Wikipedia:Reference desk/Archives/Mathematics/2013 March 7

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March 7

Another integral formula for the Gamma/Factorial function

$x!=\Gamma (x+1)=\int _{0}^{\infty }{\frac {dt}{e^{\sqrt[{x}]{t}}}}$ . At the same time, we already know that $x!=\Gamma (x+1)=\int _{0}^{\infty }{\frac {t^{x}}{e^{t}}}\ dt$ .

I guess my question would be four-fold: (1) did anyone notice this alternate expression before ? (2) are there other meaningful or interesting properties attached to the function $e^{\pm {\sqrt[{n}]{x}}}$ ? (3) are there any other integral expressions for the factorial/Gamma function ? and (4) what are we to make of the fact that the integrals of these two different but related exponential functions have the exact same value over R₊ ? — 79.113.241.192 (talk) 09:09, 7 March 2013 (UTC)

(1) Probably

(2) I can't see that this one is particularly meaningful but the function will surely have some unique properties.

(3) There are as many integral representations as substitutions you care to make.

(4) Maybe it's worthy of a mildly interested 'huh' noise. That's probably about it. — Preceding unsigned comment added by 202.65.245.7 (talk) 14:47, 7 March 2013 (UTC)

I guess you were refering to the fact that this formula can be derived from the former one by using the substitution t^x = T, and then calculating dt in terms of dT. (Sorry for being so dense, but I've only noticed this just now, and then remembered what you wrote...) — 79.113.196.61 (talk) 10:35, 10 March 2013 (UTC)

Resolved

Can you use propositional logic/calculus to construct mathematical models for quantitative research?

I would like to construct a model depicting the influence of the majority over the minority in election. I would argue that the majority can influence the minority groups. How can I use a simple propositional model for this? Can you give some advice or examples? — Preceding unsigned comment added by Joshua Atienza (talk • contribs) 09:25, 7 March 2013 (UTC)

Preformatted math-tags cut this page 30 seconds

I have inserted 2 sections of <pre>...</pre> tags around the slow <math> tags to cut 30 seconds (13+17) from the time needed to reformat this WP:Reference_desk/Mathematics page. The extra format time is averaging about 1 second to format each <math> tag, and I am unaware of whether that speed is unusually slow these days. An entire page must reformat within the 60-second timeout period, or else face wp:Wikimedia Foundation error, which still saves an edit but does not redisplay the page after edit-save. More later. -Wikid77 (talk) 17:49, 7 March 2013 (UTC)

So it sounds like you're saying that the math tags are incredibly slow and we should stop using them ? How about if we put them inside collapse boxes, so you only incur the slowdown when you choose to open them ? StuRat (talk) 03:21, 8 March 2013 (UTC)

That's not how it works. The problem is that the <math> tags are being processed slowly by the MediaWiki engine when the page is saved. Collapse boxes are merely a visual effect—their content is always there, it's just hidden (client-side) by the browser when they are collapsed. It isn't like the content of collapse boxes is rendered by the server when you open them. —Bkell (talk) 07:57, 8 March 2013 (UTC)

Is there a chance somebody is working to solve this problem? I have seen it perhaps a week now. YohanN7 (talk) 14:00, 8 March 2013 (UTC)

A truck and a person

A truck is trying to run over a person, and the person is trying to escape. Both are point objects. The truck has maximum speed Vt and maximum acceleration at; the person has maximum speed Vp and maximum acceleration ap. What's a necessary and sufficient condition to guarantee that the truck can always run over the person? Trivially, Vt > Vp, but what about the accelerations? Can at be smaller than ap?

