DrRocket

Yes exactky. But read my entire post rather than an isolared sentence out of context for the explanation as to how this really works.

Ontology as practiced by scientists is relevant. As practiced by philosophers in its most benign form it is merely irrelevant -- see essay by Steven Weinberg. Note quote from Wittgenstein on lack of relevance of philosophy to the practicing scientist. http://depts.washington.edu/ssnet/Weinberg_SSN_1_14.pdf

Not that I know of for certain as the theorem that you state is a bit unusual. The ingredients can be found in Rudin's Principles of mathematical Analysis. You might also look in Folland's Real Analysis: Modern Techniques and Their Applications but I am not terribly familiar with this one. Why would you care ?

1. String theory is still speculative. It is not "popular science". 2. In general relativity you do NOT have 3-spatial axes and one time axis, except locally for a chosen observer. 2a. In special relativity you have flat Minkowski spacetime, a 4-dimensional spacetime with a metric of signature (+,-,-,-) or equivalently (-,+,+,+). Neither "time" nor "space" are uniquely defined, A vector x is "timelike" if ,x,x> < 0 and ANY timelike vector give the "direction" for time for some observer. The orthogonal directions are the "space" for that observer. 2b. In general relativity the situation of special relativity is seen to be only local -- special relativity applies to the tangent space at a point, and is accurate only as a local approximation. There is NO global notion of either time or space. Time and space are "mixed together" by the curvature of the spacetime manifold. On other words there is no such thing as a "time axis" or three "space axes" except as local approximations. 2c. You may find this counter-intuitive but that is irrelevant. There is a mountain of empirical data supporting special and general relativity. 3. The idea of extra dimensions arose with the observation of Kaluza and Klein that electromagnetic theory and gravitation arose naturally from a theory like general relativity if one postulated a fifth spatial dimension. Kaluza Klein theory runs into difficulties with particle physics and has given rise to a more general approach -- Yang-Mills gauge theories. There are still unresolved issues. The point is that "extra" spatial dimensions are postulated in these theories in order to explain empirical facts -- the existence of known particles and forces -- in a unified manner. 4. It is very obvious that in our ordinary experience, locally, there are three spatial dimensions, not 5 or 11 or whatever. But as Klein observed, if extra spatial dimensions are compactified -- realized in a geometrically small compact manifold -- they would go unnoticed. This is of course speculative, but that is consistent with physics being a vibrant area of research -- there is a lot that is not understood. 5. String theories add another layer of research and speculation. There is much work needed to even clearly define in mathematical terms just what string theory is. Nevertheless it appears that additional spatial dimensions, in the form of Calabi-Yau manifolds, are needed to make string theories internally consistent, and to determine the laws of physics that come with each candidate string theory. Here spacetime is the Cartesian product of ordinary spacetime with a compact Calabi-Yau manifold, so again the existence of additional compactified dimensions can be made plausible if the Calabi-Yau manifold is sufficiently small. Yes, this came from imaginative minds -- both imaginative and disciplined. No, it is not established -- it is representative of cutting-edge and therefore speculative research. Yes, there is a body of supporting empirical evidence for some of it, and laboratory investigations continue unabated. No, it is not fantasy.

zoning ordinances

From that information you can't determine the peak force, or even the average force. In fact, since you don't know the propellant characteristics, or even mass, you can't calculate the recoil momentum.

Mathematicians would be nearly unaffected. Most theorems have nothing to do with any number base (there are a few exceptions). Balancing your checkbook would be a trick -- you would need to memorize addition tables for numbers up to 4000. This is your basic bad idea.

You need to learn a bit more physics. You are using the notion of "force" in situations in which it does not apply. When one launches a projectile recoil is the result of conservation of momentum. Recoil momentum is quite easy to determine, being equal to the combined momentum of the projectile and the propelling as. Recoil force, on the other hand is very difficult to determine. The time integral of the recoil force is equal to the recoil momentum. But the actual time history of that force is determined by the boundary conditions -- the elasticity of the absorbing media, including any non-linearities such as viscoelasticity. Very little is "robbed from the shell" in any recoil scenario. A tiny amount of recoil motion while the projectile is in the barrel may result in a minuscule pressure drop and/or reverse force on the projectile, but that effect is of little importance. There is no such thing as "the force of the shell in motion". There is of course kinetic energy and momentum. If you know the height attained and the initial launch angle you can determine the muzzle velocity -- easily if air resistance is neglected, and with numerical aid if not (given knowledge of atmospheric properties and shape of the projectile.) If you are launching vertically mgh=1/2 mv^2 so v = sqrt(2gh).

