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In a sense, neither is "real". Suppose you tie a weight to one end of a rope, grab hold of the other end of the rope, and swing the rope so the weighted end is rotating around you. There are several real forces involved in this example. You are exerting a force on the rope, pulling it toward you. The rock's inertia pulls on the rope in the other direction, putting a tension in the rope. Some old-timers (older than me!) call that outward pull of the rope on you (a real force) a centrifugal reaction force. This nomenclature is very outdated and is not what is usually meant by the centrifugal force. This is a fictitious force. It appears as a result of doing physics in a rotating frame and pretending that Newton's laws are still valid. In the rotating frame in which the rock is stationary, the tension in the rope exerts a real inward force on the rock. Pretending that Newton's laws apply in this frame, there must be some outward force acting on the rock to keep the rock stationary. This apparent outward force is called centrifugal force. Note well: the centrifugal force only appears as a result of trying to explain motion in a rotating frame in terms of forces. All centrifugal forces vanish when one explains motion from the perspective of an inertial reference frame. Centripetal force is simply centripetal acceleration (a kinematics concept) dressed up as a force. Kinematics doesn't care what causes motion to occur. All kinematics says is that the rock is perpetually accelerating toward you: the rock is undergoing centripetal acceleration. While that centripetal acceleration results in this case because you have tied the to the end of a flexible rope, that is irrelevant. The same motion can occur if you glue the rock to the end of a rigid rod, or if you give yourself and the rock opposite electrical charges, or if you make the rock orbit a very heavy mass, or if you use a Klingon tractor beam on the rock. Kinematics doesn't care about forces. Centripetal force is simply centripetal acceleration times mass. In other words, it is kinematics dressed up as dynamics.

It most certainly does apply to waves. The superposition principle applies to forces, potential energy, water waves, ... It applies to any linear system, of which Schrödinger's wave equation is but one example.

Its about physicists trying to take science fiction seriously ... ... maybe.

Because both observers have to agree on the final result. Time dilation and length contraction are flip sides of the same coin. The observer on the train sees the train as having the same length at all times. Its all relative ... The postulates are easy to understand. The math is simple algebra. The consequences are admittedly bizarre and counterintuitive. Just follow the math. Its not like they're doing anything hairy, after all. They haven't touched GR yet because with GR it is the math that makes your head explode.

Yeah, yeah, yeah. I know the difference, too. Reminder to self: Never post in a thread on grammar. It is a tautology that will make a boneheaded mistake.

Those who don't understand the distinction between "loose" and "lose" are a bunch of losers who are way to loose with the grammar. I can easily distinguish between people who say "practise" from those who say "practice". Hint: The British accent is a dead giveaway. Affect/effect. I can get discombobulated with this one. In most cases, one should use "effect" if the word in question is a noun, "affect" if the word in question is a verb. But effect can be a verb ("the arbiter effected a settlement in the dispute") and affect can be a noun (the patients displayed normal affects). My rule: Don't be affectatious and then you don't have to worry (the obvious choice is correct).

How a pseudo random number generator (PRNG) is initialized is up to the caller. The library interface simply calls for an integer (or set of integers, depending on which PRNG is used). In particular, most PRNGs do not provide an option to set the PRNG from the system clock. There is nothing to stop the caller from using the clock, however. This is a bad idea, and with some PRNGs this is a notoriously bad idea. The low-order bits are not nearly as random as the high order bits. In one infamously bad implementation the low order bit alternated between zero and one on subsequent calls. Using the mod N approach with N=2 would yield a sequence 0,1,0,1,0,1,... --- not very random at all. A better approach is to normalize the PRNG output to a floating point number between 0 and 1, multiply by N, and take the integral part of the result.

: That post deserves the loftiest kind of praise! (Rep points granted).

Somebody gave me some reputation points, but the whole point of doing so was so they could spam me. The comment is "Embroidery Corporate clothing tee shirt printing t shirts printing" in the form of a bunch of hyperlinks. Is there any way to delete that garbage?

I already did. Let x and y be two complex numbers. One could define [math]0^0 = \lim_{x,y\to 0}x^y[/math] if the limit exists and is the same along all possible paths that can be taken by x and y as they approach zero. However, by choosing different paths one can come up with any complex value one desires (not just zero or one) as the value of the limit. Attaining any positive real value Let a be some positive real number. Define [math]y(x) = \frac{\ln a}{\ln x}[/math]. Since [math]\lim_{x\to0^+} y(x) = 0[/math], this defines one family of paths for evaluating xy as x and y simultaneously approach zero. Note that for all positive x, [math]x^y=a[/math] with y defined as [math]y(x) = \frac{\ln a}{\ln x}[/math]. This family of curves generates all positive reals as values for [math]\lim_{x,y\to 0}x^y[/math] Attaining any complex value Use the same definition for y(x) as above, here using the principal value of the complex logarithm. Now one can chose a to be any complex value whatsoever, with [math]\lim_{x,y\to 0} x^y = a[/math].

