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June 10

Recursion, probability and max

Given $a_{0,j}=\phi$ where $\phi$ is a random number drawn from some uniform (or normal?) distribution, does there exist an analytical solution for expressions such as: $a_{i,j}=\phi +max(a_{i-1,j-1},a_{i-1,j},a_{i-1,j+1})$ ?

This represents the accumulation of noise in parallel programs, with $i$ being the iteration and $j$ being the processor ID. The above illustrates the simple case of a 3-stencil communication pattern: step $i$ on processor $j$ cannot begin until all three previous tasks on processors $(j-1,j,j+1)$ have completed.

I expect the answer is going to be a bit too complicated for a succinct reply here; RTFM is fine, given a pointer to the appropriate M to FR. I've been poking around in the PERT/makespan literature and it looks like the general problem is both hard and open, but I'm hoping that restricting the problem to stencils will make it more tractable.

Garamond Lethe (talk) 09:04, 10 June 2012 (UTC)[reply]

If φ has the uniform distribution on the interval 0<φ<1, then the max of three samples has a beta distribution. I don't understand your last paragraph. Bo Jacoby (talk) 16:06, 10 June 2012 (UTC).[reply]

Thanks! I don't know if that's the piece I was missing, but the PERT references in the beta distribution article that look promising that I didn't know about before. Garamond Lethe (talk) 19:20, 10 June 2012 (UTC)[reply]

Why are modern mathematicians so underrated?

I mean mathematicians after 1900, when abstraction began to take place and things got rigorous. They seem to be only famous in their own fields. Also why is it the general public think mathematics is all about mental calculations? That stuff is totally useless since we got computers, mathematics needs people with creativity not doing what computers can do a billion times faster. Money is tight (talk) 10:38, 10 June 2012 (UTC)[reply]

To your first question, you might enjoy reading Davis and Hersh's thoughts on the "Ideal Mathematician". The primary culprit is specialization. Garamond Lethe (talk) 10:53, 10 June 2012 (UTC)[reply]

I don't think you're right about the premises, mathematicians have had some press just like lots of other people in other areas. How many people have heard about Gauss compared to Andrew Wiles? It never has been a major headline hitter like physics, but people generally would be hard pressed to name more than five people in particular any area of study. How many names do you know associated with evolution, DNA, or heredity and that's a pretty hot area of general interest currently. Dmcq (talk) 10:55, 10 June 2012 (UTC)[reply]

I think the link provided by Garamond Lethe explains my question. The problem is the outsider can't understand what a specialist is doing. Andrew Wiles is most noted for his proof of Fermat's last theorem, which any layman can understand (the theorem itself, not the proof), where as Gauss' best work (to me) is his differential geometry which very few layman understands. Stephen is so famous to the public because of his conjectured Hawking radiation, which is very interesting to the layman, and one the other hand Ed Witten isn't known at all to the public because his work is all too technical. However it does bother me that many layman claim people who are mental calculators as geniuses, when those mental calculators have done nothing other than trying to beat the computer. If they know they aren't experts, don't make such a comment. Money is tight (talk) 11:30, 10 June 2012 (UTC)[reply]

Many interesting problems that Newton or Gauss might have called "mathematics" centuries ago are, in 2012, recategorized as physics, computer science, and other related topics. As a result of the reorganization of academic subject-matter, and the emergence of new fields of study, "mathematics" today refers to a subset of topics: those most abstract and most distilled, purest forms of structured analysis. It is my opinion that most of the interesting and practical mathematical problems are today called "computer science" - areas like the classification problem, numerical methods, advanced techniques of statistical analysis, and essentially any part of mathematics that requires arithmetic calculation. What is left in mathematics are those problems that are too abstract to be directly computed; or too intangible to be modeled physically; and so, most people outside of the field do not find these problems "interesting." Needless to say, there is still some great work in modern pure mathematics, but unlike a few centuries ago, (when every subject was either "natural philosophy" or "theology"), today's interesting ideas and problems are often labeled or categorized into a different field. As an example: Edsger Dijkstra had a degree in physics and was employed at a very theoretical mathematics institute, but ask anyone what his field was, and they will claim "computer science." Alan Turing was formally a mathematician, and his work preceded the use of the term "computer" in common parlance; but his important contributions to the mathematical theories of computability are today called computer science. Nimur (talk) 15:06, 10 June 2012 (UTC)[reply]

Well, that depends on who's doing the calling. To me the undecidability of the halting problem is mathematical logic, not computer science. Of course the "to me" is the key phrase here; I'm not going to say that anyone calling it "computer science" is wrong, though I will say that my categorization is not a personal idiosyncrasy.

Reminds me of the arguments about whether von Neumann was a physicist or a mathematician: "Oh, it was easy; I just summed an infinite series". --Trovatore (talk) 21:14, 11 June 2012 (UTC)[reply]

Would it not simply be the case that the working mathematician is generally interested in answering questions that don't directly impact practical matters, analogous to the reason that most people are averse to becoming mathematicians (or mathematically skilled)? Its questions cannot be associated with the 'real' world without the concept of modeling, and this concept's value has to be taught well or latched onto by the person in some other way or mathematics looks at first like a stupid (impractical) activity and then later like a thorough mystery. As a part of popular culture, mathematics is marginal in the extreme aside from direct efforts against this; and as a part of intellectual culture it is very specialized (and intellectual culture is averse to valuing fame to a degree, as well, for both good and not-so-good reasons; so its feed to popular culture might be diminished beyond some ideal if what we want is for people to understand things).173.15.152.77 (talk) 03:27, 12 June 2012 (UTC)[reply]

On the first question, there are certainly some 20th century mathematical developments that have found widespread fame, such as some of the work on chaos theory, fractals, graph theory (like small-world networks), and game theory. 81.98.43.107 (talk) 23:32, 13 June 2012 (UTC)[reply]

converting foot pounds of torque to Horse Power

I'm wondering if you could please help me out. I'm sure there are plenty of others out there that would like to know the same answer as I am trying to find. I see that lawn mower engines are now rated in foot pounds of torque instead of the good old fashioned way of Horse Power. So I was wondering if you could please enlighten me as to how much horse power their 4.50 foot pounds of torque lawnmower really is. —Preceding unsigned comment added by 67.251.81.144 (talk) 18:53, 10 June 2012‎

Torque is actually measured in pound-feet in Imperial measure, and to convert to horsepower the particular engine speed is needed. See [1] for an explanation of the conversion.→109.148.243.127 (talk) 19:19, 10 June 2012 (UTC)[reply]

At Power (physics)#Mechanical power it explains that power is the product of torque and angular velocity in radians per second. If torque is in pound-feet then this equation will deliver mechanical power in pound-feet per second. One horsepower is equal to 550 pound-feet per second.

But there is an added complication - the speed at which maximum torque occurs is slower than the speed at which maximum power occurs. To find the maximum horsepower you need to know the speed at which this power occurs, and the torque available at that speed, with wide-open throttle. That won't be easy so it would be best to contact the engine manufacturer. Dolphin (t) 03:47, 12 June 2012 (UTC)[reply]