I thought that I'd start this thread to get a little game started. What anyone can do is post a question (about physics & maths obviously) and the first one who can get it wins. I understand that not all of us are on the forum 24-7 but who cares! I think it will be fun and a good break from all the arguing about relativity. Please Register or Log in to view the hidden image! Try not to make them too hard, but a good tricky question will go a long way (and maybe a few good remarks!). If someone gets the right answer it would be appreciated if the question-asker replies as quickly as possible. (Oh, and try and keep it to 1 question per pesron at a time.) Anyway, I'll begin. Q) What is so special about the following two numbers... 153 and 548834 Good Luck.
If somebody should come with a (plausible) theory how to destroy the universe (iniciating some chained/unstoppable process), should we test that theory?
Calculate this integral. I know the answer, but have no idea how to calculate it. integrals{exp[-r]/r dr} between the limits 0->infinity. For the more inclined, the integral of the Yukawa potential.
Calculate this integral. I know the answer, but have no idea how to calculate it. integrals{exp[-r]/r dr} between the limits 0->infinity. For the more inclined, the integral of the Yukawa potential, or the Laplace transform of 1/r.
Two roommates share a 2-bedroom $600-a-month apartment. One bedroom is larger than the other. Either would pay an extra $50 for the larger room. How should they split the rent?
zanket, if they settle on who shall get the big bedroom, then the occupant of that room will pay $325 a month and the other shall pay $275.
Yes. Seems simple eh? Yet you’ll find a lot of people who think the roomie paying $325 a month is paying only $25 more, since if the bedrooms were the same size then each would pay $300. Here’s another real-life problem: You’re 10 years into a 30-year fixed (interest rate is constant) mortgage at 7.0%. Rates have since lowered to the point where you can refinance at no cost whatsoever (excluding your time) at 6.5%, also 30-year fixed. Your monthly payment will be less, but instead of 20 years of payments left you’d be 30 years out again. All other considerations being equal, should you refinance?
Whether we should or not we will - why change the habits of a lifetime. I believe that some people thought that the detonation the first A-bomb would lead to an unstoppable chain reaction. Pity we didn't believe them, for it has.
You have twelve coins identical in appearance and a set of scales (balances). One of the coins is fake and is either lighter or heavier that the other eleven. Using the scales just three times find which coin is fake and whether it is heavier or lighter.
integral{exp[-r]/r} Here is how to solve the inverse LaPlace transform: Okay, suppose that a function (we'll call F) of the complex variable (s) is analytic throughout the finite 's' plane except for a finite number of isolated singularities, obviously. Now let L denote a vertical line from s = g - iR to s = g + iR, where the constant 'g' is positive and large enough that the singularities of 'F' all lie to the left of that segment. A new function 'f' of the real variable 't' is defined for positive values of 't' by means of the equation... f(t) = 1/2(pi)i(lim R-goes-to-infinity)of(the integral[L]exp[st]F(s)ds...1 I hope you can extract that! It is a little hard to write this stuff on the forum! Now provided that the limit exists this equation is usually written... f(t) = 1/2(pi)i P.V. (the integral from (g - i(inf) to g + i(inf)) of exp[st[F(s)ds...2 It can be shown that, when fairly general conditions are imposed on the functions involved, f(t) is the INVERSE LaPlace Transform of F(s). That is, if F(s) is the LaPlace transform of f(t), defined by the following equation... F(s) = integral from 0 to infinity of (exp[-st]f(t)...3 ...then f(t) is retrieved by means of the second equation (2), where the choice of the positive number 'g' is immaterial as long as the singularities of F all lie to the left of L. Now we have to use Residues to evaluate the limit when the function F(s) is specified. We let Sn ( n = 1, 2, 3,..., N) denote the singularities of F(s). We the let R0 denote the target of their moduli and consider a semicircle C with parametric representation... s = g + Re^i(theta) between pi/2 and 3(pi)/2...4 ...where R > R0 + g. Hence the singularities all lie in the interior of the semicircular region bounded by CR and LR and the residue theorem tells us that... integral[CR][exp[st]F(s) = 2(pi)iSIGMA(n=1 to N){Res(s=sn)[exp[st]F(s)} - integral[CR]exp[st]F(s)ds...5 Suppose now that for all points 's' on CR, there is a positive constant MR such