The answer is that [math]0^{ 0}[/math] is indeterminate. In power series, mathematicians denote [math]x^{ 0}[/math] to be 1 for all x. This notation is not definitional. It is a very convenient abuse of notation. It is convenient because we can compactly write things like [math]f(x) = \sum_{n=0}^{\infty} a_n x^n[/math] and [math]f'(x) = \sum_{n=0}^{\infty} (n+1) a_{n+1} x^n[/math]. This notation is abuse of notation because defining [math]0^{ 0}[/math] to be 1 (or any other value) would lead to contradictions. Any mathematical system must, above all else, be consistent with itself. Lack of consistency (e.g. a contradiction) leads to all kinds of problems. For example, contradictory statements let one can prove any statement to be true or false. One single contradiction makes an entire mathematical system worthless. Consider the expression [math]x^y[/math], and make x and y simultaneously approach zero. Given how you make x and y approach zero, you can make [math]x^y[/math] take on any value. For example, let [math]y=\ln(a)/\ln(x)[/math]. Making x approach zero from above makes y approach zero, as [math]\lim_{x\to 0^+} y = 0[/math]. With this definition of y, [math]x^y = a[/math] for all positive x, and thus [math]\lim_{x\to 0^+} x^y = a[/math]. In short, you can make [math]0^0[/math] be any value whatsoever. To avoid problems, one must say that [math]0^0[/math] is indeterminate.

You're totally wrong. First, a nitpick: Gravitational force would decrease with depth if the Earth had a constant density. It doesn't. The density at the center of the Earth is significantly greater than surface rocks for a couple of reasons. The Earth's core is mostly iron, which by itself is much denser than rock. The extreme pressure at the center makes the density even greater. Starting from the surface, the gravitational force first increases with increasing depth, reaches a maximum, and then declines to zero acceleration at the center of the Earth. The reason you are totally wrong is that you have forgotten about the pressure inside the Earth. This increases with depth, and very quickly. Think of it this way: pressure in a body of water increases by about one atmosphere for every ten meters of depth. That's just water, and that is ignoring compressibility. Rock is a lot denser than water. The pressure at the center of the Earth is estimated to be about 360 gigapascal, or about 3.6 million atmospheres. This compressive forces resulting from this huge pressure swamps the tiny centrifugal force resulting from Earth rotation by many, many orders of magnitude.

You don't need GR to explain the twin paradox. For example, see http://en.wikipedia.org/wiki/Twin_paradox#Resolution_of_the_paradox_in_special_relativity or http://www.phys.vt.edu/~jhs/faq/twins.html. Einstein didn't even view this as a paradox. He just viewed it as an interesting consequence of special relativity. Historically, special relativity was accepted rather quickly. The evidence was already in, and not just in the form of the Michelson-Morley experiment. Einstein didn't even refer to that experiment in his 1905 paper. Einstein did make extensive reference to Maxwell's equations in that paper. The main problem with special relativity was attribution rather than acceptance.

Of course science has limits, and rightfully so. Without limits on ethical concerns, some future Dr. Josef Mengele or Taliaferro Clark would have free rein to once again experiment on how to create children with blue eyes or passively watch a curable disease run rampant through a population. Without very strict rules on biohazard research, some dufus in what should be a biosafety level 4 facility might well unleash the disease that kills us all. Holding science sacrosanct leads to all kinds of abuses. Example: Killing whales for commercial reasons is illegal; killing whales in the name of science is perfectly fine. The Japanese have flaunted the rules against commercial whaling by calling their whaling science.

No paradox. You just did many things that are wrong. You ignored that [math]x^2=1[/math] has two solutions and [math]x^4=1[/math] has four and you assumed [math](ab)^c=a^c\,b^c[/math], which is valid only for real c and positive real a and b.

Man goes for a walk around Dublin. Nothing happens.

