back, you would have to eliminate all outside forces. Then get a measuring device into the hollow sphere. All this assuming you have made a perfect sphere with completely uniform density. But, I do not understand how people are still arguing over the theoretical result -- pull out a decent 1st semester university physics text and it would show you how to do the calculations for yourself. In fact, if you google it, ignoring all the pseudoscience stuff about the hollow earth theory, you can find a few websites that explain this even further.

There is plenty of understanding... open up an undergraduate level thermodynamics or physical chemistry textbook. It comes down to this: the chemical potential of each phase must be equal. Otherwise, there will be a change. It is not possible at the moment to predict the rate of change, just that there will be a change. The polynomial fit is for ease of use, the complete thermodynamics is a complex issue. Especially with non-ideal mixtures, azeotropes, polar molecules, charged molecules, asymmetrical molecules, etc. Nevertheless, there have been several theories proffered. It comes down to properly describing the chemical potential of these complicated cases. I guess if pressed, there is no complete first principles basis for all of thermodynamics, but it is being worked on pretty fervently. There are several models and ideas today that come pretty close.