This is not homework, just an amusing question I thought about while thinking of deliberate run-overs. --140.180.243.114 (talk) 19:00, 7 March 2013 (UTC)

These problems are generally known as pursuit problems or pursuit-evasion problems. I believe you will find answers to your case with a bit more googling on those terms. We do have an article on the Homicidal_chauffeur_problem, but no solution is posted there (there are promising links to analytic treatments). SemanticMantis (talk) 20:30, 7 March 2013 (UTC)

As stated, so long as the truck has some positive acceleration, and it's max speed is greater than the person, then it will eventually catch him. Now, if you add in a time or distance limit, then you get a more interesting case. That is, can the truck catch the person before the limit is hit. (I assumed no turning is allowed.) StuRat (talk) 03:13, 8 March 2013 (UTC)

Hmm. It seems intuitively true that if the truck's maximum speed is greater than that of the person, and there aren't any additional constraints (such as maneuverability), then the truck can eventually hit the person. But it's interesting to try to prove it. —Bkell (talk) 08:03, 8 March 2013 (UTC)

The acceleration is such a constraint with speed considered as a vector. Dmcq (talk) 08:34, 8 March 2013 (UTC)

That's true. I don't know what the answer is in that case. —Bkell (talk) 10:49, 8 March 2013 (UTC)

Just had a think about this and it certainly isn't obvious. The truck can't catch a person by trying to follow them from a short distance, it just can't go round such a small circle at the same rate. However if it goes some distance away, fixes the unfortunate person in its headlights and gets to high speed then I think it can get the person if the max speed times acceleration is greater than that of the person's max speed times acceleration. It needs a bit of checking though. Dmcq (talk) 12:00, 8 March 2013 (UTC)

That sounds very much like the real life strategy seals use to avoid the sharp ends of sharks - they turn inside the sharks minimum turn radius and stay on the shark's tail. Roger (talk) 12:22, 8 March 2013 (UTC)

Do your players (a truck and a person) move along a line or on a plane...? On a one-dimensional track V_truck > V_person would be enough to cath the person. However, on an airport ramp or a large city square the person might escape the truck whatever the vehicle's velocity, simply by jumping aside just in front of it, so in two-dimensional problem the vehicle must also have its acceleration greater than the person. --CiaPan (talk) 15:49, 8 March 2013 (UTC)

No, in a 2-dimensional plane (or even in 3-dimensional space, with a flying truck and a person with a jetpack), the truck can always catch the person if its velocity is greater than the person's, as long as there are no additional maneuverability constraints. The slower person cannot avoid the truck simply by moving aside when the truck gets close.

Here's the truck's strategy. Without loss of generality, by scaling appropriately, we may assume the maximum speed of the person is 1 and the maximum speed of the truck is 1 + ε for some ε > 0. We can also assume that the truck begins at its maximum speed; otherwise it can just accelerate to its maximum speed, ignoring whatever the person is doing, before it begins this strategy. Suppose the distance between the truck and the person is initially d₀. The truck observes the position P₀ of the person at that moment in time and drives at top speed in a straight line toward P₀ for time d₀/(2 + ε). At the end of this step, the distance from the truck to the point P₀ is d₀ − (1 + ε)d₀/(2 + ε), and the distance from P₀ to the person is at most d₀/(2 + ε), so the distance from the truck to the person is at most d₀ − (1 + ε)d₀/(2 + ε) + d₀/(2 + ε) = 2d₀/(2 + ε) < d₀. Now the truck observes the new position P₁ of the person and repeats this process: it drives at top speed in a straight line toward P₁ for time d₁/(2 + ε), where d₁ is the distance between the truck and P₁ at the beginning of this step. And so on.

Step k of this process takes time d_k/(2 + ε). So the time required by the entire process is

$\sum _{k=0}^{\infty }{\frac {d_{k}}{2+\epsilon }}\leq \sum _{k=0}^{\infty }{\frac {[2/(2+\epsilon )]^{k}d_{0}}{2+\epsilon }}={\frac {d_{0}}{2+\epsilon }}\sum _{k=0}^{\infty }\left({\frac {2}{2+\epsilon }}\right)^{k}=\left({\frac {d_{0}}{2+\epsilon }}\right)\left({\frac {1}{1-2/(2+\epsilon )}}\right)<\infty ,$

which means that the process takes only finitely much time, so the truck will eventually hit the person. —Bkell (talk) 18:17, 8 March 2013 (UTC)