Why calculate the square one digit at a time when Newton's method more than doubles the number of digits each step? Newton's method to compute the square root of some number a is a iterative algorithm, [math]x_{n+1} = \frac 1 2 \left(\frac a {x_n} + x_n\right)[/math] You need an initial guess, and it does not need to be particularly good. (Newton's method will converge for any initial guess.) For example, [math]\surd 0.9=0.948683298505138154[/math]. x1=0.95. The square root of 0.9 is obviously somewhere between 0.9 and 1, so a good place to start is the average of these two. This happens to be a very good initial guess. Let [math]e_n=-\log_10\left(\left|1-x_n/\surd 0.9\right|\right)[/math], which is essentially the number of significant digits in the nth[/su] estimate. So, right off the bat, the initial guess is good to 2.86 decimal places. x2=(0.9/0.95+0.95)/2=0.948684210526315769. This next step more than doubles the precision, as e2=6.02. x3=(0.9/0.948684210526315769+0.948684210526315769)/2=0.9486832980509526865. The precision metric is now 12.33. x4=(0.9/0.9486832980509526865 + 0.9486832980509526865)/2=0.948683298505138154. This nails the result. I would have to use an extended precision calculator beyond this point. Starting with a semi-reasonable initial guess of 0.9 or 1.0 merely adds one step (both give 0.95 as the next guess). Suppose we start with a lousy initial estimate. x1=0.3. Oops. The error metric is 0.17, not even one significant digit. x2=(0.9/0.3+0.3)/2=1.65, with an error metric of 0.13. In terms of the error metric, this is a step away from the solution. x3=(0.9/1.65+1.65)/2=1.09772727. The error metric is now 0.80; we almost have one significant digit. x4=(0.9/1.09772727+1.09772727)/2=0.958801524. The error metric is now 1.97, so we are once again in the regime of more than doubling the precision with each step. x5=(0.9/0.958801524 + 0.958801524)=0.948736687, for which e5=4.25 And so on. It took quite a few steps to get to the stage where quadratic convergence kicked in, but once there the technique zeros in on the answer.

Just because scientists write things on paper does not make them lumberjacks. Recording a bunch of experimental measurements in the form of numbers is merely accountancy. Accounting is no longer a branch of mathematics. Scientists also use calculus and statistics. The key word here is "use". Mathematicians have created some extremely useful concepts that the rest of us use, sometimes to the chagrin of mathematicians. From G.H.Hardy, A Mathematician's Apology, Cambridge University Press (1940), full text at http://www.math.ualberta.ca/~mss/misc/A%20Mathematician's%20Apology.pdf, It will probably be plain by now to what conclusions I am coming; so I will state them at once dogmatically and then elaborate them a little. It is undeniable that a good deal of elementary mathematics—and I use the word ‘elementary’ in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus—has considerable practical utility. These parts of mathematics are, on the whole, rather dull; they are just the parts which have the least aesthetic value. The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly ‘useless’ (and this is as true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his work.

Math is not the study of digits. Math is not science. Math is about constructing theorems from a set of axioms. Science is about constructing theories from a set of hypotheses. The key difference: Science is tested against reality by means of experimentation while math is tested against logic. Scientific theories can never be proven true. Mathematical theorems can be.

Even the monatomic noble gases are all at least slightly soluble in water.

A bit pedantic, but the Supreme Court can't find Bush to be guilty in the case of Al-Haramain Islamic Foundation v. Bush for the simple reason that this a civil lawsuit, not a criminal case.

Assuming your matrix G is not singular, there are many, infinitely many, matrices S such that S*S=G. The square root of a matrix is not unique. If G is symmetric positive definite it can be decomposed by singular value decomposition as G=UVU* where the matrix U is unitary and V is diagonal. The diagonal elements of V are positive if G is symmetric positive definite; these are the singular values of G. The square root of the matrix V is easily computed: Take the square root of each element. The matrix S=UV1/2V* is symmetric and is one common way to denote the square root of a matrix. Another commonly used technique is Cholesky decomposition.

The compensation in most intellectual fields is fairly egalitarian. The salary of Nobel Prize-winning professor who works a top-rated school is only a few times that of a mediocre professor at Mediocre State. The salary of an engineer or scientist who manages several dozen people and has decades of experience is only a few times that of a fresh out. There are exceptions, of course. The medical profession isn't all that egalitarian (neurosurgeons are paid a lot more than GPs), and the compensation for the upper echelon of Borg Technologies, Inc ("Resistance is futile") can be quite obscene. Other communities such as business and law are far from egalitarian. Most business and law majors do not get anywhere close to six figure salaries out of college. Most get less than engineers and scientists. Some get a whole lot more, and the disparity grows wider over time.

Just because there are fewer management slots does not mean the supply exceeds the demand. The supply is limited by the number of people qualified to do the job. That concept certainly pertains to professional sports. How many people can hit a 100 mph fastball? How many can throw one? That said, many professions, such as medical doctors, actively work to reduce supply. Would doctors in the US be paid so much if medical schools didn't have artificially low size constraints or if restrictions on doctors educated in non-US medical schools were lifted? (The regulations on who is qualified to be a doctor in the US almost certainly violates many international treaties, and yet the medical community gets away with it.) Medical doctors are just a bit more protective of their turf than other intellectual professions. It is a difference of degree, not kind. Most intellectual professions have artificial barriers. My employer, for example, most certainly would not entertain the idea of hiring someone without an appropriate degree.

The ISS most certainly does rely on shipments of oxygen, in the form of water. The water is electrolyzed, with the hydrogen vented to vacuum. People (and biological specimens) burn the oxygen to form CO2 and water. The CO2 is scrubbed and vented to vacuum. The water is recycled.