that |F(s) <or equal to MR where MR tends to zero as R tends to infinity. We may use the parametric representation (4) for CR to write... integral[CR]{exp[st]}Fsds = integral(from pi/2 to 3pi/2)exp{gt + Rte^i(theta)}F(g + Re^i(theta))Rie^i(theta)d(theta) Then since... |exp(gt + Rte^i(theta))| = (e^gt)(e^Rtcos(theta)) and... |F(g + Re^i(theta))| < or equal to MR we find that... | integral[CR]of exp[st]F(s)ds | < or equal to (e^gt)MR.R * integral(from pi/2 to 3pi/2) of e^RTcos(theta)d(theta)...6 But the substitution p = (theta) - pi/2, together with Jordan's inequality reveals that... The integral(from pi/2 to 3pi/2) of e^Rtcos(theta)d(theta) = the integral(from 0 to pi)exp[-rtsin(p)]d(p) < pi/Rt The inequality then becomes... | integral[CR] e^st F(s)ds | < or equal to (e^(gt))MR(pi))/t...6 and this shows that... the limit as R tends to infinity of the integral[CR] of (e^st)F(s)ds = 0...7 Letting R tend to infinity in equation (5), then, we see that the function f(t), defined by equation (1), exists and that it can be written as... f(t) = SIGMA(from n=1 tp infinity)Res(s=sn){exp[st]}F(s) Courtesy of James Brown (Author of "Complex Variables and Applications" this would not have been possible without the help of his excellent resources) Cheers. Ben.
Answer to my question These numbers are called Narcissistic Numbers. 154 = 1^3 + 5^3 + 4^3 548834 = 5^6 + 4^6 + 8^6 + 8^6 + 3^6 + 4^6
i m not sure, but i don t expect that this integral converges. near r=0, it will look like 1/r which diverges. maybe you want different bounds of integration?
With what two letters do all Microsoft executables begin? What do these letters mean? Not really a physics or mathematics-related question... but idiotic questions about doomsday aren't either.
Oxy, Man, that brought back some fond memories from last semester for me. Thanks for the trip down memory lane. In fact, one of my motivations for taking Complex Analysis was, among other things, to learn how to compute generalized Inverse Laplace Transforms, since when I took a required (sophomore level) EE class that covered Laplace Transforms, I heard the oft repeated phrase when generalized inverses were discussed, "It's beyond the scope of this class", coupled with, "You'll only need to use the table". Those are two phrases that simply drive me up the wall!
Dapthar and curioucity, What about "the proof of this can be found in ref. 5 in app. 6". So I went to appendix 6 and reference 5 is in another book. So I found the other book and the proof said "the proof of theroem 5.1.2 is simple and has been left as an example for the reader". Is it right? It has been a while since I posted that. No-one said anything about it so I assumed it is correct. I spent all day reading the chapter on complex integration. I eventually worked out how to do questions and apply theroems and I had a dig at it.
It depends on what you do with the money you're no longer spending on repayments. (I think the following figures and calculations are right... but use at your own risk Please Register or Log in to view the hidden image! ) Let's look at a $100,000 loan. 30 year repayments at 6.5% are $665.30/month. After 10 years, your principal is down to $85,812.82 If you continue at 6.5% you'll pay $159,675.04 over the following 20 years. If you refinance at 7%, your monthly payments are now down to $542.40, but your total payments are $195,269.13 over 30 years. Bad move, right? Not necessarily... What are you doing with the $122.90 you save each month? If you invest that excess (so you're still paying $665.30 per month, just not all to the bank), you'll build up a kitty over the next 20 years that you can use to repay the remaining 10 years... Whether it's worth it or not depends on the interest your investment earns. Let's say you can get 5% return. You refinance, pay $542.40 per month to the bank, and $122.90 per month into your investment account ($665.30 total). Twenty years later, your investment account is at $53.239.91, and your mortgage is at $47,765.67. You now stop both payments, and start paying the mortgage from your investment account. (ie Your total out-of pocket payments are exactly the same as if you didn't refinance). Ten years later, the mortage is paid off... not quite! Your investment account has run dry, and you have to pay a touch over $1000 to cover the last two payments. However, if you can get 5.5% on you investment, you'll still have $6159.61 to play with when the mortgage is all paid up. The final word? If you can get an effective low risk return of more than 5.1%, then you should refinance and invest the difference in your repayments. If you can't get that return or the risk is not very low, then you should stay with the 7% deal.