Except that changing direction involves acceleration towards the centre of curvature even if it doesn't change the speed. Dmcq (talk) 20:22, 8 March 2013 (UTC)

And my previous thoughts were just wrong. I can't see now how to catch something that can accelerates faster. Dmcq (talk) 20:54, 8 March 2013 (UTC)

Even if the target accelerates faster, if it has a lower top speed, and the pursuer has some positive acceleration, the pursuer will eventually reach a higher speed, and then will catch the target. (Again I'm assuming no turning.) StuRat (talk) 21:10, 8 March 2013 (UTC)

That would be a very cooperative target! Dmcq (talk) 22:19, 8 March 2013 (UTC)

Right, changing direction requires acceleration. The proof above assumes there is no acceleration constraint ("…as long as there are no additional maneuverability constraints"—a constraint on acceleration is a constraint on maneuverability, as noted above). It was meant to show that, with no other constraints, it is sufficient for the truck to have a greater maximum speed than the person. As I said above, I don't know what the answer to the problem is if you add in the constraint on acceleration. —Bkell (talk) 23:54, 8 March 2013 (UTC)

What do you mean by 'no additional maneuverability constraints'...? Maneuverability is directly dependent on the maximum acceleration (and acceleration is a = dv/dt with both velocity and the original question deines a constraint being |a_truck| ≤ some a_max).--CiaPan (talk) 10:11, 11 March 2013 (UTC)

Thanks for the answers! Unfortunately I haven't been able to find a solution to this specific question, although pursuit evasion was very interesting. Also, for the original question, the pursuit takes place on a 2D plane where both people are allowed to turn in any direction, so long as they don't exceed the maximum acceleration. --140.180.243.114 (talk) 02:41, 9 March 2013 (UTC)

The answer to your question may depend on exactly what you mean when you say "acceleration." If by acceleration you only mean a change in speed, so that turning corners sharply while maintaining the same speed does not count as acceleration, then my proof above shows that the truck can always catch the person if the truck's maximum speed is greater than that of the person. The acceleration of the truck is irrelevant, because if the truck uses the strategy I described above, it is always traveling at top speed. It makes lots of sharp turns, but once it gets going it never slows down.

On the other hand, in physics velocity is normally considered a vector quantity, which has a direction as well as a magnitude, so any change in the direction of motion counts as acceleration (even if the speed doesn't change). If that's what you mean by acceleration, then there is a limit on how sharply the truck and person can change directions. The strategy I described above no longer works, because the truck can't make the instantaneous sharp turns that are required. In this case I don't know what the answer to your question is. —Bkell (talk) 02:59, 9 March 2013 (UTC)

I meant acceleration in the physics sense. You're right that the problem becomes uninteresting if only changes in speed are considered acceleration. --140.180.243.114 (talk) 05:21, 9 March 2013 (UTC)

I don't think that the truck's change-of-speed acceleration is relevant, as the person's strategy would be to reduce the problem to a series of repeated encounters with them standing still as the truck approaches, presumably at top speed, then jumping aside early enough to avoid contact and late enough to make unimportant any change in direction of the truck. I wouldn't like do actually do it, but intuitively, I could dodge a truck 8 feet wide going at 70 mph with this strategy.←86.186.142.172 (talk) 11:04, 9 March 2013 (UTC)

• Not sure if anyone's still following this thread, but I'll update anyway: Here is a nice pdf that covers much of the history any many solutions to the problem. [1] (The OP's problem is exactly the homicidal chauffeur, which has been studied since 1951. Despite the fanciful nature and phrasing, a primary early interest was military in nature.) Basically, the human *can* escape indefinitely if the truck's turning (hence acceleration) is sufficiently restricted. The truck *can* hit the pedestrian otherwise. The details are in the pdf and references, but this problem requires some pretty serious background to even follow previous solutions. SemanticMantis (talk) 21:09, 11 March 2013 (UTC)