The cause of hangovers is not getting drunk. It is sobering up afterwards. The remedy is obvious.

I'm not quite sure what you have in mind, but it sounds like an old sling (David and Goliath kind) and yes, it works there.

Gravity is a conservative force. Any ncrease in velocity that occurs when approaching a body is lost again when departing from it. So, you ask, how do spacecraft take advantage of a "slingshot effect" ? The answer is that the "slingshot effect" takes advantage of the velocity, hence kinetic energy, associated with the orbit of a planet around the sun. The spacecraft approaches the planet from "behind" and gains some energy from the planet's speed, using gravity as a "rope". The key is that one is not considering the planet in isolation, but rather in a heliocentric coordinate system. While light is affected by a strong gravitational field and is red-shifted when climbing out, and blue-shifted when falling in, the effects compensate.

Standards have indeed fallen. I have not lectured in quite a while, but I have confirmed that in at least two graduate mathematics programs at major universities new grad students are no longer prepared to take measure theory. This astonishes me, since I took such a class with essentially no pre-requisites. I am also told that students are no longer willing to take notes and demand either a textbook or notes provided by the professor. I my day we worked quite hard, but did no demanding of trivia. Quite often no text existed -- anywhere -- and often we, the students, presented the bulk of the material. Grades seem to be up, but accomplishment down. Since the gene pool has not changed much I conclude that the problem lies with educators and the public at large lacking backbone. We had standards, and not everyone passed. In fact in the PhD program, relatively few made it all the way. Three or four of us passed the general exam in the year that I took it. We were the first to pass it in about five years. Even in high schooleople were held back. The fullback got a standing ovation when he received his diploma -- finally at about age 21. I once received a call from what I later learned was the "Dean in charge of football players". We had a short conversation: He: I understand that "S" was in your class. (S had been a high school All American.) Me: Yes He: How did he do ? Me: Flunked He: Why ? Me: Not enough points. That ended the conversation.

String theory and other promising but speculative ideas notwithstanding, so far as is known there are only three spatial dimensions. Gravity works the same way in all of them, The reason that many bodies tend to be spherical is related to minimization of potential energy which is related to the fact that a sphere is the shape of minimal surface area for a given volume. Elliptical orbits result from the inverse square law of gravity and ordinary Newtonian mechanics. In fact, Newton developed much of his theory -- including inventing calculus and differential equations -- in an effort to explain Kepler's laws of planetary motion. These things are relatively easily explained, but only after you have some physics and calculus under your belt.

There have been attempts to model space and time discretely. They have not thus far been successful. Current theories treat spacetime as a manifold. There is no smallest unit, no pixel.. Movement implies aq change in position over time, so quantization of movement requires both spatial and temporal measurements. -- see "differential calculus".

Ah, maybe it will help a lurker. I don't expect to get anything back. The sort of students who are typically appreciative ask deeper questions, or none at all outside of their class. It's hard to get excited about something that one can solve four different ways without thinking very hard. If he/she/it decided that I was wrong it might be interesting to see what was considered right. What intrigues me is that there seems to be a supply of people taking courses requiring proofs who apparently are clueless regarding the most basic subject matter. And it is May, so they have been in the class for quite a while.

You need to prove that P^-1AP is invertible. Because that is the assignment. There are several ways to prove this, in very few steps. Which you choose is a matter of taste and what you know about operators on finite-dimensional vector spaces at this point in your class. 1. Find the inverse. 2. Show that the determinant is not zero. 3. Show that the kernel is trivial. 4. Show that the rank is n.

In that case you would know the history of twin screw mixer/extruders in the explosives industry -- KABOOM I would be rather careful with your "mum's old meat mincer.

I presume then that you have some purely mechanical system available to: Make nitroglycerine and control the process temperature wile you do it. Make and dehydrate the nitrocelluose, keeping it solvent-wet throughout the process Mix the nitrocellulose, nitroglycerine and salts Extrude and cut the cordite rods Glaze the propellant with graphite without causing any static discharge and blowing things up Note that cordite was typically loaded into brass cases before the case neck was formed -- the neck was formed on a loaded round and then the bullet was added -- so typical reloadong procedures do not apply. The necessary know-how is not in books in the library. Much of the know-how for safe manufacture of explosives is in the minds of employees or in proprietary documents belonging to manufacturers. It can of course be re-created, but a lot of blood that was shed to develop the knowledge will be re-shed in doing it over again. Without electricity those lessons will be bloody indeed. Chemistry is important, but there is a lot more to explosive manufacture than just chemistry. Now that you have cordite, you still need primers. Priming coumpound, a real, no-kidding, primary explosive is a lot more touchy. You will also need great confidence in the consistency of your product in order to work up a load that will be effective but not blow up the gun. That usually requires instrumentation that uses some electricity. BTW if you fall back on black powder, you are going to have to also fall back to flintlocks, unless you think you can make percussion caps or primers.

I'm pretty sure that the food in my refrigerator floats when I close the door.

There is no doubt of a catastrophe in 2012. However it will not occur on Dec 21, but rather on Nov 6 -- election day. This calamity seems to be periodic, with a 4-year cycle.

"Bounded" as a metric space and having a boundary as a manifold are two completely different things. An ordinary sphere is a 2-manifold without boundary, but it is certainly bounded as a metric space. The right-half plane, including the y-axis, is an unbounded metric space, but it is a 2-manifold with boundary. I would stop worrying so much about metric spaces. The space-like slices of spacetime are metric spaces using the inherited metric, because it happens to be a Riemannian metric. But the Lorentzian metric of the full spacetime does not induce a topological metric on all of spacetime (because it is not positive-definite). Note that the word "metric" is being used in two different ways here -- a metric tensor need not be related to a topological metric (It does in Riemannian geometry, but not in pseudo-Riemannian geometry). You can, if you wish think about "finite" space as being a bounded metric space, but I think it is easier to think of it as compact. That is more natural and depends only on the topology and not the specific choice of metric. Unless you are very precise in your use of terminology, you are headed for a semantic morass. The only cure is to understand the mathematics in some depth.

Apparently there are some problems with Romanov's initial claim. http://www.computer.org/portal/web/news/home/-/blogs/3-sat-solution-flawed

A photon carries no electric charge. It does not interact with the electromagnetic field. This is despite the fact that the electromagnetic field is a bunch of photons. Photons do not interact directly with other photons. Electrons outside of the atom -- free electrons are accurately described classically and they interact with the electromagnetic field according to classical mechanics under the Lorentz force with the handbook value for electron mass. Back in the dark ages (say prior to qbout 2005) many television and computer displays used cathode ray tubes -- which are based on controlling electron trajectories using an electromagnetic field. Ditto for oscilloscopes. BTW the energy and position operators have both a discrete and a continuous spectrum. Free electrons can take on a continuum of values for both position and energy. Only confined electrons have only discrete values for thse quantities.

Modern gunpowder is based on nitrocellulose, with a high nitrogen content. Handling nitrocellulose is a job for experts. In the dry, unglazed, unstabilized form, nitrocellulose is one of the most sensitive and erratic compounds on the planet -- extremely dangerous. There is at present only one plant in the U.S. that makes weapons-grade nitrocellulose. It requires specialized equipment and lots of expertise. The basic raw ingredients are wood pulp or cotton linterrs, sulphuric acid and nitric acid. The hard part is the processing -- and that requires a lot more than laboratory equipment. No, I will not discuss the process -- there are restrictions on that information. There is only one plant in the U.S. that makes black powder. GOEX has had several accidents in the past and has moved the plant at least once following an accident. Making any kind of gun powder requires expertise. Lots of it. And knowledge of explosive safety. P.S. If you think gunpowder is bad, consider priming compound. Modern firearms require primers in the ammo in addition to gun propellant.

Use what you know about the correspondence between matrices and linear transformations.